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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC.</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.17112v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S1" title="In SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S2" title="In SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3" title="In SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>The Proof</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">SEPARATION NUMBER AND TREEWIDTH, REVISITED<span class="ltx_note ltx_role_thanks" id="id3.id1"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">thanks: </span>This research was partly funded by NSERC.</span></span></span> </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Hussein Houdrouge </span><span class="ltx_author_notes">School of Computer Science, Carleton University.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Babak Miraftab<span class="ltx_note ltx_role_footnotemark" id="footnotex1"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_note_type">footnotemark: </span><span class="ltx_tag ltx_tag_note">2</span></span></span></span> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> and Pat Morin<span class="ltx_note ltx_role_footnotemark" id="footnotex2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_note_type">footnotemark: </span><span class="ltx_tag ltx_tag_note">2</span></span></span></span> </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id2.2">We give a constructive proof of the fact that the treewidth of a graph <math alttext="G" 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">G</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">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_G</annotation></semantics></math> is bounded by a linear function of the separation number of <math alttext="G" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mi id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><ci id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_G</annotation></semantics></math>.</p> </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.p1.11">In this paper every graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_G</annotation></semantics></math> is undirected and simple with vertex set <math alttext="V(G)" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mrow id="S1.p1.2.m2.1.2" xref="S1.p1.2.m2.1.2.cmml"><mi id="S1.p1.2.m2.1.2.2" xref="S1.p1.2.m2.1.2.2.cmml">V</mi><mo id="S1.p1.2.m2.1.2.1" xref="S1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.p1.2.m2.1.2.3.2" xref="S1.p1.2.m2.1.2.cmml"><mo id="S1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S1.p1.2.m2.1.2.cmml">(</mo><mi id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml">G</mi><mo id="S1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><apply id="S1.p1.2.m2.1.2.cmml" xref="S1.p1.2.m2.1.2"><times id="S1.p1.2.m2.1.2.1.cmml" xref="S1.p1.2.m2.1.2.1"></times><ci id="S1.p1.2.m2.1.2.2.cmml" xref="S1.p1.2.m2.1.2.2">𝑉</ci><ci id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">V(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_V ( italic_G )</annotation></semantics></math> and edge set <math alttext="E(G)" 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">E</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">G</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">𝐸</ci><ci id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">E(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_E ( italic_G )</annotation></semantics></math>. A <em class="ltx_emph ltx_font_italic" id="S1.p1.11.1" style="color:#C22147;">tree decomposition</em> of a graph <math alttext="G" 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">G</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">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">italic_G</annotation></semantics></math> is a collection <math alttext="\mathcal{T}:=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="S1.p1.5.m5.1"><semantics id="S1.p1.5.m5.1a"><mrow id="S1.p1.5.m5.1b"><mi class="ltx_font_mathcaligraphic" id="S1.p1.5.m5.1.1">𝒯</mi><mo id="S1.p1.5.m5.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S1.p1.5.m5.1.3"><mo id="S1.p1.5.m5.1.3.1" stretchy="false">(</mo><msub id="S1.p1.5.m5.1.3.2"><mi id="S1.p1.5.m5.1.3.2.2">B</mi><mi id="S1.p1.5.m5.1.3.2.3">x</mi></msub><mo id="S1.p1.5.m5.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S1.p1.5.m5.1.3.4">x</mi><mo id="S1.p1.5.m5.1.3.5">∈</mo><mi id="S1.p1.5.m5.1.3.6">V</mi><mrow id="S1.p1.5.m5.1.3.7"><mo id="S1.p1.5.m5.1.3.7.1" stretchy="false">(</mo><mi id="S1.p1.5.m5.1.3.7.2">T</mi><mo id="S1.p1.5.m5.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S1.p1.5.m5.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p1.5.m5.1c">\mathcal{T}:=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> of vertex subsets of <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>, called <em class="ltx_emph ltx_font_italic" id="S1.p1.11.2" style="color:#C22147;">bags</em>, that is indexed by the vertices of a tree <math alttext="T" class="ltx_Math" display="inline" id="S1.p1.7.m7.1"><semantics id="S1.p1.7.m7.1a"><mi id="S1.p1.7.m7.1.1" xref="S1.p1.7.m7.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.p1.7.m7.1b"><ci id="S1.p1.7.m7.1.1.cmml" xref="S1.p1.7.m7.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.7.m7.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.p1.7.m7.1d">italic_T</annotation></semantics></math> and such that <span class="ltx_inline-enumerate" id="S1.I1"> <span class="ltx_inline-item" id="S1.I1.i1"><span class="ltx_tag ltx_tag_inline-item">(i)</span> <span class="ltx_text" id="S1.I1.i1.3">for each <math alttext="vw\in E(G)" class="ltx_Math" display="inline" id="S1.I1.i1.1.m1.1"><semantics id="S1.I1.i1.1.m1.1a"><mrow id="S1.I1.i1.1.m1.1.2" xref="S1.I1.i1.1.m1.1.2.cmml"><mrow id="S1.I1.i1.1.m1.1.2.2" xref="S1.I1.i1.1.m1.1.2.2.cmml"><mi id="S1.I1.i1.1.m1.1.2.2.2" xref="S1.I1.i1.1.m1.1.2.2.2.cmml">v</mi><mo id="S1.I1.i1.1.m1.1.2.2.1" xref="S1.I1.i1.1.m1.1.2.2.1.cmml">⁢</mo><mi id="S1.I1.i1.1.m1.1.2.2.3" xref="S1.I1.i1.1.m1.1.2.2.3.cmml">w</mi></mrow><mo id="S1.I1.i1.1.m1.1.2.1" xref="S1.I1.i1.1.m1.1.2.1.cmml">∈</mo><mrow id="S1.I1.i1.1.m1.1.2.3" xref="S1.I1.i1.1.m1.1.2.3.cmml"><mi id="S1.I1.i1.1.m1.1.2.3.2" xref="S1.I1.i1.1.m1.1.2.3.2.cmml">E</mi><mo id="S1.I1.i1.1.m1.1.2.3.1" xref="S1.I1.i1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S1.I1.i1.1.m1.1.2.3.3.2" xref="S1.I1.i1.1.m1.1.2.3.cmml"><mo id="S1.I1.i1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S1.I1.i1.1.m1.1.2.3.cmml">(</mo><mi id="S1.I1.i1.1.m1.1.1" xref="S1.I1.i1.1.m1.1.1.cmml">G</mi><mo id="S1.I1.i1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S1.I1.i1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.1.m1.1b"><apply id="S1.I1.i1.1.m1.1.2.cmml" xref="S1.I1.i1.1.m1.1.2"><in id="S1.I1.i1.1.m1.1.2.1.cmml" xref="S1.I1.i1.1.m1.1.2.1"></in><apply id="S1.I1.i1.1.m1.1.2.2.cmml" xref="S1.I1.i1.1.m1.1.2.2"><times id="S1.I1.i1.1.m1.1.2.2.1.cmml" xref="S1.I1.i1.1.m1.1.2.2.1"></times><ci id="S1.I1.i1.1.m1.1.2.2.2.cmml" xref="S1.I1.i1.1.m1.1.2.2.2">𝑣</ci><ci id="S1.I1.i1.1.m1.1.2.2.3.cmml" xref="S1.I1.i1.1.m1.1.2.2.3">𝑤</ci></apply><apply id="S1.I1.i1.1.m1.1.2.3.cmml" xref="S1.I1.i1.1.m1.1.2.3"><times id="S1.I1.i1.1.m1.1.2.3.1.cmml" xref="S1.I1.i1.1.m1.1.2.3.1"></times><ci id="S1.I1.i1.1.m1.1.2.3.2.cmml" xref="S1.I1.i1.1.m1.1.2.3.2">𝐸</ci><ci id="S1.I1.i1.1.m1.1.1.cmml" xref="S1.I1.i1.1.m1.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.1.m1.1c">vw\in E(G)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.1.m1.1d">italic_v italic_w ∈ italic_E ( italic_G )</annotation></semantics></math>, there exists <math alttext="x\in V(T)" class="ltx_Math" display="inline" id="S1.I1.i1.2.m2.1"><semantics id="S1.I1.i1.2.m2.1a"><mrow id="S1.I1.i1.2.m2.1.2" xref="S1.I1.i1.2.m2.1.2.cmml"><mi id="S1.I1.i1.2.m2.1.2.2" xref="S1.I1.i1.2.m2.1.2.2.cmml">x</mi><mo id="S1.I1.i1.2.m2.1.2.1" xref="S1.I1.i1.2.m2.1.2.1.cmml">∈</mo><mrow id="S1.I1.i1.2.m2.1.2.3" xref="S1.I1.i1.2.m2.1.2.3.cmml"><mi id="S1.I1.i1.2.m2.1.2.3.2" xref="S1.I1.i1.2.m2.1.2.3.2.cmml">V</mi><mo id="S1.I1.i1.2.m2.1.2.3.1" xref="S1.I1.i1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S1.I1.i1.2.m2.1.2.3.3.2" xref="S1.I1.i1.2.m2.1.2.3.cmml"><mo id="S1.I1.i1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S1.I1.i1.2.m2.1.2.3.cmml">(</mo><mi id="S1.I1.i1.2.m2.1.1" xref="S1.I1.i1.2.m2.1.1.cmml">T</mi><mo id="S1.I1.i1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S1.I1.i1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.2.m2.1b"><apply id="S1.I1.i1.2.m2.1.2.cmml" xref="S1.I1.i1.2.m2.1.2"><in id="S1.I1.i1.2.m2.1.2.1.cmml" xref="S1.I1.i1.2.m2.1.2.1"></in><ci id="S1.I1.i1.2.m2.1.2.2.cmml" xref="S1.I1.i1.2.m2.1.2.2">𝑥</ci><apply id="S1.I1.i1.2.m2.1.2.3.cmml" xref="S1.I1.i1.2.m2.1.2.3"><times id="S1.I1.i1.2.m2.1.2.3.1.cmml" xref="S1.I1.i1.2.m2.1.2.3.1"></times><ci id="S1.I1.i1.2.m2.1.2.3.2.cmml" xref="S1.I1.i1.2.m2.1.2.3.2">𝑉</ci><ci id="S1.I1.i1.2.m2.1.1.cmml" xref="S1.I1.i1.2.m2.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.2.m2.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.2.m2.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math> such that <math alttext="\{v,w\}\subseteq B_{x}" class="ltx_Math" display="inline" id="S1.I1.i1.3.m3.2"><semantics id="S1.I1.i1.3.m3.2a"><mrow id="S1.I1.i1.3.m3.2.3" xref="S1.I1.i1.3.m3.2.3.cmml"><mrow id="S1.I1.i1.3.m3.2.3.2.2" xref="S1.I1.i1.3.m3.2.3.2.1.cmml"><mo id="S1.I1.i1.3.m3.2.3.2.2.1" stretchy="false" xref="S1.I1.i1.3.m3.2.3.2.1.cmml">{</mo><mi id="S1.I1.i1.3.m3.1.1" xref="S1.I1.i1.3.m3.1.1.cmml">v</mi><mo id="S1.I1.i1.3.m3.2.3.2.2.2" xref="S1.I1.i1.3.m3.2.3.2.1.cmml">,</mo><mi id="S1.I1.i1.3.m3.2.2" xref="S1.I1.i1.3.m3.2.2.cmml">w</mi><mo id="S1.I1.i1.3.m3.2.3.2.2.3" stretchy="false" xref="S1.I1.i1.3.m3.2.3.2.1.cmml">}</mo></mrow><mo id="S1.I1.i1.3.m3.2.3.1" xref="S1.I1.i1.3.m3.2.3.1.cmml">⊆</mo><msub id="S1.I1.i1.3.m3.2.3.3" xref="S1.I1.i1.3.m3.2.3.3.cmml"><mi id="S1.I1.i1.3.m3.2.3.3.2" xref="S1.I1.i1.3.m3.2.3.3.2.cmml">B</mi><mi id="S1.I1.i1.3.m3.2.3.3.3" xref="S1.I1.i1.3.m3.2.3.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.3.m3.2b"><apply id="S1.I1.i1.3.m3.2.3.cmml" xref="S1.I1.i1.3.m3.2.3"><subset id="S1.I1.i1.3.m3.2.3.1.cmml" xref="S1.I1.i1.3.m3.2.3.1"></subset><set id="S1.I1.i1.3.m3.2.3.2.1.cmml" xref="S1.I1.i1.3.m3.2.3.2.2"><ci id="S1.I1.i1.3.m3.1.1.cmml" xref="S1.I1.i1.3.m3.1.1">𝑣</ci><ci id="S1.I1.i1.3.m3.2.2.cmml" xref="S1.I1.i1.3.m3.2.2">𝑤</ci></set><apply id="S1.I1.i1.3.m3.2.3.3.cmml" xref="S1.I1.i1.3.m3.2.3.3"><csymbol cd="ambiguous" id="S1.I1.i1.3.m3.2.3.3.1.cmml" xref="S1.I1.i1.3.m3.2.3.3">subscript</csymbol><ci id="S1.I1.i1.3.m3.2.3.3.2.cmml" xref="S1.I1.i1.3.m3.2.3.3.2">𝐵</ci><ci id="S1.I1.i1.3.m3.2.3.3.3.cmml" xref="S1.I1.i1.3.m3.2.3.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.3.m3.2c">\{v,w\}\subseteq B_{x}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.3.m3.2d">{ italic_v , italic_w } ⊆ italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>; and </span></span> <span class="ltx_inline-item" id="S1.I1.i2"><span class="ltx_tag ltx_tag_inline-item">(ii)</span> <span class="ltx_text" id="S1.I1.i2.4">for each vertex <math alttext="v" class="ltx_Math" display="inline" id="S1.I1.i2.1.m1.1"><semantics id="S1.I1.i2.1.m1.1a"><mi id="S1.I1.i2.1.m1.1.1" xref="S1.I1.i2.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.1.m1.1b"><ci id="S1.I1.i2.1.m1.1.1.cmml" xref="S1.I1.i2.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.1.m1.1d">italic_v</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S1.I1.i2.2.m2.1"><semantics id="S1.I1.i2.2.m2.1a"><mi id="S1.I1.i2.2.m2.1.1" xref="S1.I1.i2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.2.m2.1b"><ci id="S1.I1.i2.2.m2.1.1.cmml" xref="S1.I1.i2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.2.m2.1d">italic_G</annotation></semantics></math>, <math alttext="T[\{x\in V(T):v\in B_{x}\}]" class="ltx_Math" display="inline" id="S1.I1.i2.3.m3.2"><semantics id="S1.I1.i2.3.m3.2a"><mrow id="S1.I1.i2.3.m3.2.2" xref="S1.I1.i2.3.m3.2.2.cmml"><mi id="S1.I1.i2.3.m3.2.2.3" xref="S1.I1.i2.3.m3.2.2.3.cmml">T</mi><mo id="S1.I1.i2.3.m3.2.2.2" xref="S1.I1.i2.3.m3.2.2.2.cmml">⁢</mo><mrow id="S1.I1.i2.3.m3.2.2.1.1" xref="S1.I1.i2.3.m3.2.2.1.2.cmml"><mo id="S1.I1.i2.3.m3.2.2.1.1.2" stretchy="false" xref="S1.I1.i2.3.m3.2.2.1.2.1.cmml">[</mo><mrow id="S1.I1.i2.3.m3.2.2.1.1.1.2" xref="S1.I1.i2.3.m3.2.2.1.1.1.3.cmml"><mo id="S1.I1.i2.3.m3.2.2.1.1.1.2.3" stretchy="false" xref="S1.I1.i2.3.m3.2.2.1.1.1.3.1.cmml">{</mo><mrow id="S1.I1.i2.3.m3.2.2.1.1.1.1.1" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.cmml"><mi id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.2" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.2.cmml">x</mi><mo id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.1" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.1.cmml">∈</mo><mrow id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.cmml"><mi id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.2" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.2.cmml">V</mi><mo id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.1" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.3.2" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.cmml"><mo id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.3.2.1" stretchy="false" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.cmml">(</mo><mi id="S1.I1.i2.3.m3.1.1" xref="S1.I1.i2.3.m3.1.1.cmml">T</mi><mo id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.3.2.2" rspace="0.278em" stretchy="false" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S1.I1.i2.3.m3.2.2.1.1.1.2.4" rspace="0.278em" xref="S1.I1.i2.3.m3.2.2.1.1.1.3.1.cmml">:</mo><mrow id="S1.I1.i2.3.m3.2.2.1.1.1.2.2" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.cmml"><mi id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.2" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.2.cmml">v</mi><mo id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.1" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.1.cmml">∈</mo><msub id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.cmml"><mi id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.2" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.2.cmml">B</mi><mi id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.3" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.3.cmml">x</mi></msub></mrow><mo id="S1.I1.i2.3.m3.2.2.1.1.1.2.5" stretchy="false" xref="S1.I1.i2.3.m3.2.2.1.1.1.3.1.cmml">}</mo></mrow><mo id="S1.I1.i2.3.m3.2.2.1.1.3" stretchy="false" xref="S1.I1.i2.3.m3.2.2.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.3.m3.2b"><apply id="S1.I1.i2.3.m3.2.2.cmml" xref="S1.I1.i2.3.m3.2.2"><times id="S1.I1.i2.3.m3.2.2.2.cmml" xref="S1.I1.i2.3.m3.2.2.2"></times><ci id="S1.I1.i2.3.m3.2.2.3.cmml" xref="S1.I1.i2.3.m3.2.2.3">𝑇</ci><apply id="S1.I1.i2.3.m3.2.2.1.2.cmml" xref="S1.I1.i2.3.m3.2.2.1.1"><csymbol cd="latexml" id="S1.I1.i2.3.m3.2.2.1.2.1.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.2">delimited-[]</csymbol><apply id="S1.I1.i2.3.m3.2.2.1.1.1.3.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2"><csymbol cd="latexml" id="S1.I1.i2.3.m3.2.2.1.1.1.3.1.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.3">conditional-set</csymbol><apply id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1"><in id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.1.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.1"></in><ci id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.2.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.2">𝑥</ci><apply id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3"><times id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.1.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.1"></times><ci id="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.2.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.1.1.3.2">𝑉</ci><ci id="S1.I1.i2.3.m3.1.1.cmml" xref="S1.I1.i2.3.m3.1.1">𝑇</ci></apply></apply><apply id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2"><in id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.1.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.1"></in><ci id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.2.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.2">𝑣</ci><apply id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.1.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3">subscript</csymbol><ci id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.2.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.2">𝐵</ci><ci id="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.3.cmml" xref="S1.I1.i2.3.m3.2.2.1.1.1.2.2.3.3">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.3.m3.2c">T[\{x\in V(T):v\in B_{x}\}]</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.3.m3.2d">italic_T [ { italic_x ∈ italic_V ( italic_T ) : italic_v ∈ italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT } ]</annotation></semantics></math> is a non-empty (connected) subtree of <math alttext="T" class="ltx_Math" display="inline" id="S1.I1.i2.4.m4.1"><semantics id="S1.I1.i2.4.m4.1a"><mi id="S1.I1.i2.4.m4.1.1" xref="S1.I1.i2.4.m4.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.4.m4.1b"><ci id="S1.I1.i2.4.m4.1.1.cmml" xref="S1.I1.i2.4.m4.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.4.m4.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.4.m4.1d">italic_T</annotation></semantics></math>. </span></span> </span> The <em class="ltx_emph ltx_font_italic" id="S1.p1.11.3" style="color:#C22147;">width</em>, of a tree decomposition <math alttext="\mathcal{T}:=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="S1.p1.8.m8.1"><semantics id="S1.p1.8.m8.1a"><mrow id="S1.p1.8.m8.1b"><mi class="ltx_font_mathcaligraphic" id="S1.p1.8.m8.1.1">𝒯</mi><mo id="S1.p1.8.m8.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S1.p1.8.m8.1.3"><mo id="S1.p1.8.m8.1.3.1" stretchy="false">(</mo><msub id="S1.p1.8.m8.1.3.2"><mi id="S1.p1.8.m8.1.3.2.2">B</mi><mi id="S1.p1.8.m8.1.3.2.3">x</mi></msub><mo id="S1.p1.8.m8.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S1.p1.8.m8.1.3.4">x</mi><mo id="S1.p1.8.m8.1.3.5">∈</mo><mi id="S1.p1.8.m8.1.3.6">V</mi><mrow id="S1.p1.8.m8.1.3.7"><mo id="S1.p1.8.m8.1.3.7.1" stretchy="false">(</mo><mi id="S1.p1.8.m8.1.3.7.2">T</mi><mo id="S1.p1.8.m8.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S1.p1.8.m8.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p1.8.m8.1c">\mathcal{T}:=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S1.p1.8.m8.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> is <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{width}(\mathcal{T}):=\max\{|B_{x}|:x\in V(T)\}-1" class="ltx_Math" display="inline" id="S1.p1.9.m9.5"><semantics id="S1.p1.9.m9.5a"><mrow id="S1.p1.9.m9.5.5" xref="S1.p1.9.m9.5.5.cmml"><mrow id="S1.p1.9.m9.5.5.3.2" xref="S1.p1.9.m9.5.5.3.1.cmml"><mi id="S1.p1.9.m9.1.1" mathcolor="#C22147" xref="S1.p1.9.m9.1.1.cmml">width</mi><mo id="S1.p1.9.m9.5.5.3.2a" xref="S1.p1.9.m9.5.5.3.1.cmml">⁡</mo><mrow id="S1.p1.9.m9.5.5.3.2.1" xref="S1.p1.9.m9.5.5.3.1.cmml"><mo id="S1.p1.9.m9.5.5.3.2.1.1" mathcolor="#C22147" stretchy="false" xref="S1.p1.9.m9.5.5.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.9.m9.2.2" mathcolor="#C22147" xref="S1.p1.9.m9.2.2.cmml">𝒯</mi><mo id="S1.p1.9.m9.5.5.3.2.1.2" mathcolor="#C22147" rspace="0.278em" stretchy="false" xref="S1.p1.9.m9.5.5.3.1.cmml">)</mo></mrow></mrow><mo id="S1.p1.9.m9.5.5.2" rspace="0.278em" xref="S1.p1.9.m9.5.5.2.cmml">:=</mo><mrow id="S1.p1.9.m9.5.5.1" xref="S1.p1.9.m9.5.5.1.cmml"><mrow id="S1.p1.9.m9.5.5.1.1.1" xref="S1.p1.9.m9.5.5.1.1.2.cmml"><mi id="S1.p1.9.m9.4.4" xref="S1.p1.9.m9.4.4.cmml">max</mi><mo id="S1.p1.9.m9.5.5.1.1.1a" xref="S1.p1.9.m9.5.5.1.1.2.cmml">⁡</mo><mrow id="S1.p1.9.m9.5.5.1.1.1.1" xref="S1.p1.9.m9.5.5.1.1.2.cmml"><mo id="S1.p1.9.m9.5.5.1.1.1.1.2" 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xref="S1.p1.9.m9.3.3">𝑇</ci></apply></apply></apply></apply><cn id="S1.p1.9.m9.5.5.1.3.cmml" type="integer" xref="S1.p1.9.m9.5.5.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.9.m9.5c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{width}(\mathcal{T}):=\max\{|B_{x}|:x\in V(T)\}-1</annotation><annotation encoding="application/x-llamapun" id="S1.p1.9.m9.5d">roman_width ( caligraphic_T ) := roman_max { | italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | : italic_x ∈ italic_V ( italic_T ) } - 1</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S1.p1.11.4" style="color:#C22147;">treewidth</em> of a graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p1.10.m10.1"><semantics id="S1.p1.10.m10.1a"><mi id="S1.p1.10.m10.1.1" xref="S1.p1.10.m10.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p1.10.m10.1b"><ci id="S1.p1.10.m10.1.1.cmml" xref="S1.p1.10.m10.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.10.m10.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p1.10.m10.1d">italic_G</annotation></semantics></math> is <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{tw}(G):=\min\{\operatorname{width}(\mathcal{T}):% \text{$\mathcal{T}$ is a tree decomposition of $G$}\}" class="ltx_Math" display="inline" id="S1.p1.11.m11.8"><semantics id="S1.p1.11.m11.8a"><mrow id="S1.p1.11.m11.8.8" xref="S1.p1.11.m11.8.8.cmml"><mrow id="S1.p1.11.m11.8.8.3.2" xref="S1.p1.11.m11.8.8.3.1.cmml"><mi id="S1.p1.11.m11.3.3" mathcolor="#C22147" xref="S1.p1.11.m11.3.3.cmml">tw</mi><mo id="S1.p1.11.m11.8.8.3.2a" xref="S1.p1.11.m11.8.8.3.1.cmml">⁡</mo><mrow id="S1.p1.11.m11.8.8.3.2.1" xref="S1.p1.11.m11.8.8.3.1.cmml"><mo id="S1.p1.11.m11.8.8.3.2.1.1" mathcolor="#C22147" stretchy="false" xref="S1.p1.11.m11.8.8.3.1.cmml">(</mo><mi id="S1.p1.11.m11.4.4" mathcolor="#C22147" xref="S1.p1.11.m11.4.4.cmml">G</mi><mo id="S1.p1.11.m11.8.8.3.2.1.2" mathcolor="#C22147" rspace="0.278em" stretchy="false" xref="S1.p1.11.m11.8.8.3.1.cmml">)</mo></mrow></mrow><mo id="S1.p1.11.m11.8.8.2" rspace="0.278em" xref="S1.p1.11.m11.8.8.2.cmml">:=</mo><mrow id="S1.p1.11.m11.8.8.1.1" xref="S1.p1.11.m11.8.8.1.2.cmml"><mi id="S1.p1.11.m11.7.7" xref="S1.p1.11.m11.7.7.cmml">min</mi><mo id="S1.p1.11.m11.8.8.1.1a" xref="S1.p1.11.m11.8.8.1.2.cmml">⁡</mo><mrow id="S1.p1.11.m11.8.8.1.1.1" xref="S1.p1.11.m11.8.8.1.2.cmml"><mo id="S1.p1.11.m11.8.8.1.1.1.2" 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id="S1.p1.11.m11.1.1.1.m1.1.1" xref="S1.p1.11.m11.1.1.1.m1.1.1.cmml">𝒯</mi><mtext id="S1.p1.11.m11.2.2.2a" xref="S1.p1.11.m11.2.2.2b.cmml"> is a tree decomposition of </mtext><mi id="S1.p1.11.m11.2.2.2.m2.1.1" xref="S1.p1.11.m11.2.2.2.m2.1.1.cmml">G</mi></mrow></mrow><mo id="S1.p1.11.m11.8.8.1.1.1.3" stretchy="false" xref="S1.p1.11.m11.8.8.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.11.m11.8b"><apply id="S1.p1.11.m11.8.8.cmml" xref="S1.p1.11.m11.8.8"><csymbol cd="latexml" id="S1.p1.11.m11.8.8.2.cmml" xref="S1.p1.11.m11.8.8.2">assign</csymbol><apply id="S1.p1.11.m11.8.8.3.1.cmml" xref="S1.p1.11.m11.8.8.3.2"><ci id="S1.p1.11.m11.3.3.cmml" xref="S1.p1.11.m11.3.3">tw</ci><ci id="S1.p1.11.m11.4.4.cmml" xref="S1.p1.11.m11.4.4">𝐺</ci></apply><apply id="S1.p1.11.m11.8.8.1.2.cmml" xref="S1.p1.11.m11.8.8.1.1"><min id="S1.p1.11.m11.7.7.cmml" xref="S1.p1.11.m11.7.7"></min><apply id="S1.p1.11.m11.8.8.1.1.1.1.cmml" xref="S1.p1.11.m11.8.8.1.1.1.1"><ci id="S1.p1.11.m11.8.8.1.1.1.1.1.cmml" xref="S1.p1.11.m11.8.8.1.1.1.1.1">:</ci><apply id="S1.p1.11.m11.8.8.1.1.1.1.2.1.cmml" xref="S1.p1.11.m11.8.8.1.1.1.1.2.2"><ci id="S1.p1.11.m11.5.5.cmml" xref="S1.p1.11.m11.5.5">width</ci><ci id="S1.p1.11.m11.6.6.cmml" xref="S1.p1.11.m11.6.6">𝒯</ci></apply><ci id="S1.p1.11.m11.2.2.2b.cmml" xref="S1.p1.11.m11.2.2.2"><mrow id="S1.p1.11.m11.2.2.2.cmml" xref="S1.p1.11.m11.2.2.2"><mi class="ltx_font_mathcaligraphic" id="S1.p1.11.m11.1.1.1.m1.1.1.cmml" xref="S1.p1.11.m11.1.1.1.m1.1.1">𝒯</mi><mtext id="S1.p1.11.m11.2.2.2a.cmml" xref="S1.p1.11.m11.2.2.2"> is a tree decomposition of </mtext><mi id="S1.p1.11.m11.2.2.2.m2.1.1.cmml" xref="S1.p1.11.m11.2.2.2.m2.1.1">G</mi></mrow></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.11.m11.8c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{tw}(G):=\min\{\operatorname{width}(\mathcal{T}):% \text{$\mathcal{T}$ is a tree decomposition of $G$}\}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.11.m11.8d">roman_tw ( italic_G ) := roman_min { roman_width ( caligraphic_T ) : caligraphic_T is a tree decomposition of italic_G }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.18">A <em class="ltx_emph ltx_font_italic" id="S1.p2.18.1" style="color:#C22147;">separation</em> <math alttext="(A,B)" class="ltx_Math" display="inline" id="S1.p2.1.m1.2"><semantics id="S1.p2.1.m1.2a"><mrow id="S1.p2.1.m1.2.3.2" xref="S1.p2.1.m1.2.3.1.cmml"><mo id="S1.p2.1.m1.2.3.2.1" stretchy="false" xref="S1.p2.1.m1.2.3.1.cmml">(</mo><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">A</mi><mo id="S1.p2.1.m1.2.3.2.2" xref="S1.p2.1.m1.2.3.1.cmml">,</mo><mi id="S1.p2.1.m1.2.2" xref="S1.p2.1.m1.2.2.cmml">B</mi><mo id="S1.p2.1.m1.2.3.2.3" stretchy="false" xref="S1.p2.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.2b"><interval closure="open" id="S1.p2.1.m1.2.3.1.cmml" xref="S1.p2.1.m1.2.3.2"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝐴</ci><ci id="S1.p2.1.m1.2.2.cmml" xref="S1.p2.1.m1.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.2d">( italic_A , italic_B )</annotation></semantics></math> of a graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_G</annotation></semantics></math> is a pair of subsets of <math alttext="V(G)" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mrow id="S1.p2.3.m3.1.2" xref="S1.p2.3.m3.1.2.cmml"><mi id="S1.p2.3.m3.1.2.2" xref="S1.p2.3.m3.1.2.2.cmml">V</mi><mo id="S1.p2.3.m3.1.2.1" xref="S1.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.p2.3.m3.1.2.3.2" xref="S1.p2.3.m3.1.2.cmml"><mo id="S1.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S1.p2.3.m3.1.2.cmml">(</mo><mi id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml">G</mi><mo id="S1.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S1.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.2.cmml" xref="S1.p2.3.m3.1.2"><times id="S1.p2.3.m3.1.2.1.cmml" xref="S1.p2.3.m3.1.2.1"></times><ci id="S1.p2.3.m3.1.2.2.cmml" xref="S1.p2.3.m3.1.2.2">𝑉</ci><ci id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">V(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">italic_V ( italic_G )</annotation></semantics></math> with <math alttext="A\cup B=V(G)" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mrow id="S1.p2.4.m4.1.2" xref="S1.p2.4.m4.1.2.cmml"><mrow id="S1.p2.4.m4.1.2.2" xref="S1.p2.4.m4.1.2.2.cmml"><mi id="S1.p2.4.m4.1.2.2.2" xref="S1.p2.4.m4.1.2.2.2.cmml">A</mi><mo id="S1.p2.4.m4.1.2.2.1" xref="S1.p2.4.m4.1.2.2.1.cmml">∪</mo><mi id="S1.p2.4.m4.1.2.2.3" xref="S1.p2.4.m4.1.2.2.3.cmml">B</mi></mrow><mo id="S1.p2.4.m4.1.2.1" xref="S1.p2.4.m4.1.2.1.cmml">=</mo><mrow id="S1.p2.4.m4.1.2.3" xref="S1.p2.4.m4.1.2.3.cmml"><mi id="S1.p2.4.m4.1.2.3.2" xref="S1.p2.4.m4.1.2.3.2.cmml">V</mi><mo id="S1.p2.4.m4.1.2.3.1" xref="S1.p2.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S1.p2.4.m4.1.2.3.3.2" xref="S1.p2.4.m4.1.2.3.cmml"><mo id="S1.p2.4.m4.1.2.3.3.2.1" stretchy="false" xref="S1.p2.4.m4.1.2.3.cmml">(</mo><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">G</mi><mo id="S1.p2.4.m4.1.2.3.3.2.2" stretchy="false" xref="S1.p2.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><apply id="S1.p2.4.m4.1.2.cmml" xref="S1.p2.4.m4.1.2"><eq id="S1.p2.4.m4.1.2.1.cmml" xref="S1.p2.4.m4.1.2.1"></eq><apply id="S1.p2.4.m4.1.2.2.cmml" xref="S1.p2.4.m4.1.2.2"><union id="S1.p2.4.m4.1.2.2.1.cmml" xref="S1.p2.4.m4.1.2.2.1"></union><ci id="S1.p2.4.m4.1.2.2.2.cmml" xref="S1.p2.4.m4.1.2.2.2">𝐴</ci><ci id="S1.p2.4.m4.1.2.2.3.cmml" xref="S1.p2.4.m4.1.2.2.3">𝐵</ci></apply><apply id="S1.p2.4.m4.1.2.3.cmml" xref="S1.p2.4.m4.1.2.3"><times id="S1.p2.4.m4.1.2.3.1.cmml" xref="S1.p2.4.m4.1.2.3.1"></times><ci id="S1.p2.4.m4.1.2.3.2.cmml" xref="S1.p2.4.m4.1.2.3.2">𝑉</ci><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">A\cup B=V(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">italic_A ∪ italic_B = italic_V ( italic_G )</annotation></semantics></math> and such that, for each edge <math alttext="vw" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mrow id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml"><mi id="S1.p2.5.m5.1.1.2" xref="S1.p2.5.m5.1.1.2.cmml">v</mi><mo id="S1.p2.5.m5.1.1.1" xref="S1.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.p2.5.m5.1.1.3" xref="S1.p2.5.m5.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><apply id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1"><times id="S1.p2.5.m5.1.1.1.cmml" xref="S1.p2.5.m5.1.1.1"></times><ci id="S1.p2.5.m5.1.1.2.cmml" xref="S1.p2.5.m5.1.1.2">𝑣</ci><ci id="S1.p2.5.m5.1.1.3.cmml" xref="S1.p2.5.m5.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">vw</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_v italic_w</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S1.p2.6.m6.1"><semantics id="S1.p2.6.m6.1a"><mi id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.1b"><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.1d">italic_G</annotation></semantics></math>, <math alttext="\{v,w\}\subseteq A" class="ltx_Math" display="inline" id="S1.p2.7.m7.2"><semantics id="S1.p2.7.m7.2a"><mrow id="S1.p2.7.m7.2.3" xref="S1.p2.7.m7.2.3.cmml"><mrow id="S1.p2.7.m7.2.3.2.2" xref="S1.p2.7.m7.2.3.2.1.cmml"><mo id="S1.p2.7.m7.2.3.2.2.1" stretchy="false" xref="S1.p2.7.m7.2.3.2.1.cmml">{</mo><mi id="S1.p2.7.m7.1.1" xref="S1.p2.7.m7.1.1.cmml">v</mi><mo id="S1.p2.7.m7.2.3.2.2.2" xref="S1.p2.7.m7.2.3.2.1.cmml">,</mo><mi id="S1.p2.7.m7.2.2" xref="S1.p2.7.m7.2.2.cmml">w</mi><mo id="S1.p2.7.m7.2.3.2.2.3" stretchy="false" xref="S1.p2.7.m7.2.3.2.1.cmml">}</mo></mrow><mo id="S1.p2.7.m7.2.3.1" xref="S1.p2.7.m7.2.3.1.cmml">⊆</mo><mi id="S1.p2.7.m7.2.3.3" xref="S1.p2.7.m7.2.3.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.7.m7.2b"><apply id="S1.p2.7.m7.2.3.cmml" xref="S1.p2.7.m7.2.3"><subset id="S1.p2.7.m7.2.3.1.cmml" xref="S1.p2.7.m7.2.3.1"></subset><set id="S1.p2.7.m7.2.3.2.1.cmml" xref="S1.p2.7.m7.2.3.2.2"><ci id="S1.p2.7.m7.1.1.cmml" xref="S1.p2.7.m7.1.1">𝑣</ci><ci id="S1.p2.7.m7.2.2.cmml" xref="S1.p2.7.m7.2.2">𝑤</ci></set><ci id="S1.p2.7.m7.2.3.3.cmml" xref="S1.p2.7.m7.2.3.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.7.m7.2c">\{v,w\}\subseteq A</annotation><annotation encoding="application/x-llamapun" id="S1.p2.7.m7.2d">{ italic_v , italic_w } ⊆ italic_A</annotation></semantics></math> or <math alttext="\{v,w\}\subseteq B" class="ltx_Math" display="inline" id="S1.p2.8.m8.2"><semantics id="S1.p2.8.m8.2a"><mrow id="S1.p2.8.m8.2.3" xref="S1.p2.8.m8.2.3.cmml"><mrow id="S1.p2.8.m8.2.3.2.2" xref="S1.p2.8.m8.2.3.2.1.cmml"><mo id="S1.p2.8.m8.2.3.2.2.1" stretchy="false" xref="S1.p2.8.m8.2.3.2.1.cmml">{</mo><mi id="S1.p2.8.m8.1.1" xref="S1.p2.8.m8.1.1.cmml">v</mi><mo id="S1.p2.8.m8.2.3.2.2.2" xref="S1.p2.8.m8.2.3.2.1.cmml">,</mo><mi id="S1.p2.8.m8.2.2" xref="S1.p2.8.m8.2.2.cmml">w</mi><mo id="S1.p2.8.m8.2.3.2.2.3" stretchy="false" xref="S1.p2.8.m8.2.3.2.1.cmml">}</mo></mrow><mo id="S1.p2.8.m8.2.3.1" xref="S1.p2.8.m8.2.3.1.cmml">⊆</mo><mi id="S1.p2.8.m8.2.3.3" xref="S1.p2.8.m8.2.3.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.8.m8.2b"><apply id="S1.p2.8.m8.2.3.cmml" xref="S1.p2.8.m8.2.3"><subset id="S1.p2.8.m8.2.3.1.cmml" xref="S1.p2.8.m8.2.3.1"></subset><set id="S1.p2.8.m8.2.3.2.1.cmml" xref="S1.p2.8.m8.2.3.2.2"><ci id="S1.p2.8.m8.1.1.cmml" xref="S1.p2.8.m8.1.1">𝑣</ci><ci id="S1.p2.8.m8.2.2.cmml" xref="S1.p2.8.m8.2.2">𝑤</ci></set><ci id="S1.p2.8.m8.2.3.3.cmml" xref="S1.p2.8.m8.2.3.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.8.m8.2c">\{v,w\}\subseteq B</annotation><annotation encoding="application/x-llamapun" id="S1.p2.8.m8.2d">{ italic_v , italic_w } ⊆ italic_B</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S1.p2.18.2" style="color:#C22147;">order</em> of a separation <math alttext="(A,B)" class="ltx_Math" display="inline" id="S1.p2.9.m9.2"><semantics id="S1.p2.9.m9.2a"><mrow id="S1.p2.9.m9.2.3.2" xref="S1.p2.9.m9.2.3.1.cmml"><mo id="S1.p2.9.m9.2.3.2.1" stretchy="false" xref="S1.p2.9.m9.2.3.1.cmml">(</mo><mi id="S1.p2.9.m9.1.1" xref="S1.p2.9.m9.1.1.cmml">A</mi><mo id="S1.p2.9.m9.2.3.2.2" xref="S1.p2.9.m9.2.3.1.cmml">,</mo><mi id="S1.p2.9.m9.2.2" xref="S1.p2.9.m9.2.2.cmml">B</mi><mo id="S1.p2.9.m9.2.3.2.3" stretchy="false" xref="S1.p2.9.m9.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.9.m9.2b"><interval closure="open" id="S1.p2.9.m9.2.3.1.cmml" xref="S1.p2.9.m9.2.3.2"><ci id="S1.p2.9.m9.1.1.cmml" xref="S1.p2.9.m9.1.1">𝐴</ci><ci id="S1.p2.9.m9.2.2.cmml" xref="S1.p2.9.m9.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.9.m9.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.9.m9.2d">( italic_A , italic_B )</annotation></semantics></math> is <math alttext="|A\cap B|" class="ltx_Math" display="inline" id="S1.p2.10.m10.1"><semantics id="S1.p2.10.m10.1a"><mrow id="S1.p2.10.m10.1.1.1" xref="S1.p2.10.m10.1.1.2.cmml"><mo id="S1.p2.10.m10.1.1.1.2" stretchy="false" xref="S1.p2.10.m10.1.1.2.1.cmml">|</mo><mrow id="S1.p2.10.m10.1.1.1.1" xref="S1.p2.10.m10.1.1.1.1.cmml"><mi id="S1.p2.10.m10.1.1.1.1.2" xref="S1.p2.10.m10.1.1.1.1.2.cmml">A</mi><mo id="S1.p2.10.m10.1.1.1.1.1" xref="S1.p2.10.m10.1.1.1.1.1.cmml">∩</mo><mi id="S1.p2.10.m10.1.1.1.1.3" xref="S1.p2.10.m10.1.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p2.10.m10.1.1.1.3" stretchy="false" xref="S1.p2.10.m10.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.10.m10.1b"><apply id="S1.p2.10.m10.1.1.2.cmml" xref="S1.p2.10.m10.1.1.1"><abs id="S1.p2.10.m10.1.1.2.1.cmml" xref="S1.p2.10.m10.1.1.1.2"></abs><apply id="S1.p2.10.m10.1.1.1.1.cmml" xref="S1.p2.10.m10.1.1.1.1"><intersect id="S1.p2.10.m10.1.1.1.1.1.cmml" xref="S1.p2.10.m10.1.1.1.1.1"></intersect><ci id="S1.p2.10.m10.1.1.1.1.2.cmml" xref="S1.p2.10.m10.1.1.1.1.2">𝐴</ci><ci id="S1.p2.10.m10.1.1.1.1.3.cmml" xref="S1.p2.10.m10.1.1.1.1.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.10.m10.1c">|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S1.p2.10.m10.1d">| italic_A ∩ italic_B |</annotation></semantics></math>. A separation <math alttext="(A,B)" class="ltx_Math" display="inline" id="S1.p2.11.m11.2"><semantics id="S1.p2.11.m11.2a"><mrow id="S1.p2.11.m11.2.3.2" xref="S1.p2.11.m11.2.3.1.cmml"><mo id="S1.p2.11.m11.2.3.2.1" stretchy="false" xref="S1.p2.11.m11.2.3.1.cmml">(</mo><mi id="S1.p2.11.m11.1.1" xref="S1.p2.11.m11.1.1.cmml">A</mi><mo id="S1.p2.11.m11.2.3.2.2" xref="S1.p2.11.m11.2.3.1.cmml">,</mo><mi id="S1.p2.11.m11.2.2" xref="S1.p2.11.m11.2.2.cmml">B</mi><mo id="S1.p2.11.m11.2.3.2.3" stretchy="false" xref="S1.p2.11.m11.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.11.m11.2b"><interval closure="open" id="S1.p2.11.m11.2.3.1.cmml" xref="S1.p2.11.m11.2.3.2"><ci id="S1.p2.11.m11.1.1.cmml" xref="S1.p2.11.m11.1.1">𝐴</ci><ci id="S1.p2.11.m11.2.2.cmml" xref="S1.p2.11.m11.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.11.m11.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.11.m11.2d">( italic_A , italic_B )</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S1.p2.18.3" style="color:#C22147;">balanced</em> if <math alttext="|A\setminus B|\leq\tfrac{2}{3}|V(G)|" class="ltx_Math" display="inline" id="S1.p2.12.m12.3"><semantics id="S1.p2.12.m12.3a"><mrow id="S1.p2.12.m12.3.3" xref="S1.p2.12.m12.3.3.cmml"><mrow id="S1.p2.12.m12.2.2.1.1" xref="S1.p2.12.m12.2.2.1.2.cmml"><mo id="S1.p2.12.m12.2.2.1.1.2" stretchy="false" xref="S1.p2.12.m12.2.2.1.2.1.cmml">|</mo><mrow id="S1.p2.12.m12.2.2.1.1.1" xref="S1.p2.12.m12.2.2.1.1.1.cmml"><mi id="S1.p2.12.m12.2.2.1.1.1.2" xref="S1.p2.12.m12.2.2.1.1.1.2.cmml">A</mi><mo id="S1.p2.12.m12.2.2.1.1.1.1" xref="S1.p2.12.m12.2.2.1.1.1.1.cmml">∖</mo><mi id="S1.p2.12.m12.2.2.1.1.1.3" xref="S1.p2.12.m12.2.2.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p2.12.m12.2.2.1.1.3" stretchy="false" xref="S1.p2.12.m12.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p2.12.m12.3.3.3" xref="S1.p2.12.m12.3.3.3.cmml">≤</mo><mrow id="S1.p2.12.m12.3.3.2" xref="S1.p2.12.m12.3.3.2.cmml"><mfrac id="S1.p2.12.m12.3.3.2.3" xref="S1.p2.12.m12.3.3.2.3.cmml"><mn id="S1.p2.12.m12.3.3.2.3.2" xref="S1.p2.12.m12.3.3.2.3.2.cmml">2</mn><mn id="S1.p2.12.m12.3.3.2.3.3" xref="S1.p2.12.m12.3.3.2.3.3.cmml">3</mn></mfrac><mo id="S1.p2.12.m12.3.3.2.2" xref="S1.p2.12.m12.3.3.2.2.cmml">⁢</mo><mrow id="S1.p2.12.m12.3.3.2.1.1" xref="S1.p2.12.m12.3.3.2.1.2.cmml"><mo id="S1.p2.12.m12.3.3.2.1.1.2" stretchy="false" xref="S1.p2.12.m12.3.3.2.1.2.1.cmml">|</mo><mrow id="S1.p2.12.m12.3.3.2.1.1.1" xref="S1.p2.12.m12.3.3.2.1.1.1.cmml"><mi id="S1.p2.12.m12.3.3.2.1.1.1.2" xref="S1.p2.12.m12.3.3.2.1.1.1.2.cmml">V</mi><mo id="S1.p2.12.m12.3.3.2.1.1.1.1" xref="S1.p2.12.m12.3.3.2.1.1.1.1.cmml">⁢</mo><mrow id="S1.p2.12.m12.3.3.2.1.1.1.3.2" xref="S1.p2.12.m12.3.3.2.1.1.1.cmml"><mo id="S1.p2.12.m12.3.3.2.1.1.1.3.2.1" stretchy="false" xref="S1.p2.12.m12.3.3.2.1.1.1.cmml">(</mo><mi id="S1.p2.12.m12.1.1" xref="S1.p2.12.m12.1.1.cmml">G</mi><mo id="S1.p2.12.m12.3.3.2.1.1.1.3.2.2" stretchy="false" xref="S1.p2.12.m12.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p2.12.m12.3.3.2.1.1.3" stretchy="false" xref="S1.p2.12.m12.3.3.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.12.m12.3b"><apply id="S1.p2.12.m12.3.3.cmml" xref="S1.p2.12.m12.3.3"><leq id="S1.p2.12.m12.3.3.3.cmml" xref="S1.p2.12.m12.3.3.3"></leq><apply id="S1.p2.12.m12.2.2.1.2.cmml" xref="S1.p2.12.m12.2.2.1.1"><abs id="S1.p2.12.m12.2.2.1.2.1.cmml" xref="S1.p2.12.m12.2.2.1.1.2"></abs><apply id="S1.p2.12.m12.2.2.1.1.1.cmml" xref="S1.p2.12.m12.2.2.1.1.1"><setdiff id="S1.p2.12.m12.2.2.1.1.1.1.cmml" xref="S1.p2.12.m12.2.2.1.1.1.1"></setdiff><ci id="S1.p2.12.m12.2.2.1.1.1.2.cmml" xref="S1.p2.12.m12.2.2.1.1.1.2">𝐴</ci><ci id="S1.p2.12.m12.2.2.1.1.1.3.cmml" xref="S1.p2.12.m12.2.2.1.1.1.3">𝐵</ci></apply></apply><apply id="S1.p2.12.m12.3.3.2.cmml" xref="S1.p2.12.m12.3.3.2"><times id="S1.p2.12.m12.3.3.2.2.cmml" xref="S1.p2.12.m12.3.3.2.2"></times><apply id="S1.p2.12.m12.3.3.2.3.cmml" xref="S1.p2.12.m12.3.3.2.3"><divide id="S1.p2.12.m12.3.3.2.3.1.cmml" xref="S1.p2.12.m12.3.3.2.3"></divide><cn id="S1.p2.12.m12.3.3.2.3.2.cmml" type="integer" xref="S1.p2.12.m12.3.3.2.3.2">2</cn><cn id="S1.p2.12.m12.3.3.2.3.3.cmml" type="integer" xref="S1.p2.12.m12.3.3.2.3.3">3</cn></apply><apply id="S1.p2.12.m12.3.3.2.1.2.cmml" xref="S1.p2.12.m12.3.3.2.1.1"><abs id="S1.p2.12.m12.3.3.2.1.2.1.cmml" xref="S1.p2.12.m12.3.3.2.1.1.2"></abs><apply id="S1.p2.12.m12.3.3.2.1.1.1.cmml" xref="S1.p2.12.m12.3.3.2.1.1.1"><times id="S1.p2.12.m12.3.3.2.1.1.1.1.cmml" xref="S1.p2.12.m12.3.3.2.1.1.1.1"></times><ci id="S1.p2.12.m12.3.3.2.1.1.1.2.cmml" xref="S1.p2.12.m12.3.3.2.1.1.1.2">𝑉</ci><ci id="S1.p2.12.m12.1.1.cmml" xref="S1.p2.12.m12.1.1">𝐺</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.12.m12.3c">|A\setminus B|\leq\tfrac{2}{3}|V(G)|</annotation><annotation encoding="application/x-llamapun" id="S1.p2.12.m12.3d">| italic_A ∖ italic_B | ≤ divide start_ARG 2 end_ARG start_ARG 3 end_ARG | italic_V ( italic_G ) |</annotation></semantics></math> and <math alttext="|B\setminus A|\leq\tfrac{2}{3}|V(G)|" class="ltx_Math" display="inline" id="S1.p2.13.m13.3"><semantics id="S1.p2.13.m13.3a"><mrow id="S1.p2.13.m13.3.3" xref="S1.p2.13.m13.3.3.cmml"><mrow id="S1.p2.13.m13.2.2.1.1" xref="S1.p2.13.m13.2.2.1.2.cmml"><mo id="S1.p2.13.m13.2.2.1.1.2" stretchy="false" xref="S1.p2.13.m13.2.2.1.2.1.cmml">|</mo><mrow id="S1.p2.13.m13.2.2.1.1.1" xref="S1.p2.13.m13.2.2.1.1.1.cmml"><mi id="S1.p2.13.m13.2.2.1.1.1.2" xref="S1.p2.13.m13.2.2.1.1.1.2.cmml">B</mi><mo id="S1.p2.13.m13.2.2.1.1.1.1" xref="S1.p2.13.m13.2.2.1.1.1.1.cmml">∖</mo><mi id="S1.p2.13.m13.2.2.1.1.1.3" xref="S1.p2.13.m13.2.2.1.1.1.3.cmml">A</mi></mrow><mo id="S1.p2.13.m13.2.2.1.1.3" stretchy="false" xref="S1.p2.13.m13.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p2.13.m13.3.3.3" xref="S1.p2.13.m13.3.3.3.cmml">≤</mo><mrow id="S1.p2.13.m13.3.3.2" xref="S1.p2.13.m13.3.3.2.cmml"><mfrac id="S1.p2.13.m13.3.3.2.3" xref="S1.p2.13.m13.3.3.2.3.cmml"><mn id="S1.p2.13.m13.3.3.2.3.2" xref="S1.p2.13.m13.3.3.2.3.2.cmml">2</mn><mn id="S1.p2.13.m13.3.3.2.3.3" xref="S1.p2.13.m13.3.3.2.3.3.cmml">3</mn></mfrac><mo id="S1.p2.13.m13.3.3.2.2" xref="S1.p2.13.m13.3.3.2.2.cmml">⁢</mo><mrow id="S1.p2.13.m13.3.3.2.1.1" xref="S1.p2.13.m13.3.3.2.1.2.cmml"><mo id="S1.p2.13.m13.3.3.2.1.1.2" stretchy="false" xref="S1.p2.13.m13.3.3.2.1.2.1.cmml">|</mo><mrow id="S1.p2.13.m13.3.3.2.1.1.1" xref="S1.p2.13.m13.3.3.2.1.1.1.cmml"><mi id="S1.p2.13.m13.3.3.2.1.1.1.2" xref="S1.p2.13.m13.3.3.2.1.1.1.2.cmml">V</mi><mo id="S1.p2.13.m13.3.3.2.1.1.1.1" xref="S1.p2.13.m13.3.3.2.1.1.1.1.cmml">⁢</mo><mrow id="S1.p2.13.m13.3.3.2.1.1.1.3.2" xref="S1.p2.13.m13.3.3.2.1.1.1.cmml"><mo id="S1.p2.13.m13.3.3.2.1.1.1.3.2.1" stretchy="false" xref="S1.p2.13.m13.3.3.2.1.1.1.cmml">(</mo><mi id="S1.p2.13.m13.1.1" xref="S1.p2.13.m13.1.1.cmml">G</mi><mo id="S1.p2.13.m13.3.3.2.1.1.1.3.2.2" stretchy="false" xref="S1.p2.13.m13.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p2.13.m13.3.3.2.1.1.3" stretchy="false" xref="S1.p2.13.m13.3.3.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.13.m13.3b"><apply id="S1.p2.13.m13.3.3.cmml" xref="S1.p2.13.m13.3.3"><leq id="S1.p2.13.m13.3.3.3.cmml" xref="S1.p2.13.m13.3.3.3"></leq><apply id="S1.p2.13.m13.2.2.1.2.cmml" xref="S1.p2.13.m13.2.2.1.1"><abs id="S1.p2.13.m13.2.2.1.2.1.cmml" xref="S1.p2.13.m13.2.2.1.1.2"></abs><apply id="S1.p2.13.m13.2.2.1.1.1.cmml" xref="S1.p2.13.m13.2.2.1.1.1"><setdiff id="S1.p2.13.m13.2.2.1.1.1.1.cmml" xref="S1.p2.13.m13.2.2.1.1.1.1"></setdiff><ci id="S1.p2.13.m13.2.2.1.1.1.2.cmml" xref="S1.p2.13.m13.2.2.1.1.1.2">𝐵</ci><ci id="S1.p2.13.m13.2.2.1.1.1.3.cmml" xref="S1.p2.13.m13.2.2.1.1.1.3">𝐴</ci></apply></apply><apply id="S1.p2.13.m13.3.3.2.cmml" xref="S1.p2.13.m13.3.3.2"><times id="S1.p2.13.m13.3.3.2.2.cmml" xref="S1.p2.13.m13.3.3.2.2"></times><apply id="S1.p2.13.m13.3.3.2.3.cmml" xref="S1.p2.13.m13.3.3.2.3"><divide id="S1.p2.13.m13.3.3.2.3.1.cmml" xref="S1.p2.13.m13.3.3.2.3"></divide><cn id="S1.p2.13.m13.3.3.2.3.2.cmml" type="integer" xref="S1.p2.13.m13.3.3.2.3.2">2</cn><cn id="S1.p2.13.m13.3.3.2.3.3.cmml" type="integer" xref="S1.p2.13.m13.3.3.2.3.3">3</cn></apply><apply id="S1.p2.13.m13.3.3.2.1.2.cmml" xref="S1.p2.13.m13.3.3.2.1.1"><abs id="S1.p2.13.m13.3.3.2.1.2.1.cmml" xref="S1.p2.13.m13.3.3.2.1.1.2"></abs><apply id="S1.p2.13.m13.3.3.2.1.1.1.cmml" xref="S1.p2.13.m13.3.3.2.1.1.1"><times id="S1.p2.13.m13.3.3.2.1.1.1.1.cmml" xref="S1.p2.13.m13.3.3.2.1.1.1.1"></times><ci id="S1.p2.13.m13.3.3.2.1.1.1.2.cmml" xref="S1.p2.13.m13.3.3.2.1.1.1.2">𝑉</ci><ci id="S1.p2.13.m13.1.1.cmml" xref="S1.p2.13.m13.1.1">𝐺</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.13.m13.3c">|B\setminus A|\leq\tfrac{2}{3}|V(G)|</annotation><annotation encoding="application/x-llamapun" id="S1.p2.13.m13.3d">| italic_B ∖ italic_A | ≤ divide start_ARG 2 end_ARG start_ARG 3 end_ARG | italic_V ( italic_G ) |</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S1.p2.18.4" style="color:#C22147;">separation number</em> <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{sn}(G)" class="ltx_Math" display="inline" id="S1.p2.14.m14.2"><semantics id="S1.p2.14.m14.2a"><mrow id="S1.p2.14.m14.2.3.2" xref="S1.p2.14.m14.2.3.1.cmml"><mi id="S1.p2.14.m14.1.1" mathcolor="#C22147" xref="S1.p2.14.m14.1.1.cmml">sn</mi><mo id="S1.p2.14.m14.2.3.2a" xref="S1.p2.14.m14.2.3.1.cmml">⁡</mo><mrow id="S1.p2.14.m14.2.3.2.1" xref="S1.p2.14.m14.2.3.1.cmml"><mo id="S1.p2.14.m14.2.3.2.1.1" mathcolor="#C22147" stretchy="false" xref="S1.p2.14.m14.2.3.1.cmml">(</mo><mi id="S1.p2.14.m14.2.2" mathcolor="#C22147" xref="S1.p2.14.m14.2.2.cmml">G</mi><mo id="S1.p2.14.m14.2.3.2.1.2" mathcolor="#C22147" stretchy="false" xref="S1.p2.14.m14.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.14.m14.2b"><apply id="S1.p2.14.m14.2.3.1.cmml" xref="S1.p2.14.m14.2.3.2"><ci id="S1.p2.14.m14.1.1.cmml" xref="S1.p2.14.m14.1.1">sn</ci><ci id="S1.p2.14.m14.2.2.cmml" xref="S1.p2.14.m14.2.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.14.m14.2c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{sn}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.14.m14.2d">roman_sn ( italic_G )</annotation></semantics></math> of a graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p2.15.m15.1"><semantics id="S1.p2.15.m15.1a"><mi id="S1.p2.15.m15.1.1" xref="S1.p2.15.m15.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p2.15.m15.1b"><ci id="S1.p2.15.m15.1.1.cmml" xref="S1.p2.15.m15.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.15.m15.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p2.15.m15.1d">italic_G</annotation></semantics></math> is the minimum integer <math alttext="a" class="ltx_Math" display="inline" id="S1.p2.16.m16.1"><semantics id="S1.p2.16.m16.1a"><mi id="S1.p2.16.m16.1.1" xref="S1.p2.16.m16.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S1.p2.16.m16.1b"><ci id="S1.p2.16.m16.1.1.cmml" xref="S1.p2.16.m16.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.16.m16.1c">a</annotation><annotation encoding="application/x-llamapun" id="S1.p2.16.m16.1d">italic_a</annotation></semantics></math> such that every subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S1.p2.17.m17.1"><semantics id="S1.p2.17.m17.1a"><mi id="S1.p2.17.m17.1.1" xref="S1.p2.17.m17.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p2.17.m17.1b"><ci id="S1.p2.17.m17.1.1.cmml" xref="S1.p2.17.m17.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.17.m17.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p2.17.m17.1d">italic_G</annotation></semantics></math> has a balanced separation of order at most <math alttext="a" class="ltx_Math" display="inline" id="S1.p2.18.m18.1"><semantics id="S1.p2.18.m18.1a"><mi id="S1.p2.18.m18.1.1" xref="S1.p2.18.m18.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S1.p2.18.m18.1b"><ci id="S1.p2.18.m18.1.1.cmml" xref="S1.p2.18.m18.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.18.m18.1c">a</annotation><annotation encoding="application/x-llamapun" id="S1.p2.18.m18.1d">italic_a</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.2">A short argument due to <cite class="ltx_cite ltx_citemacro_citet">Robertson and Seymour [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib5" title="">5</a>]</cite>, which has many generalizations, shows that for any graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_G</annotation></semantics></math>, <math alttext="\operatorname{sn}(G)\leq\operatorname{tw}(G)+1" class="ltx_Math" display="inline" id="S1.p3.2.m2.4"><semantics id="S1.p3.2.m2.4a"><mrow id="S1.p3.2.m2.4.5" xref="S1.p3.2.m2.4.5.cmml"><mrow id="S1.p3.2.m2.4.5.2.2" xref="S1.p3.2.m2.4.5.2.1.cmml"><mi id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">sn</mi><mo id="S1.p3.2.m2.4.5.2.2a" xref="S1.p3.2.m2.4.5.2.1.cmml">⁡</mo><mrow id="S1.p3.2.m2.4.5.2.2.1" xref="S1.p3.2.m2.4.5.2.1.cmml"><mo id="S1.p3.2.m2.4.5.2.2.1.1" stretchy="false" xref="S1.p3.2.m2.4.5.2.1.cmml">(</mo><mi id="S1.p3.2.m2.2.2" xref="S1.p3.2.m2.2.2.cmml">G</mi><mo id="S1.p3.2.m2.4.5.2.2.1.2" stretchy="false" xref="S1.p3.2.m2.4.5.2.1.cmml">)</mo></mrow></mrow><mo id="S1.p3.2.m2.4.5.1" xref="S1.p3.2.m2.4.5.1.cmml">≤</mo><mrow id="S1.p3.2.m2.4.5.3" xref="S1.p3.2.m2.4.5.3.cmml"><mrow id="S1.p3.2.m2.4.5.3.2.2" xref="S1.p3.2.m2.4.5.3.2.1.cmml"><mi id="S1.p3.2.m2.3.3" xref="S1.p3.2.m2.3.3.cmml">tw</mi><mo id="S1.p3.2.m2.4.5.3.2.2a" xref="S1.p3.2.m2.4.5.3.2.1.cmml">⁡</mo><mrow id="S1.p3.2.m2.4.5.3.2.2.1" xref="S1.p3.2.m2.4.5.3.2.1.cmml"><mo id="S1.p3.2.m2.4.5.3.2.2.1.1" stretchy="false" xref="S1.p3.2.m2.4.5.3.2.1.cmml">(</mo><mi id="S1.p3.2.m2.4.4" xref="S1.p3.2.m2.4.4.cmml">G</mi><mo id="S1.p3.2.m2.4.5.3.2.2.1.2" stretchy="false" xref="S1.p3.2.m2.4.5.3.2.1.cmml">)</mo></mrow></mrow><mo id="S1.p3.2.m2.4.5.3.1" xref="S1.p3.2.m2.4.5.3.1.cmml">+</mo><mn id="S1.p3.2.m2.4.5.3.3" xref="S1.p3.2.m2.4.5.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.4b"><apply id="S1.p3.2.m2.4.5.cmml" xref="S1.p3.2.m2.4.5"><leq id="S1.p3.2.m2.4.5.1.cmml" xref="S1.p3.2.m2.4.5.1"></leq><apply id="S1.p3.2.m2.4.5.2.1.cmml" xref="S1.p3.2.m2.4.5.2.2"><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">sn</ci><ci id="S1.p3.2.m2.2.2.cmml" xref="S1.p3.2.m2.2.2">𝐺</ci></apply><apply id="S1.p3.2.m2.4.5.3.cmml" xref="S1.p3.2.m2.4.5.3"><plus id="S1.p3.2.m2.4.5.3.1.cmml" xref="S1.p3.2.m2.4.5.3.1"></plus><apply id="S1.p3.2.m2.4.5.3.2.1.cmml" xref="S1.p3.2.m2.4.5.3.2.2"><ci id="S1.p3.2.m2.3.3.cmml" xref="S1.p3.2.m2.3.3">tw</ci><ci id="S1.p3.2.m2.4.4.cmml" xref="S1.p3.2.m2.4.4">𝐺</ci></apply><cn id="S1.p3.2.m2.4.5.3.3.cmml" type="integer" xref="S1.p3.2.m2.4.5.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.4c">\operatorname{sn}(G)\leq\operatorname{tw}(G)+1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.4d">roman_sn ( italic_G ) ≤ roman_tw ( italic_G ) + 1</annotation></semantics></math> . We reprove a weak converse of this fact, first proven by <cite class="ltx_cite ltx_citemacro_citet">Dvořák and Norin [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib4" title="">4</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_thm" id="Thmthm1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm1.1.1.1">Theorem 1</span></span><span class="ltx_text ltx_font_bold" id="Thmthm1.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm1.p1"> <p class="ltx_p" id="Thmthm1.p1.3"><span class="ltx_text ltx_font_italic" id="Thmthm1.p1.3.3">There exists a constant <math alttext="c" class="ltx_Math" display="inline" id="Thmthm1.p1.1.1.m1.1"><semantics id="Thmthm1.p1.1.1.m1.1a"><mi id="Thmthm1.p1.1.1.m1.1.1" xref="Thmthm1.p1.1.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="Thmthm1.p1.1.1.m1.1b"><ci id="Thmthm1.p1.1.1.m1.1.1.cmml" xref="Thmthm1.p1.1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm1.p1.1.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="Thmthm1.p1.1.1.m1.1d">italic_c</annotation></semantics></math> such that, for every graph <math alttext="G" class="ltx_Math" display="inline" id="Thmthm1.p1.2.2.m2.1"><semantics id="Thmthm1.p1.2.2.m2.1a"><mi id="Thmthm1.p1.2.2.m2.1.1" xref="Thmthm1.p1.2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm1.p1.2.2.m2.1b"><ci id="Thmthm1.p1.2.2.m2.1.1.cmml" xref="Thmthm1.p1.2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm1.p1.2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm1.p1.2.2.m2.1d">italic_G</annotation></semantics></math>, <math alttext="\operatorname{tw}(G)\leq c\cdot\operatorname{sn}(G)" class="ltx_Math" display="inline" id="Thmthm1.p1.3.3.m3.4"><semantics id="Thmthm1.p1.3.3.m3.4a"><mrow id="Thmthm1.p1.3.3.m3.4.5" xref="Thmthm1.p1.3.3.m3.4.5.cmml"><mrow id="Thmthm1.p1.3.3.m3.4.5.2.2" xref="Thmthm1.p1.3.3.m3.4.5.2.1.cmml"><mi id="Thmthm1.p1.3.3.m3.1.1" xref="Thmthm1.p1.3.3.m3.1.1.cmml">tw</mi><mo id="Thmthm1.p1.3.3.m3.4.5.2.2a" xref="Thmthm1.p1.3.3.m3.4.5.2.1.cmml">⁡</mo><mrow id="Thmthm1.p1.3.3.m3.4.5.2.2.1" xref="Thmthm1.p1.3.3.m3.4.5.2.1.cmml"><mo id="Thmthm1.p1.3.3.m3.4.5.2.2.1.1" stretchy="false" xref="Thmthm1.p1.3.3.m3.4.5.2.1.cmml">(</mo><mi id="Thmthm1.p1.3.3.m3.2.2" xref="Thmthm1.p1.3.3.m3.2.2.cmml">G</mi><mo id="Thmthm1.p1.3.3.m3.4.5.2.2.1.2" stretchy="false" xref="Thmthm1.p1.3.3.m3.4.5.2.1.cmml">)</mo></mrow></mrow><mo id="Thmthm1.p1.3.3.m3.4.5.1" xref="Thmthm1.p1.3.3.m3.4.5.1.cmml">≤</mo><mrow id="Thmthm1.p1.3.3.m3.4.5.3" xref="Thmthm1.p1.3.3.m3.4.5.3.cmml"><mi id="Thmthm1.p1.3.3.m3.4.5.3.2" xref="Thmthm1.p1.3.3.m3.4.5.3.2.cmml">c</mi><mo id="Thmthm1.p1.3.3.m3.4.5.3.1" lspace="0.222em" rspace="0.222em" xref="Thmthm1.p1.3.3.m3.4.5.3.1.cmml">⋅</mo><mrow id="Thmthm1.p1.3.3.m3.4.5.3.3.2" xref="Thmthm1.p1.3.3.m3.4.5.3.3.1.cmml"><mi id="Thmthm1.p1.3.3.m3.3.3" xref="Thmthm1.p1.3.3.m3.3.3.cmml">sn</mi><mo id="Thmthm1.p1.3.3.m3.4.5.3.3.2a" xref="Thmthm1.p1.3.3.m3.4.5.3.3.1.cmml">⁡</mo><mrow id="Thmthm1.p1.3.3.m3.4.5.3.3.2.1" xref="Thmthm1.p1.3.3.m3.4.5.3.3.1.cmml"><mo id="Thmthm1.p1.3.3.m3.4.5.3.3.2.1.1" stretchy="false" xref="Thmthm1.p1.3.3.m3.4.5.3.3.1.cmml">(</mo><mi id="Thmthm1.p1.3.3.m3.4.4" xref="Thmthm1.p1.3.3.m3.4.4.cmml">G</mi><mo id="Thmthm1.p1.3.3.m3.4.5.3.3.2.1.2" stretchy="false" xref="Thmthm1.p1.3.3.m3.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm1.p1.3.3.m3.4b"><apply id="Thmthm1.p1.3.3.m3.4.5.cmml" xref="Thmthm1.p1.3.3.m3.4.5"><leq id="Thmthm1.p1.3.3.m3.4.5.1.cmml" xref="Thmthm1.p1.3.3.m3.4.5.1"></leq><apply id="Thmthm1.p1.3.3.m3.4.5.2.1.cmml" xref="Thmthm1.p1.3.3.m3.4.5.2.2"><ci id="Thmthm1.p1.3.3.m3.1.1.cmml" xref="Thmthm1.p1.3.3.m3.1.1">tw</ci><ci id="Thmthm1.p1.3.3.m3.2.2.cmml" xref="Thmthm1.p1.3.3.m3.2.2">𝐺</ci></apply><apply id="Thmthm1.p1.3.3.m3.4.5.3.cmml" xref="Thmthm1.p1.3.3.m3.4.5.3"><ci id="Thmthm1.p1.3.3.m3.4.5.3.1.cmml" xref="Thmthm1.p1.3.3.m3.4.5.3.1">⋅</ci><ci id="Thmthm1.p1.3.3.m3.4.5.3.2.cmml" xref="Thmthm1.p1.3.3.m3.4.5.3.2">𝑐</ci><apply id="Thmthm1.p1.3.3.m3.4.5.3.3.1.cmml" xref="Thmthm1.p1.3.3.m3.4.5.3.3.2"><ci id="Thmthm1.p1.3.3.m3.3.3.cmml" xref="Thmthm1.p1.3.3.m3.3.3">sn</ci><ci id="Thmthm1.p1.3.3.m3.4.4.cmml" xref="Thmthm1.p1.3.3.m3.4.4">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm1.p1.3.3.m3.4c">\operatorname{tw}(G)\leq c\cdot\operatorname{sn}(G)</annotation><annotation encoding="application/x-llamapun" id="Thmthm1.p1.3.3.m3.4d">roman_tw ( italic_G ) ≤ italic_c ⋅ roman_sn ( italic_G )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.7">To put <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm1" title="Theorem 1. ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1</span></a> into context, consider the following classic result of <cite class="ltx_cite ltx_citemacro_citet">Robertson and Seymour [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib5" title="">5</a>]</cite>. Let <math alttext="W" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mi id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_W</annotation></semantics></math> be a subset of the vertices in a graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">italic_G</annotation></semantics></math>. A separation <math alttext="(A,B)" class="ltx_Math" display="inline" id="S1.p4.3.m3.2"><semantics id="S1.p4.3.m3.2a"><mrow id="S1.p4.3.m3.2.3.2" xref="S1.p4.3.m3.2.3.1.cmml"><mo id="S1.p4.3.m3.2.3.2.1" stretchy="false" xref="S1.p4.3.m3.2.3.1.cmml">(</mo><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">A</mi><mo id="S1.p4.3.m3.2.3.2.2" xref="S1.p4.3.m3.2.3.1.cmml">,</mo><mi id="S1.p4.3.m3.2.2" xref="S1.p4.3.m3.2.2.cmml">B</mi><mo id="S1.p4.3.m3.2.3.2.3" stretchy="false" xref="S1.p4.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.2b"><interval closure="open" id="S1.p4.3.m3.2.3.1.cmml" xref="S1.p4.3.m3.2.3.2"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">𝐴</ci><ci id="S1.p4.3.m3.2.2.cmml" xref="S1.p4.3.m3.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.2d">( italic_A , italic_B )</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><mi id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><ci id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">italic_G</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S1.p4.5.1"><math alttext="W" class="ltx_Math" display="inline" id="S1.p4.5.1.m1.1"><semantics id="S1.p4.5.1.m1.1a"><mi id="S1.p4.5.1.m1.1.1" mathcolor="#C22147" xref="S1.p4.5.1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.p4.5.1.m1.1b"><ci id="S1.p4.5.1.m1.1.1.cmml" xref="S1.p4.5.1.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.1.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.1.m1.1d">italic_W</annotation></semantics></math><span class="ltx_text" id="S1.p4.5.1.1" style="color:#C22147;">-balanced</span></em> if <math alttext="|(A\setminus B)\cap W|\leq\tfrac{2}{3}|W|" class="ltx_Math" display="inline" id="S1.p4.6.m5.2"><semantics id="S1.p4.6.m5.2a"><mrow id="S1.p4.6.m5.2.2" xref="S1.p4.6.m5.2.2.cmml"><mrow id="S1.p4.6.m5.2.2.1.1" xref="S1.p4.6.m5.2.2.1.2.cmml"><mo id="S1.p4.6.m5.2.2.1.1.2" stretchy="false" xref="S1.p4.6.m5.2.2.1.2.1.cmml">|</mo><mrow id="S1.p4.6.m5.2.2.1.1.1" xref="S1.p4.6.m5.2.2.1.1.1.cmml"><mrow id="S1.p4.6.m5.2.2.1.1.1.1.1" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.cmml"><mo id="S1.p4.6.m5.2.2.1.1.1.1.1.2" stretchy="false" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p4.6.m5.2.2.1.1.1.1.1.1" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.cmml"><mi id="S1.p4.6.m5.2.2.1.1.1.1.1.1.2" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.2.cmml">A</mi><mo id="S1.p4.6.m5.2.2.1.1.1.1.1.1.1" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S1.p4.6.m5.2.2.1.1.1.1.1.1.3" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p4.6.m5.2.2.1.1.1.1.1.3" stretchy="false" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p4.6.m5.2.2.1.1.1.2" xref="S1.p4.6.m5.2.2.1.1.1.2.cmml">∩</mo><mi id="S1.p4.6.m5.2.2.1.1.1.3" xref="S1.p4.6.m5.2.2.1.1.1.3.cmml">W</mi></mrow><mo id="S1.p4.6.m5.2.2.1.1.3" stretchy="false" xref="S1.p4.6.m5.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p4.6.m5.2.2.2" xref="S1.p4.6.m5.2.2.2.cmml">≤</mo><mrow id="S1.p4.6.m5.2.2.3" xref="S1.p4.6.m5.2.2.3.cmml"><mfrac id="S1.p4.6.m5.2.2.3.2" xref="S1.p4.6.m5.2.2.3.2.cmml"><mn id="S1.p4.6.m5.2.2.3.2.2" xref="S1.p4.6.m5.2.2.3.2.2.cmml">2</mn><mn id="S1.p4.6.m5.2.2.3.2.3" xref="S1.p4.6.m5.2.2.3.2.3.cmml">3</mn></mfrac><mo id="S1.p4.6.m5.2.2.3.1" xref="S1.p4.6.m5.2.2.3.1.cmml">⁢</mo><mrow id="S1.p4.6.m5.2.2.3.3.2" xref="S1.p4.6.m5.2.2.3.3.1.cmml"><mo id="S1.p4.6.m5.2.2.3.3.2.1" stretchy="false" xref="S1.p4.6.m5.2.2.3.3.1.1.cmml">|</mo><mi id="S1.p4.6.m5.1.1" xref="S1.p4.6.m5.1.1.cmml">W</mi><mo id="S1.p4.6.m5.2.2.3.3.2.2" stretchy="false" xref="S1.p4.6.m5.2.2.3.3.1.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.6.m5.2b"><apply id="S1.p4.6.m5.2.2.cmml" xref="S1.p4.6.m5.2.2"><leq id="S1.p4.6.m5.2.2.2.cmml" xref="S1.p4.6.m5.2.2.2"></leq><apply id="S1.p4.6.m5.2.2.1.2.cmml" xref="S1.p4.6.m5.2.2.1.1"><abs id="S1.p4.6.m5.2.2.1.2.1.cmml" xref="S1.p4.6.m5.2.2.1.1.2"></abs><apply id="S1.p4.6.m5.2.2.1.1.1.cmml" xref="S1.p4.6.m5.2.2.1.1.1"><intersect id="S1.p4.6.m5.2.2.1.1.1.2.cmml" xref="S1.p4.6.m5.2.2.1.1.1.2"></intersect><apply id="S1.p4.6.m5.2.2.1.1.1.1.1.1.cmml" xref="S1.p4.6.m5.2.2.1.1.1.1.1"><setdiff id="S1.p4.6.m5.2.2.1.1.1.1.1.1.1.cmml" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.1"></setdiff><ci id="S1.p4.6.m5.2.2.1.1.1.1.1.1.2.cmml" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.2">𝐴</ci><ci id="S1.p4.6.m5.2.2.1.1.1.1.1.1.3.cmml" xref="S1.p4.6.m5.2.2.1.1.1.1.1.1.3">𝐵</ci></apply><ci id="S1.p4.6.m5.2.2.1.1.1.3.cmml" xref="S1.p4.6.m5.2.2.1.1.1.3">𝑊</ci></apply></apply><apply id="S1.p4.6.m5.2.2.3.cmml" xref="S1.p4.6.m5.2.2.3"><times id="S1.p4.6.m5.2.2.3.1.cmml" xref="S1.p4.6.m5.2.2.3.1"></times><apply id="S1.p4.6.m5.2.2.3.2.cmml" xref="S1.p4.6.m5.2.2.3.2"><divide id="S1.p4.6.m5.2.2.3.2.1.cmml" xref="S1.p4.6.m5.2.2.3.2"></divide><cn id="S1.p4.6.m5.2.2.3.2.2.cmml" type="integer" xref="S1.p4.6.m5.2.2.3.2.2">2</cn><cn id="S1.p4.6.m5.2.2.3.2.3.cmml" type="integer" xref="S1.p4.6.m5.2.2.3.2.3">3</cn></apply><apply id="S1.p4.6.m5.2.2.3.3.1.cmml" xref="S1.p4.6.m5.2.2.3.3.2"><abs id="S1.p4.6.m5.2.2.3.3.1.1.cmml" xref="S1.p4.6.m5.2.2.3.3.2.1"></abs><ci id="S1.p4.6.m5.1.1.cmml" xref="S1.p4.6.m5.1.1">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m5.2c">|(A\setminus B)\cap W|\leq\tfrac{2}{3}|W|</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m5.2d">| ( italic_A ∖ italic_B ) ∩ italic_W | ≤ divide start_ARG 2 end_ARG start_ARG 3 end_ARG | italic_W |</annotation></semantics></math> and <math alttext="|(B\setminus A)\cap W|\leq\tfrac{2}{3}|W|" class="ltx_Math" display="inline" id="S1.p4.7.m6.2"><semantics id="S1.p4.7.m6.2a"><mrow id="S1.p4.7.m6.2.2" xref="S1.p4.7.m6.2.2.cmml"><mrow id="S1.p4.7.m6.2.2.1.1" xref="S1.p4.7.m6.2.2.1.2.cmml"><mo id="S1.p4.7.m6.2.2.1.1.2" stretchy="false" xref="S1.p4.7.m6.2.2.1.2.1.cmml">|</mo><mrow id="S1.p4.7.m6.2.2.1.1.1" xref="S1.p4.7.m6.2.2.1.1.1.cmml"><mrow id="S1.p4.7.m6.2.2.1.1.1.1.1" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.cmml"><mo id="S1.p4.7.m6.2.2.1.1.1.1.1.2" stretchy="false" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p4.7.m6.2.2.1.1.1.1.1.1" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.cmml"><mi id="S1.p4.7.m6.2.2.1.1.1.1.1.1.2" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.2.cmml">B</mi><mo id="S1.p4.7.m6.2.2.1.1.1.1.1.1.1" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S1.p4.7.m6.2.2.1.1.1.1.1.1.3" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.3.cmml">A</mi></mrow><mo id="S1.p4.7.m6.2.2.1.1.1.1.1.3" stretchy="false" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p4.7.m6.2.2.1.1.1.2" xref="S1.p4.7.m6.2.2.1.1.1.2.cmml">∩</mo><mi id="S1.p4.7.m6.2.2.1.1.1.3" xref="S1.p4.7.m6.2.2.1.1.1.3.cmml">W</mi></mrow><mo id="S1.p4.7.m6.2.2.1.1.3" stretchy="false" xref="S1.p4.7.m6.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p4.7.m6.2.2.2" xref="S1.p4.7.m6.2.2.2.cmml">≤</mo><mrow id="S1.p4.7.m6.2.2.3" xref="S1.p4.7.m6.2.2.3.cmml"><mfrac id="S1.p4.7.m6.2.2.3.2" xref="S1.p4.7.m6.2.2.3.2.cmml"><mn id="S1.p4.7.m6.2.2.3.2.2" xref="S1.p4.7.m6.2.2.3.2.2.cmml">2</mn><mn id="S1.p4.7.m6.2.2.3.2.3" xref="S1.p4.7.m6.2.2.3.2.3.cmml">3</mn></mfrac><mo id="S1.p4.7.m6.2.2.3.1" xref="S1.p4.7.m6.2.2.3.1.cmml">⁢</mo><mrow id="S1.p4.7.m6.2.2.3.3.2" xref="S1.p4.7.m6.2.2.3.3.1.cmml"><mo id="S1.p4.7.m6.2.2.3.3.2.1" stretchy="false" xref="S1.p4.7.m6.2.2.3.3.1.1.cmml">|</mo><mi id="S1.p4.7.m6.1.1" xref="S1.p4.7.m6.1.1.cmml">W</mi><mo id="S1.p4.7.m6.2.2.3.3.2.2" stretchy="false" xref="S1.p4.7.m6.2.2.3.3.1.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.7.m6.2b"><apply id="S1.p4.7.m6.2.2.cmml" xref="S1.p4.7.m6.2.2"><leq id="S1.p4.7.m6.2.2.2.cmml" xref="S1.p4.7.m6.2.2.2"></leq><apply id="S1.p4.7.m6.2.2.1.2.cmml" xref="S1.p4.7.m6.2.2.1.1"><abs id="S1.p4.7.m6.2.2.1.2.1.cmml" xref="S1.p4.7.m6.2.2.1.1.2"></abs><apply id="S1.p4.7.m6.2.2.1.1.1.cmml" xref="S1.p4.7.m6.2.2.1.1.1"><intersect id="S1.p4.7.m6.2.2.1.1.1.2.cmml" xref="S1.p4.7.m6.2.2.1.1.1.2"></intersect><apply id="S1.p4.7.m6.2.2.1.1.1.1.1.1.cmml" xref="S1.p4.7.m6.2.2.1.1.1.1.1"><setdiff id="S1.p4.7.m6.2.2.1.1.1.1.1.1.1.cmml" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.1"></setdiff><ci id="S1.p4.7.m6.2.2.1.1.1.1.1.1.2.cmml" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.2">𝐵</ci><ci id="S1.p4.7.m6.2.2.1.1.1.1.1.1.3.cmml" xref="S1.p4.7.m6.2.2.1.1.1.1.1.1.3">𝐴</ci></apply><ci id="S1.p4.7.m6.2.2.1.1.1.3.cmml" xref="S1.p4.7.m6.2.2.1.1.1.3">𝑊</ci></apply></apply><apply id="S1.p4.7.m6.2.2.3.cmml" xref="S1.p4.7.m6.2.2.3"><times id="S1.p4.7.m6.2.2.3.1.cmml" xref="S1.p4.7.m6.2.2.3.1"></times><apply id="S1.p4.7.m6.2.2.3.2.cmml" xref="S1.p4.7.m6.2.2.3.2"><divide id="S1.p4.7.m6.2.2.3.2.1.cmml" xref="S1.p4.7.m6.2.2.3.2"></divide><cn id="S1.p4.7.m6.2.2.3.2.2.cmml" type="integer" xref="S1.p4.7.m6.2.2.3.2.2">2</cn><cn id="S1.p4.7.m6.2.2.3.2.3.cmml" type="integer" xref="S1.p4.7.m6.2.2.3.2.3">3</cn></apply><apply id="S1.p4.7.m6.2.2.3.3.1.cmml" xref="S1.p4.7.m6.2.2.3.3.2"><abs id="S1.p4.7.m6.2.2.3.3.1.1.cmml" xref="S1.p4.7.m6.2.2.3.3.2.1"></abs><ci id="S1.p4.7.m6.1.1.cmml" xref="S1.p4.7.m6.1.1">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.7.m6.2c">|(B\setminus A)\cap W|\leq\tfrac{2}{3}|W|</annotation><annotation encoding="application/x-llamapun" id="S1.p4.7.m6.2d">| ( italic_B ∖ italic_A ) ∩ italic_W | ≤ divide start_ARG 2 end_ARG start_ARG 3 end_ARG | italic_W |</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_thm" id="Thmthm2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm2.1.1.1">Theorem 2</span></span><span class="ltx_text ltx_font_bold" id="Thmthm2.2.2"> </span>(<cite class="ltx_cite ltx_citemacro_citet">Robertson and Seymour [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib5" title="">5</a>]</cite>)<span class="ltx_text ltx_font_bold" id="Thmthm2.3.3">.</span> </h6> <div class="ltx_para" id="Thmthm2.p1"> <p class="ltx_p" id="Thmthm2.p1.6"><span class="ltx_text ltx_font_italic" id="Thmthm2.p1.6.6">Let <math alttext="G" class="ltx_Math" display="inline" id="Thmthm2.p1.1.1.m1.1"><semantics id="Thmthm2.p1.1.1.m1.1a"><mi id="Thmthm2.p1.1.1.m1.1.1" xref="Thmthm2.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm2.p1.1.1.m1.1b"><ci id="Thmthm2.p1.1.1.m1.1.1.cmml" xref="Thmthm2.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm2.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm2.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be a graph with the property that, for each <math alttext="W\subseteq V(G)" class="ltx_Math" display="inline" id="Thmthm2.p1.2.2.m2.1"><semantics id="Thmthm2.p1.2.2.m2.1a"><mrow id="Thmthm2.p1.2.2.m2.1.2" xref="Thmthm2.p1.2.2.m2.1.2.cmml"><mi id="Thmthm2.p1.2.2.m2.1.2.2" xref="Thmthm2.p1.2.2.m2.1.2.2.cmml">W</mi><mo id="Thmthm2.p1.2.2.m2.1.2.1" xref="Thmthm2.p1.2.2.m2.1.2.1.cmml">⊆</mo><mrow id="Thmthm2.p1.2.2.m2.1.2.3" xref="Thmthm2.p1.2.2.m2.1.2.3.cmml"><mi id="Thmthm2.p1.2.2.m2.1.2.3.2" xref="Thmthm2.p1.2.2.m2.1.2.3.2.cmml">V</mi><mo id="Thmthm2.p1.2.2.m2.1.2.3.1" xref="Thmthm2.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmthm2.p1.2.2.m2.1.2.3.3.2" xref="Thmthm2.p1.2.2.m2.1.2.3.cmml"><mo id="Thmthm2.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmthm2.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="Thmthm2.p1.2.2.m2.1.1" xref="Thmthm2.p1.2.2.m2.1.1.cmml">G</mi><mo id="Thmthm2.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmthm2.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm2.p1.2.2.m2.1b"><apply id="Thmthm2.p1.2.2.m2.1.2.cmml" xref="Thmthm2.p1.2.2.m2.1.2"><subset id="Thmthm2.p1.2.2.m2.1.2.1.cmml" xref="Thmthm2.p1.2.2.m2.1.2.1"></subset><ci id="Thmthm2.p1.2.2.m2.1.2.2.cmml" xref="Thmthm2.p1.2.2.m2.1.2.2">𝑊</ci><apply id="Thmthm2.p1.2.2.m2.1.2.3.cmml" xref="Thmthm2.p1.2.2.m2.1.2.3"><times id="Thmthm2.p1.2.2.m2.1.2.3.1.cmml" xref="Thmthm2.p1.2.2.m2.1.2.3.1"></times><ci id="Thmthm2.p1.2.2.m2.1.2.3.2.cmml" xref="Thmthm2.p1.2.2.m2.1.2.3.2">𝑉</ci><ci id="Thmthm2.p1.2.2.m2.1.1.cmml" xref="Thmthm2.p1.2.2.m2.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm2.p1.2.2.m2.1c">W\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="Thmthm2.p1.2.2.m2.1d">italic_W ⊆ italic_V ( italic_G )</annotation></semantics></math>, <math alttext="G" class="ltx_Math" display="inline" id="Thmthm2.p1.3.3.m3.1"><semantics id="Thmthm2.p1.3.3.m3.1a"><mi id="Thmthm2.p1.3.3.m3.1.1" xref="Thmthm2.p1.3.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm2.p1.3.3.m3.1b"><ci id="Thmthm2.p1.3.3.m3.1.1.cmml" xref="Thmthm2.p1.3.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm2.p1.3.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm2.p1.3.3.m3.1d">italic_G</annotation></semantics></math> has a <math alttext="W" class="ltx_Math" display="inline" id="Thmthm2.p1.4.4.m4.1"><semantics id="Thmthm2.p1.4.4.m4.1a"><mi id="Thmthm2.p1.4.4.m4.1.1" xref="Thmthm2.p1.4.4.m4.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="Thmthm2.p1.4.4.m4.1b"><ci id="Thmthm2.p1.4.4.m4.1.1.cmml" xref="Thmthm2.p1.4.4.m4.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm2.p1.4.4.m4.1c">W</annotation><annotation encoding="application/x-llamapun" id="Thmthm2.p1.4.4.m4.1d">italic_W</annotation></semantics></math>-balanced separation of order at most <math alttext="a" class="ltx_Math" display="inline" id="Thmthm2.p1.5.5.m5.1"><semantics id="Thmthm2.p1.5.5.m5.1a"><mi id="Thmthm2.p1.5.5.m5.1.1" xref="Thmthm2.p1.5.5.m5.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="Thmthm2.p1.5.5.m5.1b"><ci id="Thmthm2.p1.5.5.m5.1.1.cmml" xref="Thmthm2.p1.5.5.m5.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm2.p1.5.5.m5.1c">a</annotation><annotation encoding="application/x-llamapun" id="Thmthm2.p1.5.5.m5.1d">italic_a</annotation></semantics></math>. Then <math alttext="\operatorname{tw}(G)<4a" class="ltx_Math" display="inline" id="Thmthm2.p1.6.6.m6.2"><semantics id="Thmthm2.p1.6.6.m6.2a"><mrow id="Thmthm2.p1.6.6.m6.2.3" xref="Thmthm2.p1.6.6.m6.2.3.cmml"><mrow id="Thmthm2.p1.6.6.m6.2.3.2.2" xref="Thmthm2.p1.6.6.m6.2.3.2.1.cmml"><mi id="Thmthm2.p1.6.6.m6.1.1" xref="Thmthm2.p1.6.6.m6.1.1.cmml">tw</mi><mo id="Thmthm2.p1.6.6.m6.2.3.2.2a" xref="Thmthm2.p1.6.6.m6.2.3.2.1.cmml">⁡</mo><mrow id="Thmthm2.p1.6.6.m6.2.3.2.2.1" xref="Thmthm2.p1.6.6.m6.2.3.2.1.cmml"><mo id="Thmthm2.p1.6.6.m6.2.3.2.2.1.1" stretchy="false" xref="Thmthm2.p1.6.6.m6.2.3.2.1.cmml">(</mo><mi id="Thmthm2.p1.6.6.m6.2.2" xref="Thmthm2.p1.6.6.m6.2.2.cmml">G</mi><mo id="Thmthm2.p1.6.6.m6.2.3.2.2.1.2" stretchy="false" xref="Thmthm2.p1.6.6.m6.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="Thmthm2.p1.6.6.m6.2.3.1" xref="Thmthm2.p1.6.6.m6.2.3.1.cmml"><</mo><mrow id="Thmthm2.p1.6.6.m6.2.3.3" xref="Thmthm2.p1.6.6.m6.2.3.3.cmml"><mn id="Thmthm2.p1.6.6.m6.2.3.3.2" xref="Thmthm2.p1.6.6.m6.2.3.3.2.cmml">4</mn><mo id="Thmthm2.p1.6.6.m6.2.3.3.1" xref="Thmthm2.p1.6.6.m6.2.3.3.1.cmml">⁢</mo><mi id="Thmthm2.p1.6.6.m6.2.3.3.3" xref="Thmthm2.p1.6.6.m6.2.3.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm2.p1.6.6.m6.2b"><apply id="Thmthm2.p1.6.6.m6.2.3.cmml" xref="Thmthm2.p1.6.6.m6.2.3"><lt id="Thmthm2.p1.6.6.m6.2.3.1.cmml" xref="Thmthm2.p1.6.6.m6.2.3.1"></lt><apply id="Thmthm2.p1.6.6.m6.2.3.2.1.cmml" xref="Thmthm2.p1.6.6.m6.2.3.2.2"><ci id="Thmthm2.p1.6.6.m6.1.1.cmml" xref="Thmthm2.p1.6.6.m6.1.1">tw</ci><ci id="Thmthm2.p1.6.6.m6.2.2.cmml" xref="Thmthm2.p1.6.6.m6.2.2">𝐺</ci></apply><apply id="Thmthm2.p1.6.6.m6.2.3.3.cmml" xref="Thmthm2.p1.6.6.m6.2.3.3"><times id="Thmthm2.p1.6.6.m6.2.3.3.1.cmml" xref="Thmthm2.p1.6.6.m6.2.3.3.1"></times><cn id="Thmthm2.p1.6.6.m6.2.3.3.2.cmml" type="integer" xref="Thmthm2.p1.6.6.m6.2.3.3.2">4</cn><ci id="Thmthm2.p1.6.6.m6.2.3.3.3.cmml" xref="Thmthm2.p1.6.6.m6.2.3.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm2.p1.6.6.m6.2c">\operatorname{tw}(G)<4a</annotation><annotation encoding="application/x-llamapun" id="Thmthm2.p1.6.6.m6.2d">roman_tw ( italic_G ) < 4 italic_a</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.52">The proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm2" title="Theorem 2 (Robertson and Seymour [5]). ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2</span></a> is constructive and intuitive. Indeed, a tree decomposition of <math alttext="G" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mi id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><ci id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_G</annotation></semantics></math> can be constructed iteratively by an algorithm that maintains a separation <math alttext="(X,Y)" class="ltx_Math" display="inline" id="S1.p5.2.m2.2"><semantics id="S1.p5.2.m2.2a"><mrow id="S1.p5.2.m2.2.3.2" xref="S1.p5.2.m2.2.3.1.cmml"><mo id="S1.p5.2.m2.2.3.2.1" stretchy="false" xref="S1.p5.2.m2.2.3.1.cmml">(</mo><mi id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml">X</mi><mo id="S1.p5.2.m2.2.3.2.2" xref="S1.p5.2.m2.2.3.1.cmml">,</mo><mi id="S1.p5.2.m2.2.2" xref="S1.p5.2.m2.2.2.cmml">Y</mi><mo id="S1.p5.2.m2.2.3.2.3" stretchy="false" xref="S1.p5.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.2b"><interval closure="open" id="S1.p5.2.m2.2.3.1.cmml" xref="S1.p5.2.m2.2.3.2"><ci id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1">𝑋</ci><ci id="S1.p5.2.m2.2.2.cmml" xref="S1.p5.2.m2.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.2d">( italic_X , italic_Y )</annotation></semantics></math> of order at most <math alttext="3a" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mrow id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml"><mn id="S1.p5.3.m3.1.1.2" xref="S1.p5.3.m3.1.1.2.cmml">3</mn><mo id="S1.p5.3.m3.1.1.1" xref="S1.p5.3.m3.1.1.1.cmml">⁢</mo><mi id="S1.p5.3.m3.1.1.3" xref="S1.p5.3.m3.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><apply id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1"><times id="S1.p5.3.m3.1.1.1.cmml" xref="S1.p5.3.m3.1.1.1"></times><cn id="S1.p5.3.m3.1.1.2.cmml" type="integer" xref="S1.p5.3.m3.1.1.2">3</cn><ci id="S1.p5.3.m3.1.1.3.cmml" xref="S1.p5.3.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">3a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">3 italic_a</annotation></semantics></math>, and a tree decomposition <math alttext="\mathcal{T}:=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mrow id="S1.p5.4.m4.1b"><mi class="ltx_font_mathcaligraphic" id="S1.p5.4.m4.1.1">𝒯</mi><mo id="S1.p5.4.m4.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S1.p5.4.m4.1.3"><mo id="S1.p5.4.m4.1.3.1" stretchy="false">(</mo><msub id="S1.p5.4.m4.1.3.2"><mi id="S1.p5.4.m4.1.3.2.2">B</mi><mi id="S1.p5.4.m4.1.3.2.3">x</mi></msub><mo id="S1.p5.4.m4.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S1.p5.4.m4.1.3.4">x</mi><mo id="S1.p5.4.m4.1.3.5">∈</mo><mi id="S1.p5.4.m4.1.3.6">V</mi><mrow id="S1.p5.4.m4.1.3.7"><mo id="S1.p5.4.m4.1.3.7.1" stretchy="false">(</mo><mi id="S1.p5.4.m4.1.3.7.2">T</mi><mo id="S1.p5.4.m4.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S1.p5.4.m4.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">\mathcal{T}:=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> of <math alttext="G[Y]" class="ltx_Math" display="inline" id="S1.p5.5.m5.1"><semantics id="S1.p5.5.m5.1a"><mrow id="S1.p5.5.m5.1.2" xref="S1.p5.5.m5.1.2.cmml"><mi id="S1.p5.5.m5.1.2.2" xref="S1.p5.5.m5.1.2.2.cmml">G</mi><mo id="S1.p5.5.m5.1.2.1" xref="S1.p5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S1.p5.5.m5.1.2.3.2" xref="S1.p5.5.m5.1.2.3.1.cmml"><mo id="S1.p5.5.m5.1.2.3.2.1" stretchy="false" xref="S1.p5.5.m5.1.2.3.1.1.cmml">[</mo><mi id="S1.p5.5.m5.1.1" xref="S1.p5.5.m5.1.1.cmml">Y</mi><mo id="S1.p5.5.m5.1.2.3.2.2" stretchy="false" xref="S1.p5.5.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.5.m5.1b"><apply id="S1.p5.5.m5.1.2.cmml" xref="S1.p5.5.m5.1.2"><times id="S1.p5.5.m5.1.2.1.cmml" xref="S1.p5.5.m5.1.2.1"></times><ci id="S1.p5.5.m5.1.2.2.cmml" xref="S1.p5.5.m5.1.2.2">𝐺</ci><apply id="S1.p5.5.m5.1.2.3.1.cmml" xref="S1.p5.5.m5.1.2.3.2"><csymbol cd="latexml" id="S1.p5.5.m5.1.2.3.1.1.cmml" xref="S1.p5.5.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="S1.p5.5.m5.1.1.cmml" xref="S1.p5.5.m5.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.5.m5.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="S1.p5.5.m5.1d">italic_G [ italic_Y ]</annotation></semantics></math> of width less than <math alttext="4a" class="ltx_Math" display="inline" id="S1.p5.6.m6.1"><semantics id="S1.p5.6.m6.1a"><mrow id="S1.p5.6.m6.1.1" xref="S1.p5.6.m6.1.1.cmml"><mn id="S1.p5.6.m6.1.1.2" xref="S1.p5.6.m6.1.1.2.cmml">4</mn><mo id="S1.p5.6.m6.1.1.1" xref="S1.p5.6.m6.1.1.1.cmml">⁢</mo><mi id="S1.p5.6.m6.1.1.3" xref="S1.p5.6.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.6.m6.1b"><apply id="S1.p5.6.m6.1.1.cmml" xref="S1.p5.6.m6.1.1"><times id="S1.p5.6.m6.1.1.1.cmml" xref="S1.p5.6.m6.1.1.1"></times><cn id="S1.p5.6.m6.1.1.2.cmml" type="integer" xref="S1.p5.6.m6.1.1.2">4</cn><ci id="S1.p5.6.m6.1.1.3.cmml" xref="S1.p5.6.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.6.m6.1c">4a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.6.m6.1d">4 italic_a</annotation></semantics></math> in which some bag <math alttext="B_{x}" class="ltx_Math" display="inline" id="S1.p5.7.m7.1"><semantics id="S1.p5.7.m7.1a"><msub id="S1.p5.7.m7.1.1" xref="S1.p5.7.m7.1.1.cmml"><mi id="S1.p5.7.m7.1.1.2" xref="S1.p5.7.m7.1.1.2.cmml">B</mi><mi id="S1.p5.7.m7.1.1.3" xref="S1.p5.7.m7.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p5.7.m7.1b"><apply id="S1.p5.7.m7.1.1.cmml" xref="S1.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S1.p5.7.m7.1.1.1.cmml" xref="S1.p5.7.m7.1.1">subscript</csymbol><ci id="S1.p5.7.m7.1.1.2.cmml" xref="S1.p5.7.m7.1.1.2">𝐵</ci><ci id="S1.p5.7.m7.1.1.3.cmml" xref="S1.p5.7.m7.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.7.m7.1c">B_{x}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.7.m7.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> contains all vertices in <math alttext="W:=X\cap Y" class="ltx_Math" display="inline" id="S1.p5.8.m8.1"><semantics id="S1.p5.8.m8.1a"><mrow id="S1.p5.8.m8.1.1" xref="S1.p5.8.m8.1.1.cmml"><mi id="S1.p5.8.m8.1.1.2" xref="S1.p5.8.m8.1.1.2.cmml">W</mi><mo id="S1.p5.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p5.8.m8.1.1.1.cmml">:=</mo><mrow id="S1.p5.8.m8.1.1.3" xref="S1.p5.8.m8.1.1.3.cmml"><mi id="S1.p5.8.m8.1.1.3.2" xref="S1.p5.8.m8.1.1.3.2.cmml">X</mi><mo id="S1.p5.8.m8.1.1.3.1" xref="S1.p5.8.m8.1.1.3.1.cmml">∩</mo><mi id="S1.p5.8.m8.1.1.3.3" xref="S1.p5.8.m8.1.1.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.8.m8.1b"><apply id="S1.p5.8.m8.1.1.cmml" xref="S1.p5.8.m8.1.1"><csymbol cd="latexml" id="S1.p5.8.m8.1.1.1.cmml" xref="S1.p5.8.m8.1.1.1">assign</csymbol><ci id="S1.p5.8.m8.1.1.2.cmml" xref="S1.p5.8.m8.1.1.2">𝑊</ci><apply id="S1.p5.8.m8.1.1.3.cmml" xref="S1.p5.8.m8.1.1.3"><intersect id="S1.p5.8.m8.1.1.3.1.cmml" xref="S1.p5.8.m8.1.1.3.1"></intersect><ci id="S1.p5.8.m8.1.1.3.2.cmml" xref="S1.p5.8.m8.1.1.3.2">𝑋</ci><ci id="S1.p5.8.m8.1.1.3.3.cmml" xref="S1.p5.8.m8.1.1.3.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.8.m8.1c">W:=X\cap Y</annotation><annotation encoding="application/x-llamapun" id="S1.p5.8.m8.1d">italic_W := italic_X ∩ italic_Y</annotation></semantics></math>. To extend <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S1.p5.9.m9.1"><semantics id="S1.p5.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.9.m9.1.1" xref="S1.p5.9.m9.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S1.p5.9.m9.1b"><ci id="S1.p5.9.m9.1.1.cmml" xref="S1.p5.9.m9.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.9.m9.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.9.m9.1d">caligraphic_T</annotation></semantics></math>, the algorithm takes a <math alttext="W" class="ltx_Math" display="inline" id="S1.p5.10.m10.1"><semantics id="S1.p5.10.m10.1a"><mi id="S1.p5.10.m10.1.1" xref="S1.p5.10.m10.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.p5.10.m10.1b"><ci id="S1.p5.10.m10.1.1.cmml" xref="S1.p5.10.m10.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.10.m10.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.p5.10.m10.1d">italic_W</annotation></semantics></math>-balanced separation <math alttext="(A,B)" class="ltx_Math" display="inline" id="S1.p5.11.m11.2"><semantics id="S1.p5.11.m11.2a"><mrow id="S1.p5.11.m11.2.3.2" xref="S1.p5.11.m11.2.3.1.cmml"><mo id="S1.p5.11.m11.2.3.2.1" stretchy="false" xref="S1.p5.11.m11.2.3.1.cmml">(</mo><mi id="S1.p5.11.m11.1.1" xref="S1.p5.11.m11.1.1.cmml">A</mi><mo id="S1.p5.11.m11.2.3.2.2" xref="S1.p5.11.m11.2.3.1.cmml">,</mo><mi id="S1.p5.11.m11.2.2" xref="S1.p5.11.m11.2.2.cmml">B</mi><mo id="S1.p5.11.m11.2.3.2.3" stretchy="false" xref="S1.p5.11.m11.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.11.m11.2b"><interval closure="open" id="S1.p5.11.m11.2.3.1.cmml" xref="S1.p5.11.m11.2.3.2"><ci id="S1.p5.11.m11.1.1.cmml" xref="S1.p5.11.m11.1.1">𝐴</ci><ci id="S1.p5.11.m11.2.2.cmml" xref="S1.p5.11.m11.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.11.m11.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.11.m11.2d">( italic_A , italic_B )</annotation></semantics></math> of <math alttext="G[X]" class="ltx_Math" display="inline" id="S1.p5.12.m12.1"><semantics id="S1.p5.12.m12.1a"><mrow id="S1.p5.12.m12.1.2" xref="S1.p5.12.m12.1.2.cmml"><mi id="S1.p5.12.m12.1.2.2" xref="S1.p5.12.m12.1.2.2.cmml">G</mi><mo id="S1.p5.12.m12.1.2.1" xref="S1.p5.12.m12.1.2.1.cmml">⁢</mo><mrow id="S1.p5.12.m12.1.2.3.2" xref="S1.p5.12.m12.1.2.3.1.cmml"><mo id="S1.p5.12.m12.1.2.3.2.1" stretchy="false" xref="S1.p5.12.m12.1.2.3.1.1.cmml">[</mo><mi id="S1.p5.12.m12.1.1" xref="S1.p5.12.m12.1.1.cmml">X</mi><mo id="S1.p5.12.m12.1.2.3.2.2" stretchy="false" xref="S1.p5.12.m12.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.12.m12.1b"><apply id="S1.p5.12.m12.1.2.cmml" xref="S1.p5.12.m12.1.2"><times id="S1.p5.12.m12.1.2.1.cmml" xref="S1.p5.12.m12.1.2.1"></times><ci id="S1.p5.12.m12.1.2.2.cmml" xref="S1.p5.12.m12.1.2.2">𝐺</ci><apply id="S1.p5.12.m12.1.2.3.1.cmml" xref="S1.p5.12.m12.1.2.3.2"><csymbol cd="latexml" id="S1.p5.12.m12.1.2.3.1.1.cmml" xref="S1.p5.12.m12.1.2.3.2.1">delimited-[]</csymbol><ci id="S1.p5.12.m12.1.1.cmml" xref="S1.p5.12.m12.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.12.m12.1c">G[X]</annotation><annotation encoding="application/x-llamapun" id="S1.p5.12.m12.1d">italic_G [ italic_X ]</annotation></semantics></math> of order at most <math alttext="a" class="ltx_Math" display="inline" id="S1.p5.13.m13.1"><semantics id="S1.p5.13.m13.1a"><mi id="S1.p5.13.m13.1.1" xref="S1.p5.13.m13.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S1.p5.13.m13.1b"><ci id="S1.p5.13.m13.1.1.cmml" xref="S1.p5.13.m13.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.13.m13.1c">a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.13.m13.1d">italic_a</annotation></semantics></math> and creates a leaf <math alttext="x^{\prime}" class="ltx_Math" display="inline" id="S1.p5.14.m14.1"><semantics id="S1.p5.14.m14.1a"><msup id="S1.p5.14.m14.1.1" xref="S1.p5.14.m14.1.1.cmml"><mi id="S1.p5.14.m14.1.1.2" xref="S1.p5.14.m14.1.1.2.cmml">x</mi><mo id="S1.p5.14.m14.1.1.3" xref="S1.p5.14.m14.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.p5.14.m14.1b"><apply id="S1.p5.14.m14.1.1.cmml" xref="S1.p5.14.m14.1.1"><csymbol cd="ambiguous" id="S1.p5.14.m14.1.1.1.cmml" xref="S1.p5.14.m14.1.1">superscript</csymbol><ci id="S1.p5.14.m14.1.1.2.cmml" xref="S1.p5.14.m14.1.1.2">𝑥</ci><ci id="S1.p5.14.m14.1.1.3.cmml" xref="S1.p5.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.14.m14.1c">x^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.14.m14.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="T" class="ltx_Math" display="inline" id="S1.p5.15.m15.1"><semantics id="S1.p5.15.m15.1a"><mi id="S1.p5.15.m15.1.1" xref="S1.p5.15.m15.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.p5.15.m15.1b"><ci id="S1.p5.15.m15.1.1.cmml" xref="S1.p5.15.m15.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.15.m15.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.p5.15.m15.1d">italic_T</annotation></semantics></math> adjacent to <math alttext="x" class="ltx_Math" display="inline" id="S1.p5.16.m16.1"><semantics id="S1.p5.16.m16.1a"><mi id="S1.p5.16.m16.1.1" xref="S1.p5.16.m16.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.p5.16.m16.1b"><ci id="S1.p5.16.m16.1.1.cmml" xref="S1.p5.16.m16.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.16.m16.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.p5.16.m16.1d">italic_x</annotation></semantics></math> with <math alttext="B_{x^{\prime}}:=W\cup(A\cap B)" class="ltx_Math" display="inline" id="S1.p5.17.m17.1"><semantics id="S1.p5.17.m17.1a"><mrow id="S1.p5.17.m17.1.1" xref="S1.p5.17.m17.1.1.cmml"><msub id="S1.p5.17.m17.1.1.3" xref="S1.p5.17.m17.1.1.3.cmml"><mi id="S1.p5.17.m17.1.1.3.2" xref="S1.p5.17.m17.1.1.3.2.cmml">B</mi><msup id="S1.p5.17.m17.1.1.3.3" xref="S1.p5.17.m17.1.1.3.3.cmml"><mi id="S1.p5.17.m17.1.1.3.3.2" xref="S1.p5.17.m17.1.1.3.3.2.cmml">x</mi><mo id="S1.p5.17.m17.1.1.3.3.3" xref="S1.p5.17.m17.1.1.3.3.3.cmml">′</mo></msup></msub><mo id="S1.p5.17.m17.1.1.2" lspace="0.278em" rspace="0.278em" xref="S1.p5.17.m17.1.1.2.cmml">:=</mo><mrow id="S1.p5.17.m17.1.1.1" xref="S1.p5.17.m17.1.1.1.cmml"><mi id="S1.p5.17.m17.1.1.1.3" xref="S1.p5.17.m17.1.1.1.3.cmml">W</mi><mo id="S1.p5.17.m17.1.1.1.2" xref="S1.p5.17.m17.1.1.1.2.cmml">∪</mo><mrow id="S1.p5.17.m17.1.1.1.1.1" xref="S1.p5.17.m17.1.1.1.1.1.1.cmml"><mo id="S1.p5.17.m17.1.1.1.1.1.2" stretchy="false" xref="S1.p5.17.m17.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.17.m17.1.1.1.1.1.1" xref="S1.p5.17.m17.1.1.1.1.1.1.cmml"><mi id="S1.p5.17.m17.1.1.1.1.1.1.2" xref="S1.p5.17.m17.1.1.1.1.1.1.2.cmml">A</mi><mo id="S1.p5.17.m17.1.1.1.1.1.1.1" xref="S1.p5.17.m17.1.1.1.1.1.1.1.cmml">∩</mo><mi id="S1.p5.17.m17.1.1.1.1.1.1.3" xref="S1.p5.17.m17.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p5.17.m17.1.1.1.1.1.3" stretchy="false" xref="S1.p5.17.m17.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.17.m17.1b"><apply id="S1.p5.17.m17.1.1.cmml" xref="S1.p5.17.m17.1.1"><csymbol cd="latexml" id="S1.p5.17.m17.1.1.2.cmml" xref="S1.p5.17.m17.1.1.2">assign</csymbol><apply id="S1.p5.17.m17.1.1.3.cmml" xref="S1.p5.17.m17.1.1.3"><csymbol cd="ambiguous" id="S1.p5.17.m17.1.1.3.1.cmml" xref="S1.p5.17.m17.1.1.3">subscript</csymbol><ci id="S1.p5.17.m17.1.1.3.2.cmml" xref="S1.p5.17.m17.1.1.3.2">𝐵</ci><apply id="S1.p5.17.m17.1.1.3.3.cmml" xref="S1.p5.17.m17.1.1.3.3"><csymbol cd="ambiguous" id="S1.p5.17.m17.1.1.3.3.1.cmml" xref="S1.p5.17.m17.1.1.3.3">superscript</csymbol><ci id="S1.p5.17.m17.1.1.3.3.2.cmml" xref="S1.p5.17.m17.1.1.3.3.2">𝑥</ci><ci id="S1.p5.17.m17.1.1.3.3.3.cmml" xref="S1.p5.17.m17.1.1.3.3.3">′</ci></apply></apply><apply id="S1.p5.17.m17.1.1.1.cmml" xref="S1.p5.17.m17.1.1.1"><union id="S1.p5.17.m17.1.1.1.2.cmml" xref="S1.p5.17.m17.1.1.1.2"></union><ci id="S1.p5.17.m17.1.1.1.3.cmml" xref="S1.p5.17.m17.1.1.1.3">𝑊</ci><apply id="S1.p5.17.m17.1.1.1.1.1.1.cmml" xref="S1.p5.17.m17.1.1.1.1.1"><intersect id="S1.p5.17.m17.1.1.1.1.1.1.1.cmml" xref="S1.p5.17.m17.1.1.1.1.1.1.1"></intersect><ci id="S1.p5.17.m17.1.1.1.1.1.1.2.cmml" xref="S1.p5.17.m17.1.1.1.1.1.1.2">𝐴</ci><ci id="S1.p5.17.m17.1.1.1.1.1.1.3.cmml" xref="S1.p5.17.m17.1.1.1.1.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.17.m17.1c">B_{x^{\prime}}:=W\cup(A\cap B)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.17.m17.1d">italic_B start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT := italic_W ∪ ( italic_A ∩ italic_B )</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>A <math alttext="W" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mi id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><ci id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">W</annotation><annotation encoding="application/x-llamapun" id="footnote1.m1.1e">italic_W</annotation></semantics></math>-balanced separation <math alttext="(A,B)" class="ltx_Math" display="inline" id="footnote1.m2.2"><semantics id="footnote1.m2.2b"><mrow id="footnote1.m2.2.3.2" xref="footnote1.m2.2.3.1.cmml"><mo id="footnote1.m2.2.3.2.1" stretchy="false" xref="footnote1.m2.2.3.1.cmml">(</mo><mi id="footnote1.m2.1.1" xref="footnote1.m2.1.1.cmml">A</mi><mo id="footnote1.m2.2.3.2.2" xref="footnote1.m2.2.3.1.cmml">,</mo><mi id="footnote1.m2.2.2" xref="footnote1.m2.2.2.cmml">B</mi><mo id="footnote1.m2.2.3.2.3" stretchy="false" xref="footnote1.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m2.2c"><interval closure="open" id="footnote1.m2.2.3.1.cmml" xref="footnote1.m2.2.3.2"><ci id="footnote1.m2.1.1.cmml" xref="footnote1.m2.1.1">𝐴</ci><ci id="footnote1.m2.2.2.cmml" xref="footnote1.m2.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m2.2d">(A,B)</annotation><annotation encoding="application/x-llamapun" id="footnote1.m2.2e">( italic_A , italic_B )</annotation></semantics></math> of <math alttext="G[X]" class="ltx_Math" display="inline" id="footnote1.m3.1"><semantics id="footnote1.m3.1b"><mrow id="footnote1.m3.1.2" xref="footnote1.m3.1.2.cmml"><mi id="footnote1.m3.1.2.2" xref="footnote1.m3.1.2.2.cmml">G</mi><mo id="footnote1.m3.1.2.1" xref="footnote1.m3.1.2.1.cmml">⁢</mo><mrow id="footnote1.m3.1.2.3.2" xref="footnote1.m3.1.2.3.1.cmml"><mo id="footnote1.m3.1.2.3.2.1" stretchy="false" xref="footnote1.m3.1.2.3.1.1.cmml">[</mo><mi id="footnote1.m3.1.1" xref="footnote1.m3.1.1.cmml">X</mi><mo id="footnote1.m3.1.2.3.2.2" stretchy="false" xref="footnote1.m3.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m3.1c"><apply id="footnote1.m3.1.2.cmml" xref="footnote1.m3.1.2"><times id="footnote1.m3.1.2.1.cmml" xref="footnote1.m3.1.2.1"></times><ci id="footnote1.m3.1.2.2.cmml" xref="footnote1.m3.1.2.2">𝐺</ci><apply id="footnote1.m3.1.2.3.1.cmml" xref="footnote1.m3.1.2.3.2"><csymbol cd="latexml" id="footnote1.m3.1.2.3.1.1.cmml" xref="footnote1.m3.1.2.3.2.1">delimited-[]</csymbol><ci id="footnote1.m3.1.1.cmml" xref="footnote1.m3.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m3.1d">G[X]</annotation><annotation encoding="application/x-llamapun" id="footnote1.m3.1e">italic_G [ italic_X ]</annotation></semantics></math> can be obtained from a <math alttext="W" class="ltx_Math" display="inline" id="footnote1.m4.1"><semantics id="footnote1.m4.1b"><mi id="footnote1.m4.1.1" xref="footnote1.m4.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="footnote1.m4.1c"><ci id="footnote1.m4.1.1.cmml" xref="footnote1.m4.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m4.1d">W</annotation><annotation encoding="application/x-llamapun" id="footnote1.m4.1e">italic_W</annotation></semantics></math>-balanced separation <math alttext="(A^{\prime},B^{\prime})" class="ltx_Math" display="inline" id="footnote1.m5.2"><semantics id="footnote1.m5.2b"><mrow id="footnote1.m5.2.2.2" xref="footnote1.m5.2.2.3.cmml"><mo id="footnote1.m5.2.2.2.3" stretchy="false" xref="footnote1.m5.2.2.3.cmml">(</mo><msup id="footnote1.m5.1.1.1.1" xref="footnote1.m5.1.1.1.1.cmml"><mi id="footnote1.m5.1.1.1.1.2" xref="footnote1.m5.1.1.1.1.2.cmml">A</mi><mo id="footnote1.m5.1.1.1.1.3" xref="footnote1.m5.1.1.1.1.3.cmml">′</mo></msup><mo id="footnote1.m5.2.2.2.4" xref="footnote1.m5.2.2.3.cmml">,</mo><msup id="footnote1.m5.2.2.2.2" xref="footnote1.m5.2.2.2.2.cmml"><mi id="footnote1.m5.2.2.2.2.2" xref="footnote1.m5.2.2.2.2.2.cmml">B</mi><mo id="footnote1.m5.2.2.2.2.3" xref="footnote1.m5.2.2.2.2.3.cmml">′</mo></msup><mo id="footnote1.m5.2.2.2.5" stretchy="false" xref="footnote1.m5.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m5.2c"><interval closure="open" id="footnote1.m5.2.2.3.cmml" xref="footnote1.m5.2.2.2"><apply id="footnote1.m5.1.1.1.1.cmml" xref="footnote1.m5.1.1.1.1"><csymbol cd="ambiguous" id="footnote1.m5.1.1.1.1.1.cmml" xref="footnote1.m5.1.1.1.1">superscript</csymbol><ci id="footnote1.m5.1.1.1.1.2.cmml" xref="footnote1.m5.1.1.1.1.2">𝐴</ci><ci id="footnote1.m5.1.1.1.1.3.cmml" xref="footnote1.m5.1.1.1.1.3">′</ci></apply><apply id="footnote1.m5.2.2.2.2.cmml" xref="footnote1.m5.2.2.2.2"><csymbol cd="ambiguous" id="footnote1.m5.2.2.2.2.1.cmml" xref="footnote1.m5.2.2.2.2">superscript</csymbol><ci id="footnote1.m5.2.2.2.2.2.cmml" xref="footnote1.m5.2.2.2.2.2">𝐵</ci><ci id="footnote1.m5.2.2.2.2.3.cmml" xref="footnote1.m5.2.2.2.2.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m5.2d">(A^{\prime},B^{\prime})</annotation><annotation encoding="application/x-llamapun" id="footnote1.m5.2e">( italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="footnote1.m6.1"><semantics id="footnote1.m6.1b"><mi id="footnote1.m6.1.1" xref="footnote1.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="footnote1.m6.1c"><ci id="footnote1.m6.1.1.cmml" xref="footnote1.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m6.1d">G</annotation><annotation encoding="application/x-llamapun" id="footnote1.m6.1e">italic_G</annotation></semantics></math> by setting <math alttext="(A,B):=(A^{\prime}\cap X,B^{\prime}\cap X)" class="ltx_Math" display="inline" id="footnote1.m7.4"><semantics id="footnote1.m7.4b"><mrow id="footnote1.m7.4.4" xref="footnote1.m7.4.4.cmml"><mrow id="footnote1.m7.4.4.4.2" xref="footnote1.m7.4.4.4.1.cmml"><mo id="footnote1.m7.4.4.4.2.1" stretchy="false" xref="footnote1.m7.4.4.4.1.cmml">(</mo><mi id="footnote1.m7.1.1" xref="footnote1.m7.1.1.cmml">A</mi><mo id="footnote1.m7.4.4.4.2.2" xref="footnote1.m7.4.4.4.1.cmml">,</mo><mi id="footnote1.m7.2.2" xref="footnote1.m7.2.2.cmml">B</mi><mo id="footnote1.m7.4.4.4.2.3" rspace="0.278em" stretchy="false" xref="footnote1.m7.4.4.4.1.cmml">)</mo></mrow><mo id="footnote1.m7.4.4.3" rspace="0.278em" xref="footnote1.m7.4.4.3.cmml">:=</mo><mrow id="footnote1.m7.4.4.2.2" xref="footnote1.m7.4.4.2.3.cmml"><mo id="footnote1.m7.4.4.2.2.3" stretchy="false" xref="footnote1.m7.4.4.2.3.cmml">(</mo><mrow id="footnote1.m7.3.3.1.1.1" xref="footnote1.m7.3.3.1.1.1.cmml"><msup id="footnote1.m7.3.3.1.1.1.2" xref="footnote1.m7.3.3.1.1.1.2.cmml"><mi id="footnote1.m7.3.3.1.1.1.2.2" xref="footnote1.m7.3.3.1.1.1.2.2.cmml">A</mi><mo id="footnote1.m7.3.3.1.1.1.2.3" xref="footnote1.m7.3.3.1.1.1.2.3.cmml">′</mo></msup><mo id="footnote1.m7.3.3.1.1.1.1" xref="footnote1.m7.3.3.1.1.1.1.cmml">∩</mo><mi id="footnote1.m7.3.3.1.1.1.3" xref="footnote1.m7.3.3.1.1.1.3.cmml">X</mi></mrow><mo id="footnote1.m7.4.4.2.2.4" xref="footnote1.m7.4.4.2.3.cmml">,</mo><mrow id="footnote1.m7.4.4.2.2.2" xref="footnote1.m7.4.4.2.2.2.cmml"><msup id="footnote1.m7.4.4.2.2.2.2" xref="footnote1.m7.4.4.2.2.2.2.cmml"><mi id="footnote1.m7.4.4.2.2.2.2.2" xref="footnote1.m7.4.4.2.2.2.2.2.cmml">B</mi><mo id="footnote1.m7.4.4.2.2.2.2.3" xref="footnote1.m7.4.4.2.2.2.2.3.cmml">′</mo></msup><mo id="footnote1.m7.4.4.2.2.2.1" xref="footnote1.m7.4.4.2.2.2.1.cmml">∩</mo><mi id="footnote1.m7.4.4.2.2.2.3" xref="footnote1.m7.4.4.2.2.2.3.cmml">X</mi></mrow><mo id="footnote1.m7.4.4.2.2.5" stretchy="false" xref="footnote1.m7.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m7.4c"><apply id="footnote1.m7.4.4.cmml" xref="footnote1.m7.4.4"><csymbol cd="latexml" id="footnote1.m7.4.4.3.cmml" xref="footnote1.m7.4.4.3">assign</csymbol><interval closure="open" id="footnote1.m7.4.4.4.1.cmml" xref="footnote1.m7.4.4.4.2"><ci id="footnote1.m7.1.1.cmml" xref="footnote1.m7.1.1">𝐴</ci><ci id="footnote1.m7.2.2.cmml" xref="footnote1.m7.2.2">𝐵</ci></interval><interval closure="open" id="footnote1.m7.4.4.2.3.cmml" xref="footnote1.m7.4.4.2.2"><apply id="footnote1.m7.3.3.1.1.1.cmml" xref="footnote1.m7.3.3.1.1.1"><intersect id="footnote1.m7.3.3.1.1.1.1.cmml" xref="footnote1.m7.3.3.1.1.1.1"></intersect><apply id="footnote1.m7.3.3.1.1.1.2.cmml" xref="footnote1.m7.3.3.1.1.1.2"><csymbol cd="ambiguous" id="footnote1.m7.3.3.1.1.1.2.1.cmml" xref="footnote1.m7.3.3.1.1.1.2">superscript</csymbol><ci id="footnote1.m7.3.3.1.1.1.2.2.cmml" xref="footnote1.m7.3.3.1.1.1.2.2">𝐴</ci><ci id="footnote1.m7.3.3.1.1.1.2.3.cmml" xref="footnote1.m7.3.3.1.1.1.2.3">′</ci></apply><ci id="footnote1.m7.3.3.1.1.1.3.cmml" xref="footnote1.m7.3.3.1.1.1.3">𝑋</ci></apply><apply id="footnote1.m7.4.4.2.2.2.cmml" xref="footnote1.m7.4.4.2.2.2"><intersect id="footnote1.m7.4.4.2.2.2.1.cmml" xref="footnote1.m7.4.4.2.2.2.1"></intersect><apply id="footnote1.m7.4.4.2.2.2.2.cmml" xref="footnote1.m7.4.4.2.2.2.2"><csymbol cd="ambiguous" id="footnote1.m7.4.4.2.2.2.2.1.cmml" xref="footnote1.m7.4.4.2.2.2.2">superscript</csymbol><ci id="footnote1.m7.4.4.2.2.2.2.2.cmml" xref="footnote1.m7.4.4.2.2.2.2.2">𝐵</ci><ci id="footnote1.m7.4.4.2.2.2.2.3.cmml" xref="footnote1.m7.4.4.2.2.2.2.3">′</ci></apply><ci id="footnote1.m7.4.4.2.2.2.3.cmml" xref="footnote1.m7.4.4.2.2.2.3">𝑋</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m7.4d">(A,B):=(A^{\prime}\cap X,B^{\prime}\cap X)</annotation><annotation encoding="application/x-llamapun" id="footnote1.m7.4e">( italic_A , italic_B ) := ( italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ italic_X , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ italic_X )</annotation></semantics></math>.</span></span></span> Note that <math alttext="|B_{x^{\prime}}|\leq|W|+|A\cap B|\leq 4a" class="ltx_Math" display="inline" id="S1.p5.18.m18.3"><semantics id="S1.p5.18.m18.3a"><mrow id="S1.p5.18.m18.3.3" xref="S1.p5.18.m18.3.3.cmml"><mrow id="S1.p5.18.m18.2.2.1.1" xref="S1.p5.18.m18.2.2.1.2.cmml"><mo id="S1.p5.18.m18.2.2.1.1.2" stretchy="false" xref="S1.p5.18.m18.2.2.1.2.1.cmml">|</mo><msub id="S1.p5.18.m18.2.2.1.1.1" xref="S1.p5.18.m18.2.2.1.1.1.cmml"><mi id="S1.p5.18.m18.2.2.1.1.1.2" xref="S1.p5.18.m18.2.2.1.1.1.2.cmml">B</mi><msup id="S1.p5.18.m18.2.2.1.1.1.3" xref="S1.p5.18.m18.2.2.1.1.1.3.cmml"><mi id="S1.p5.18.m18.2.2.1.1.1.3.2" xref="S1.p5.18.m18.2.2.1.1.1.3.2.cmml">x</mi><mo id="S1.p5.18.m18.2.2.1.1.1.3.3" xref="S1.p5.18.m18.2.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S1.p5.18.m18.2.2.1.1.3" stretchy="false" xref="S1.p5.18.m18.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p5.18.m18.3.3.4" xref="S1.p5.18.m18.3.3.4.cmml">≤</mo><mrow id="S1.p5.18.m18.3.3.2" xref="S1.p5.18.m18.3.3.2.cmml"><mrow id="S1.p5.18.m18.3.3.2.3.2" xref="S1.p5.18.m18.3.3.2.3.1.cmml"><mo id="S1.p5.18.m18.3.3.2.3.2.1" stretchy="false" xref="S1.p5.18.m18.3.3.2.3.1.1.cmml">|</mo><mi id="S1.p5.18.m18.1.1" xref="S1.p5.18.m18.1.1.cmml">W</mi><mo id="S1.p5.18.m18.3.3.2.3.2.2" stretchy="false" xref="S1.p5.18.m18.3.3.2.3.1.1.cmml">|</mo></mrow><mo id="S1.p5.18.m18.3.3.2.2" xref="S1.p5.18.m18.3.3.2.2.cmml">+</mo><mrow id="S1.p5.18.m18.3.3.2.1.1" xref="S1.p5.18.m18.3.3.2.1.2.cmml"><mo id="S1.p5.18.m18.3.3.2.1.1.2" stretchy="false" xref="S1.p5.18.m18.3.3.2.1.2.1.cmml">|</mo><mrow id="S1.p5.18.m18.3.3.2.1.1.1" xref="S1.p5.18.m18.3.3.2.1.1.1.cmml"><mi id="S1.p5.18.m18.3.3.2.1.1.1.2" xref="S1.p5.18.m18.3.3.2.1.1.1.2.cmml">A</mi><mo id="S1.p5.18.m18.3.3.2.1.1.1.1" xref="S1.p5.18.m18.3.3.2.1.1.1.1.cmml">∩</mo><mi id="S1.p5.18.m18.3.3.2.1.1.1.3" xref="S1.p5.18.m18.3.3.2.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p5.18.m18.3.3.2.1.1.3" stretchy="false" xref="S1.p5.18.m18.3.3.2.1.2.1.cmml">|</mo></mrow></mrow><mo id="S1.p5.18.m18.3.3.5" xref="S1.p5.18.m18.3.3.5.cmml">≤</mo><mrow id="S1.p5.18.m18.3.3.6" xref="S1.p5.18.m18.3.3.6.cmml"><mn id="S1.p5.18.m18.3.3.6.2" xref="S1.p5.18.m18.3.3.6.2.cmml">4</mn><mo id="S1.p5.18.m18.3.3.6.1" xref="S1.p5.18.m18.3.3.6.1.cmml">⁢</mo><mi id="S1.p5.18.m18.3.3.6.3" xref="S1.p5.18.m18.3.3.6.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.18.m18.3b"><apply id="S1.p5.18.m18.3.3.cmml" xref="S1.p5.18.m18.3.3"><and id="S1.p5.18.m18.3.3a.cmml" xref="S1.p5.18.m18.3.3"></and><apply id="S1.p5.18.m18.3.3b.cmml" xref="S1.p5.18.m18.3.3"><leq id="S1.p5.18.m18.3.3.4.cmml" xref="S1.p5.18.m18.3.3.4"></leq><apply id="S1.p5.18.m18.2.2.1.2.cmml" xref="S1.p5.18.m18.2.2.1.1"><abs id="S1.p5.18.m18.2.2.1.2.1.cmml" xref="S1.p5.18.m18.2.2.1.1.2"></abs><apply id="S1.p5.18.m18.2.2.1.1.1.cmml" xref="S1.p5.18.m18.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.p5.18.m18.2.2.1.1.1.1.cmml" xref="S1.p5.18.m18.2.2.1.1.1">subscript</csymbol><ci id="S1.p5.18.m18.2.2.1.1.1.2.cmml" xref="S1.p5.18.m18.2.2.1.1.1.2">𝐵</ci><apply id="S1.p5.18.m18.2.2.1.1.1.3.cmml" xref="S1.p5.18.m18.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S1.p5.18.m18.2.2.1.1.1.3.1.cmml" xref="S1.p5.18.m18.2.2.1.1.1.3">superscript</csymbol><ci id="S1.p5.18.m18.2.2.1.1.1.3.2.cmml" xref="S1.p5.18.m18.2.2.1.1.1.3.2">𝑥</ci><ci id="S1.p5.18.m18.2.2.1.1.1.3.3.cmml" xref="S1.p5.18.m18.2.2.1.1.1.3.3">′</ci></apply></apply></apply><apply id="S1.p5.18.m18.3.3.2.cmml" xref="S1.p5.18.m18.3.3.2"><plus id="S1.p5.18.m18.3.3.2.2.cmml" xref="S1.p5.18.m18.3.3.2.2"></plus><apply id="S1.p5.18.m18.3.3.2.3.1.cmml" xref="S1.p5.18.m18.3.3.2.3.2"><abs id="S1.p5.18.m18.3.3.2.3.1.1.cmml" xref="S1.p5.18.m18.3.3.2.3.2.1"></abs><ci id="S1.p5.18.m18.1.1.cmml" xref="S1.p5.18.m18.1.1">𝑊</ci></apply><apply id="S1.p5.18.m18.3.3.2.1.2.cmml" xref="S1.p5.18.m18.3.3.2.1.1"><abs id="S1.p5.18.m18.3.3.2.1.2.1.cmml" xref="S1.p5.18.m18.3.3.2.1.1.2"></abs><apply id="S1.p5.18.m18.3.3.2.1.1.1.cmml" xref="S1.p5.18.m18.3.3.2.1.1.1"><intersect id="S1.p5.18.m18.3.3.2.1.1.1.1.cmml" xref="S1.p5.18.m18.3.3.2.1.1.1.1"></intersect><ci id="S1.p5.18.m18.3.3.2.1.1.1.2.cmml" xref="S1.p5.18.m18.3.3.2.1.1.1.2">𝐴</ci><ci id="S1.p5.18.m18.3.3.2.1.1.1.3.cmml" xref="S1.p5.18.m18.3.3.2.1.1.1.3">𝐵</ci></apply></apply></apply></apply><apply id="S1.p5.18.m18.3.3c.cmml" xref="S1.p5.18.m18.3.3"><leq id="S1.p5.18.m18.3.3.5.cmml" xref="S1.p5.18.m18.3.3.5"></leq><share href="https://arxiv.org/html/2503.17112v1#S1.p5.18.m18.3.3.2.cmml" id="S1.p5.18.m18.3.3d.cmml" xref="S1.p5.18.m18.3.3"></share><apply id="S1.p5.18.m18.3.3.6.cmml" xref="S1.p5.18.m18.3.3.6"><times id="S1.p5.18.m18.3.3.6.1.cmml" xref="S1.p5.18.m18.3.3.6.1"></times><cn id="S1.p5.18.m18.3.3.6.2.cmml" type="integer" xref="S1.p5.18.m18.3.3.6.2">4</cn><ci id="S1.p5.18.m18.3.3.6.3.cmml" xref="S1.p5.18.m18.3.3.6.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.18.m18.3c">|B_{x^{\prime}}|\leq|W|+|A\cap B|\leq 4a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.18.m18.3d">| italic_B start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | ≤ | italic_W | + | italic_A ∩ italic_B | ≤ 4 italic_a</annotation></semantics></math>, so the width of <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S1.p5.19.m19.1"><semantics id="S1.p5.19.m19.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.19.m19.1.1" xref="S1.p5.19.m19.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S1.p5.19.m19.1b"><ci id="S1.p5.19.m19.1.1.cmml" xref="S1.p5.19.m19.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.19.m19.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.19.m19.1d">caligraphic_T</annotation></semantics></math> is still less than <math alttext="4a" class="ltx_Math" display="inline" id="S1.p5.20.m20.1"><semantics id="S1.p5.20.m20.1a"><mrow id="S1.p5.20.m20.1.1" xref="S1.p5.20.m20.1.1.cmml"><mn id="S1.p5.20.m20.1.1.2" xref="S1.p5.20.m20.1.1.2.cmml">4</mn><mo id="S1.p5.20.m20.1.1.1" xref="S1.p5.20.m20.1.1.1.cmml">⁢</mo><mi id="S1.p5.20.m20.1.1.3" xref="S1.p5.20.m20.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.20.m20.1b"><apply id="S1.p5.20.m20.1.1.cmml" xref="S1.p5.20.m20.1.1"><times id="S1.p5.20.m20.1.1.1.cmml" xref="S1.p5.20.m20.1.1.1"></times><cn id="S1.p5.20.m20.1.1.2.cmml" type="integer" xref="S1.p5.20.m20.1.1.2">4</cn><ci id="S1.p5.20.m20.1.1.3.cmml" xref="S1.p5.20.m20.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.20.m20.1c">4a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.20.m20.1d">4 italic_a</annotation></semantics></math>. Let <math alttext="X_{A}:=A" class="ltx_Math" display="inline" id="S1.p5.21.m21.1"><semantics id="S1.p5.21.m21.1a"><mrow id="S1.p5.21.m21.1.1" xref="S1.p5.21.m21.1.1.cmml"><msub id="S1.p5.21.m21.1.1.2" xref="S1.p5.21.m21.1.1.2.cmml"><mi id="S1.p5.21.m21.1.1.2.2" xref="S1.p5.21.m21.1.1.2.2.cmml">X</mi><mi id="S1.p5.21.m21.1.1.2.3" xref="S1.p5.21.m21.1.1.2.3.cmml">A</mi></msub><mo id="S1.p5.21.m21.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p5.21.m21.1.1.1.cmml">:=</mo><mi id="S1.p5.21.m21.1.1.3" xref="S1.p5.21.m21.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.21.m21.1b"><apply id="S1.p5.21.m21.1.1.cmml" xref="S1.p5.21.m21.1.1"><csymbol cd="latexml" id="S1.p5.21.m21.1.1.1.cmml" xref="S1.p5.21.m21.1.1.1">assign</csymbol><apply id="S1.p5.21.m21.1.1.2.cmml" xref="S1.p5.21.m21.1.1.2"><csymbol cd="ambiguous" id="S1.p5.21.m21.1.1.2.1.cmml" xref="S1.p5.21.m21.1.1.2">subscript</csymbol><ci id="S1.p5.21.m21.1.1.2.2.cmml" xref="S1.p5.21.m21.1.1.2.2">𝑋</ci><ci id="S1.p5.21.m21.1.1.2.3.cmml" xref="S1.p5.21.m21.1.1.2.3">𝐴</ci></apply><ci id="S1.p5.21.m21.1.1.3.cmml" xref="S1.p5.21.m21.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.21.m21.1c">X_{A}:=A</annotation><annotation encoding="application/x-llamapun" id="S1.p5.21.m21.1d">italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT := italic_A</annotation></semantics></math>, <math alttext="Y_{A}:=Y\cup(A\cap B)" class="ltx_Math" display="inline" id="S1.p5.22.m22.1"><semantics id="S1.p5.22.m22.1a"><mrow id="S1.p5.22.m22.1.1" xref="S1.p5.22.m22.1.1.cmml"><msub id="S1.p5.22.m22.1.1.3" xref="S1.p5.22.m22.1.1.3.cmml"><mi id="S1.p5.22.m22.1.1.3.2" xref="S1.p5.22.m22.1.1.3.2.cmml">Y</mi><mi id="S1.p5.22.m22.1.1.3.3" xref="S1.p5.22.m22.1.1.3.3.cmml">A</mi></msub><mo id="S1.p5.22.m22.1.1.2" lspace="0.278em" rspace="0.278em" xref="S1.p5.22.m22.1.1.2.cmml">:=</mo><mrow id="S1.p5.22.m22.1.1.1" xref="S1.p5.22.m22.1.1.1.cmml"><mi id="S1.p5.22.m22.1.1.1.3" xref="S1.p5.22.m22.1.1.1.3.cmml">Y</mi><mo id="S1.p5.22.m22.1.1.1.2" xref="S1.p5.22.m22.1.1.1.2.cmml">∪</mo><mrow id="S1.p5.22.m22.1.1.1.1.1" xref="S1.p5.22.m22.1.1.1.1.1.1.cmml"><mo id="S1.p5.22.m22.1.1.1.1.1.2" stretchy="false" xref="S1.p5.22.m22.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.22.m22.1.1.1.1.1.1" xref="S1.p5.22.m22.1.1.1.1.1.1.cmml"><mi id="S1.p5.22.m22.1.1.1.1.1.1.2" xref="S1.p5.22.m22.1.1.1.1.1.1.2.cmml">A</mi><mo id="S1.p5.22.m22.1.1.1.1.1.1.1" xref="S1.p5.22.m22.1.1.1.1.1.1.1.cmml">∩</mo><mi id="S1.p5.22.m22.1.1.1.1.1.1.3" xref="S1.p5.22.m22.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p5.22.m22.1.1.1.1.1.3" stretchy="false" xref="S1.p5.22.m22.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.22.m22.1b"><apply id="S1.p5.22.m22.1.1.cmml" xref="S1.p5.22.m22.1.1"><csymbol cd="latexml" id="S1.p5.22.m22.1.1.2.cmml" xref="S1.p5.22.m22.1.1.2">assign</csymbol><apply id="S1.p5.22.m22.1.1.3.cmml" xref="S1.p5.22.m22.1.1.3"><csymbol cd="ambiguous" id="S1.p5.22.m22.1.1.3.1.cmml" xref="S1.p5.22.m22.1.1.3">subscript</csymbol><ci id="S1.p5.22.m22.1.1.3.2.cmml" xref="S1.p5.22.m22.1.1.3.2">𝑌</ci><ci id="S1.p5.22.m22.1.1.3.3.cmml" xref="S1.p5.22.m22.1.1.3.3">𝐴</ci></apply><apply id="S1.p5.22.m22.1.1.1.cmml" xref="S1.p5.22.m22.1.1.1"><union id="S1.p5.22.m22.1.1.1.2.cmml" xref="S1.p5.22.m22.1.1.1.2"></union><ci id="S1.p5.22.m22.1.1.1.3.cmml" xref="S1.p5.22.m22.1.1.1.3">𝑌</ci><apply id="S1.p5.22.m22.1.1.1.1.1.1.cmml" xref="S1.p5.22.m22.1.1.1.1.1"><intersect id="S1.p5.22.m22.1.1.1.1.1.1.1.cmml" xref="S1.p5.22.m22.1.1.1.1.1.1.1"></intersect><ci id="S1.p5.22.m22.1.1.1.1.1.1.2.cmml" xref="S1.p5.22.m22.1.1.1.1.1.1.2">𝐴</ci><ci id="S1.p5.22.m22.1.1.1.1.1.1.3.cmml" xref="S1.p5.22.m22.1.1.1.1.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.22.m22.1c">Y_{A}:=Y\cup(A\cap B)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.22.m22.1d">italic_Y start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT := italic_Y ∪ ( italic_A ∩ italic_B )</annotation></semantics></math>, <math alttext="G_{A}:=G[X_{A}\cup Y_{A}]=G[Y\cup A]" class="ltx_Math" display="inline" id="S1.p5.23.m23.2"><semantics id="S1.p5.23.m23.2a"><mrow id="S1.p5.23.m23.2.2" xref="S1.p5.23.m23.2.2.cmml"><msub id="S1.p5.23.m23.2.2.4" xref="S1.p5.23.m23.2.2.4.cmml"><mi id="S1.p5.23.m23.2.2.4.2" xref="S1.p5.23.m23.2.2.4.2.cmml">G</mi><mi id="S1.p5.23.m23.2.2.4.3" xref="S1.p5.23.m23.2.2.4.3.cmml">A</mi></msub><mo id="S1.p5.23.m23.2.2.5" lspace="0.278em" rspace="0.278em" xref="S1.p5.23.m23.2.2.5.cmml">:=</mo><mrow id="S1.p5.23.m23.1.1.1" xref="S1.p5.23.m23.1.1.1.cmml"><mi id="S1.p5.23.m23.1.1.1.3" xref="S1.p5.23.m23.1.1.1.3.cmml">G</mi><mo id="S1.p5.23.m23.1.1.1.2" xref="S1.p5.23.m23.1.1.1.2.cmml">⁢</mo><mrow id="S1.p5.23.m23.1.1.1.1.1" xref="S1.p5.23.m23.1.1.1.1.2.cmml"><mo id="S1.p5.23.m23.1.1.1.1.1.2" stretchy="false" xref="S1.p5.23.m23.1.1.1.1.2.1.cmml">[</mo><mrow id="S1.p5.23.m23.1.1.1.1.1.1" xref="S1.p5.23.m23.1.1.1.1.1.1.cmml"><msub id="S1.p5.23.m23.1.1.1.1.1.1.2" xref="S1.p5.23.m23.1.1.1.1.1.1.2.cmml"><mi id="S1.p5.23.m23.1.1.1.1.1.1.2.2" xref="S1.p5.23.m23.1.1.1.1.1.1.2.2.cmml">X</mi><mi id="S1.p5.23.m23.1.1.1.1.1.1.2.3" xref="S1.p5.23.m23.1.1.1.1.1.1.2.3.cmml">A</mi></msub><mo id="S1.p5.23.m23.1.1.1.1.1.1.1" xref="S1.p5.23.m23.1.1.1.1.1.1.1.cmml">∪</mo><msub id="S1.p5.23.m23.1.1.1.1.1.1.3" xref="S1.p5.23.m23.1.1.1.1.1.1.3.cmml"><mi id="S1.p5.23.m23.1.1.1.1.1.1.3.2" xref="S1.p5.23.m23.1.1.1.1.1.1.3.2.cmml">Y</mi><mi id="S1.p5.23.m23.1.1.1.1.1.1.3.3" xref="S1.p5.23.m23.1.1.1.1.1.1.3.3.cmml">A</mi></msub></mrow><mo id="S1.p5.23.m23.1.1.1.1.1.3" stretchy="false" xref="S1.p5.23.m23.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S1.p5.23.m23.2.2.6" xref="S1.p5.23.m23.2.2.6.cmml">=</mo><mrow id="S1.p5.23.m23.2.2.2" xref="S1.p5.23.m23.2.2.2.cmml"><mi id="S1.p5.23.m23.2.2.2.3" xref="S1.p5.23.m23.2.2.2.3.cmml">G</mi><mo id="S1.p5.23.m23.2.2.2.2" xref="S1.p5.23.m23.2.2.2.2.cmml">⁢</mo><mrow id="S1.p5.23.m23.2.2.2.1.1" xref="S1.p5.23.m23.2.2.2.1.2.cmml"><mo id="S1.p5.23.m23.2.2.2.1.1.2" stretchy="false" xref="S1.p5.23.m23.2.2.2.1.2.1.cmml">[</mo><mrow id="S1.p5.23.m23.2.2.2.1.1.1" xref="S1.p5.23.m23.2.2.2.1.1.1.cmml"><mi id="S1.p5.23.m23.2.2.2.1.1.1.2" xref="S1.p5.23.m23.2.2.2.1.1.1.2.cmml">Y</mi><mo id="S1.p5.23.m23.2.2.2.1.1.1.1" xref="S1.p5.23.m23.2.2.2.1.1.1.1.cmml">∪</mo><mi id="S1.p5.23.m23.2.2.2.1.1.1.3" xref="S1.p5.23.m23.2.2.2.1.1.1.3.cmml">A</mi></mrow><mo id="S1.p5.23.m23.2.2.2.1.1.3" stretchy="false" xref="S1.p5.23.m23.2.2.2.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.23.m23.2b"><apply id="S1.p5.23.m23.2.2.cmml" xref="S1.p5.23.m23.2.2"><and id="S1.p5.23.m23.2.2a.cmml" xref="S1.p5.23.m23.2.2"></and><apply id="S1.p5.23.m23.2.2b.cmml" xref="S1.p5.23.m23.2.2"><csymbol cd="latexml" id="S1.p5.23.m23.2.2.5.cmml" xref="S1.p5.23.m23.2.2.5">assign</csymbol><apply id="S1.p5.23.m23.2.2.4.cmml" xref="S1.p5.23.m23.2.2.4"><csymbol cd="ambiguous" id="S1.p5.23.m23.2.2.4.1.cmml" xref="S1.p5.23.m23.2.2.4">subscript</csymbol><ci id="S1.p5.23.m23.2.2.4.2.cmml" xref="S1.p5.23.m23.2.2.4.2">𝐺</ci><ci id="S1.p5.23.m23.2.2.4.3.cmml" xref="S1.p5.23.m23.2.2.4.3">𝐴</ci></apply><apply id="S1.p5.23.m23.1.1.1.cmml" xref="S1.p5.23.m23.1.1.1"><times id="S1.p5.23.m23.1.1.1.2.cmml" xref="S1.p5.23.m23.1.1.1.2"></times><ci id="S1.p5.23.m23.1.1.1.3.cmml" xref="S1.p5.23.m23.1.1.1.3">𝐺</ci><apply id="S1.p5.23.m23.1.1.1.1.2.cmml" xref="S1.p5.23.m23.1.1.1.1.1"><csymbol cd="latexml" id="S1.p5.23.m23.1.1.1.1.2.1.cmml" xref="S1.p5.23.m23.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S1.p5.23.m23.1.1.1.1.1.1.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1"><union id="S1.p5.23.m23.1.1.1.1.1.1.1.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.1"></union><apply id="S1.p5.23.m23.1.1.1.1.1.1.2.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.p5.23.m23.1.1.1.1.1.1.2.1.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.2">subscript</csymbol><ci id="S1.p5.23.m23.1.1.1.1.1.1.2.2.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.2.2">𝑋</ci><ci id="S1.p5.23.m23.1.1.1.1.1.1.2.3.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.2.3">𝐴</ci></apply><apply id="S1.p5.23.m23.1.1.1.1.1.1.3.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.p5.23.m23.1.1.1.1.1.1.3.1.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.3">subscript</csymbol><ci id="S1.p5.23.m23.1.1.1.1.1.1.3.2.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.3.2">𝑌</ci><ci id="S1.p5.23.m23.1.1.1.1.1.1.3.3.cmml" xref="S1.p5.23.m23.1.1.1.1.1.1.3.3">𝐴</ci></apply></apply></apply></apply></apply><apply id="S1.p5.23.m23.2.2c.cmml" xref="S1.p5.23.m23.2.2"><eq id="S1.p5.23.m23.2.2.6.cmml" xref="S1.p5.23.m23.2.2.6"></eq><share href="https://arxiv.org/html/2503.17112v1#S1.p5.23.m23.1.1.1.cmml" id="S1.p5.23.m23.2.2d.cmml" xref="S1.p5.23.m23.2.2"></share><apply id="S1.p5.23.m23.2.2.2.cmml" xref="S1.p5.23.m23.2.2.2"><times id="S1.p5.23.m23.2.2.2.2.cmml" xref="S1.p5.23.m23.2.2.2.2"></times><ci id="S1.p5.23.m23.2.2.2.3.cmml" xref="S1.p5.23.m23.2.2.2.3">𝐺</ci><apply id="S1.p5.23.m23.2.2.2.1.2.cmml" xref="S1.p5.23.m23.2.2.2.1.1"><csymbol cd="latexml" id="S1.p5.23.m23.2.2.2.1.2.1.cmml" xref="S1.p5.23.m23.2.2.2.1.1.2">delimited-[]</csymbol><apply id="S1.p5.23.m23.2.2.2.1.1.1.cmml" xref="S1.p5.23.m23.2.2.2.1.1.1"><union id="S1.p5.23.m23.2.2.2.1.1.1.1.cmml" xref="S1.p5.23.m23.2.2.2.1.1.1.1"></union><ci id="S1.p5.23.m23.2.2.2.1.1.1.2.cmml" xref="S1.p5.23.m23.2.2.2.1.1.1.2">𝑌</ci><ci id="S1.p5.23.m23.2.2.2.1.1.1.3.cmml" xref="S1.p5.23.m23.2.2.2.1.1.1.3">𝐴</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.23.m23.2c">G_{A}:=G[X_{A}\cup Y_{A}]=G[Y\cup A]</annotation><annotation encoding="application/x-llamapun" id="S1.p5.23.m23.2d">italic_G start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT := italic_G [ italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∪ italic_Y start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ] = italic_G [ italic_Y ∪ italic_A ]</annotation></semantics></math> and <math alttext="W_{A}:=X_{A}\cap Y_{A}" class="ltx_Math" display="inline" id="S1.p5.24.m24.1"><semantics id="S1.p5.24.m24.1a"><mrow id="S1.p5.24.m24.1.1" xref="S1.p5.24.m24.1.1.cmml"><msub id="S1.p5.24.m24.1.1.2" xref="S1.p5.24.m24.1.1.2.cmml"><mi id="S1.p5.24.m24.1.1.2.2" xref="S1.p5.24.m24.1.1.2.2.cmml">W</mi><mi id="S1.p5.24.m24.1.1.2.3" xref="S1.p5.24.m24.1.1.2.3.cmml">A</mi></msub><mo id="S1.p5.24.m24.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p5.24.m24.1.1.1.cmml">:=</mo><mrow id="S1.p5.24.m24.1.1.3" xref="S1.p5.24.m24.1.1.3.cmml"><msub id="S1.p5.24.m24.1.1.3.2" xref="S1.p5.24.m24.1.1.3.2.cmml"><mi id="S1.p5.24.m24.1.1.3.2.2" xref="S1.p5.24.m24.1.1.3.2.2.cmml">X</mi><mi id="S1.p5.24.m24.1.1.3.2.3" xref="S1.p5.24.m24.1.1.3.2.3.cmml">A</mi></msub><mo id="S1.p5.24.m24.1.1.3.1" xref="S1.p5.24.m24.1.1.3.1.cmml">∩</mo><msub id="S1.p5.24.m24.1.1.3.3" xref="S1.p5.24.m24.1.1.3.3.cmml"><mi id="S1.p5.24.m24.1.1.3.3.2" xref="S1.p5.24.m24.1.1.3.3.2.cmml">Y</mi><mi id="S1.p5.24.m24.1.1.3.3.3" xref="S1.p5.24.m24.1.1.3.3.3.cmml">A</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.24.m24.1b"><apply id="S1.p5.24.m24.1.1.cmml" xref="S1.p5.24.m24.1.1"><csymbol cd="latexml" id="S1.p5.24.m24.1.1.1.cmml" xref="S1.p5.24.m24.1.1.1">assign</csymbol><apply id="S1.p5.24.m24.1.1.2.cmml" xref="S1.p5.24.m24.1.1.2"><csymbol cd="ambiguous" id="S1.p5.24.m24.1.1.2.1.cmml" xref="S1.p5.24.m24.1.1.2">subscript</csymbol><ci id="S1.p5.24.m24.1.1.2.2.cmml" xref="S1.p5.24.m24.1.1.2.2">𝑊</ci><ci id="S1.p5.24.m24.1.1.2.3.cmml" xref="S1.p5.24.m24.1.1.2.3">𝐴</ci></apply><apply id="S1.p5.24.m24.1.1.3.cmml" xref="S1.p5.24.m24.1.1.3"><intersect id="S1.p5.24.m24.1.1.3.1.cmml" xref="S1.p5.24.m24.1.1.3.1"></intersect><apply id="S1.p5.24.m24.1.1.3.2.cmml" xref="S1.p5.24.m24.1.1.3.2"><csymbol cd="ambiguous" id="S1.p5.24.m24.1.1.3.2.1.cmml" xref="S1.p5.24.m24.1.1.3.2">subscript</csymbol><ci id="S1.p5.24.m24.1.1.3.2.2.cmml" xref="S1.p5.24.m24.1.1.3.2.2">𝑋</ci><ci id="S1.p5.24.m24.1.1.3.2.3.cmml" xref="S1.p5.24.m24.1.1.3.2.3">𝐴</ci></apply><apply id="S1.p5.24.m24.1.1.3.3.cmml" xref="S1.p5.24.m24.1.1.3.3"><csymbol cd="ambiguous" id="S1.p5.24.m24.1.1.3.3.1.cmml" xref="S1.p5.24.m24.1.1.3.3">subscript</csymbol><ci id="S1.p5.24.m24.1.1.3.3.2.cmml" xref="S1.p5.24.m24.1.1.3.3.2">𝑌</ci><ci id="S1.p5.24.m24.1.1.3.3.3.cmml" xref="S1.p5.24.m24.1.1.3.3.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.24.m24.1c">W_{A}:=X_{A}\cap Y_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.24.m24.1d">italic_W start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT := italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∩ italic_Y start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S1.p5.25.m25.1"><semantics id="S1.p5.25.m25.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.25.m25.1.1" xref="S1.p5.25.m25.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S1.p5.25.m25.1b"><ci id="S1.p5.25.m25.1.1.cmml" xref="S1.p5.25.m25.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.25.m25.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.25.m25.1d">caligraphic_T</annotation></semantics></math> is a tree decomposition <math alttext="G[Y_{A}]" class="ltx_Math" display="inline" id="S1.p5.26.m26.1"><semantics id="S1.p5.26.m26.1a"><mrow id="S1.p5.26.m26.1.1" xref="S1.p5.26.m26.1.1.cmml"><mi id="S1.p5.26.m26.1.1.3" xref="S1.p5.26.m26.1.1.3.cmml">G</mi><mo id="S1.p5.26.m26.1.1.2" xref="S1.p5.26.m26.1.1.2.cmml">⁢</mo><mrow id="S1.p5.26.m26.1.1.1.1" xref="S1.p5.26.m26.1.1.1.2.cmml"><mo id="S1.p5.26.m26.1.1.1.1.2" stretchy="false" xref="S1.p5.26.m26.1.1.1.2.1.cmml">[</mo><msub id="S1.p5.26.m26.1.1.1.1.1" xref="S1.p5.26.m26.1.1.1.1.1.cmml"><mi id="S1.p5.26.m26.1.1.1.1.1.2" xref="S1.p5.26.m26.1.1.1.1.1.2.cmml">Y</mi><mi id="S1.p5.26.m26.1.1.1.1.1.3" xref="S1.p5.26.m26.1.1.1.1.1.3.cmml">A</mi></msub><mo id="S1.p5.26.m26.1.1.1.1.3" stretchy="false" xref="S1.p5.26.m26.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.26.m26.1b"><apply id="S1.p5.26.m26.1.1.cmml" xref="S1.p5.26.m26.1.1"><times id="S1.p5.26.m26.1.1.2.cmml" xref="S1.p5.26.m26.1.1.2"></times><ci id="S1.p5.26.m26.1.1.3.cmml" xref="S1.p5.26.m26.1.1.3">𝐺</ci><apply id="S1.p5.26.m26.1.1.1.2.cmml" xref="S1.p5.26.m26.1.1.1.1"><csymbol cd="latexml" id="S1.p5.26.m26.1.1.1.2.1.cmml" xref="S1.p5.26.m26.1.1.1.1.2">delimited-[]</csymbol><apply id="S1.p5.26.m26.1.1.1.1.1.cmml" xref="S1.p5.26.m26.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p5.26.m26.1.1.1.1.1.1.cmml" xref="S1.p5.26.m26.1.1.1.1.1">subscript</csymbol><ci id="S1.p5.26.m26.1.1.1.1.1.2.cmml" xref="S1.p5.26.m26.1.1.1.1.1.2">𝑌</ci><ci id="S1.p5.26.m26.1.1.1.1.1.3.cmml" xref="S1.p5.26.m26.1.1.1.1.1.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.26.m26.1c">G[Y_{A}]</annotation><annotation encoding="application/x-llamapun" id="S1.p5.26.m26.1d">italic_G [ italic_Y start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]</annotation></semantics></math> of width less than <math alttext="4a" class="ltx_Math" display="inline" id="S1.p5.27.m27.1"><semantics id="S1.p5.27.m27.1a"><mrow id="S1.p5.27.m27.1.1" xref="S1.p5.27.m27.1.1.cmml"><mn id="S1.p5.27.m27.1.1.2" xref="S1.p5.27.m27.1.1.2.cmml">4</mn><mo id="S1.p5.27.m27.1.1.1" xref="S1.p5.27.m27.1.1.1.cmml">⁢</mo><mi id="S1.p5.27.m27.1.1.3" xref="S1.p5.27.m27.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.27.m27.1b"><apply id="S1.p5.27.m27.1.1.cmml" xref="S1.p5.27.m27.1.1"><times id="S1.p5.27.m27.1.1.1.cmml" xref="S1.p5.27.m27.1.1.1"></times><cn id="S1.p5.27.m27.1.1.2.cmml" type="integer" xref="S1.p5.27.m27.1.1.2">4</cn><ci id="S1.p5.27.m27.1.1.3.cmml" xref="S1.p5.27.m27.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.27.m27.1c">4a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.27.m27.1d">4 italic_a</annotation></semantics></math> and <math alttext="(X_{A},Y_{A})" class="ltx_Math" display="inline" id="S1.p5.28.m28.2"><semantics id="S1.p5.28.m28.2a"><mrow id="S1.p5.28.m28.2.2.2" xref="S1.p5.28.m28.2.2.3.cmml"><mo id="S1.p5.28.m28.2.2.2.3" stretchy="false" xref="S1.p5.28.m28.2.2.3.cmml">(</mo><msub id="S1.p5.28.m28.1.1.1.1" xref="S1.p5.28.m28.1.1.1.1.cmml"><mi id="S1.p5.28.m28.1.1.1.1.2" xref="S1.p5.28.m28.1.1.1.1.2.cmml">X</mi><mi id="S1.p5.28.m28.1.1.1.1.3" xref="S1.p5.28.m28.1.1.1.1.3.cmml">A</mi></msub><mo id="S1.p5.28.m28.2.2.2.4" xref="S1.p5.28.m28.2.2.3.cmml">,</mo><msub id="S1.p5.28.m28.2.2.2.2" xref="S1.p5.28.m28.2.2.2.2.cmml"><mi id="S1.p5.28.m28.2.2.2.2.2" xref="S1.p5.28.m28.2.2.2.2.2.cmml">Y</mi><mi id="S1.p5.28.m28.2.2.2.2.3" xref="S1.p5.28.m28.2.2.2.2.3.cmml">A</mi></msub><mo id="S1.p5.28.m28.2.2.2.5" stretchy="false" xref="S1.p5.28.m28.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.28.m28.2b"><interval closure="open" id="S1.p5.28.m28.2.2.3.cmml" xref="S1.p5.28.m28.2.2.2"><apply id="S1.p5.28.m28.1.1.1.1.cmml" xref="S1.p5.28.m28.1.1.1.1"><csymbol cd="ambiguous" id="S1.p5.28.m28.1.1.1.1.1.cmml" xref="S1.p5.28.m28.1.1.1.1">subscript</csymbol><ci id="S1.p5.28.m28.1.1.1.1.2.cmml" xref="S1.p5.28.m28.1.1.1.1.2">𝑋</ci><ci id="S1.p5.28.m28.1.1.1.1.3.cmml" xref="S1.p5.28.m28.1.1.1.1.3">𝐴</ci></apply><apply id="S1.p5.28.m28.2.2.2.2.cmml" xref="S1.p5.28.m28.2.2.2.2"><csymbol cd="ambiguous" id="S1.p5.28.m28.2.2.2.2.1.cmml" xref="S1.p5.28.m28.2.2.2.2">subscript</csymbol><ci id="S1.p5.28.m28.2.2.2.2.2.cmml" xref="S1.p5.28.m28.2.2.2.2.2">𝑌</ci><ci id="S1.p5.28.m28.2.2.2.2.3.cmml" xref="S1.p5.28.m28.2.2.2.2.3">𝐴</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.28.m28.2c">(X_{A},Y_{A})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.28.m28.2d">( italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT )</annotation></semantics></math> is a separation of <math alttext="G_{A}" class="ltx_Math" display="inline" id="S1.p5.29.m29.1"><semantics id="S1.p5.29.m29.1a"><msub id="S1.p5.29.m29.1.1" xref="S1.p5.29.m29.1.1.cmml"><mi id="S1.p5.29.m29.1.1.2" xref="S1.p5.29.m29.1.1.2.cmml">G</mi><mi id="S1.p5.29.m29.1.1.3" xref="S1.p5.29.m29.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p5.29.m29.1b"><apply id="S1.p5.29.m29.1.1.cmml" xref="S1.p5.29.m29.1.1"><csymbol cd="ambiguous" id="S1.p5.29.m29.1.1.1.cmml" xref="S1.p5.29.m29.1.1">subscript</csymbol><ci id="S1.p5.29.m29.1.1.2.cmml" xref="S1.p5.29.m29.1.1.2">𝐺</ci><ci id="S1.p5.29.m29.1.1.3.cmml" xref="S1.p5.29.m29.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.29.m29.1c">G_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.29.m29.1d">italic_G start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> of order <math alttext="|W_{A}|\leq|W\setminus B|+|A\cap B|\leq\tfrac{2}{3}|W|+a\leq 3a" class="ltx_Math" display="inline" id="S1.p5.30.m30.4"><semantics id="S1.p5.30.m30.4a"><mrow id="S1.p5.30.m30.4.4" xref="S1.p5.30.m30.4.4.cmml"><mrow id="S1.p5.30.m30.2.2.1.1" xref="S1.p5.30.m30.2.2.1.2.cmml"><mo id="S1.p5.30.m30.2.2.1.1.2" stretchy="false" xref="S1.p5.30.m30.2.2.1.2.1.cmml">|</mo><msub id="S1.p5.30.m30.2.2.1.1.1" xref="S1.p5.30.m30.2.2.1.1.1.cmml"><mi id="S1.p5.30.m30.2.2.1.1.1.2" xref="S1.p5.30.m30.2.2.1.1.1.2.cmml">W</mi><mi id="S1.p5.30.m30.2.2.1.1.1.3" xref="S1.p5.30.m30.2.2.1.1.1.3.cmml">A</mi></msub><mo id="S1.p5.30.m30.2.2.1.1.3" stretchy="false" xref="S1.p5.30.m30.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p5.30.m30.4.4.5" xref="S1.p5.30.m30.4.4.5.cmml">≤</mo><mrow id="S1.p5.30.m30.4.4.3" xref="S1.p5.30.m30.4.4.3.cmml"><mrow id="S1.p5.30.m30.3.3.2.1.1" xref="S1.p5.30.m30.3.3.2.1.2.cmml"><mo id="S1.p5.30.m30.3.3.2.1.1.2" stretchy="false" xref="S1.p5.30.m30.3.3.2.1.2.1.cmml">|</mo><mrow id="S1.p5.30.m30.3.3.2.1.1.1" xref="S1.p5.30.m30.3.3.2.1.1.1.cmml"><mi id="S1.p5.30.m30.3.3.2.1.1.1.2" xref="S1.p5.30.m30.3.3.2.1.1.1.2.cmml">W</mi><mo id="S1.p5.30.m30.3.3.2.1.1.1.1" xref="S1.p5.30.m30.3.3.2.1.1.1.1.cmml">∖</mo><mi id="S1.p5.30.m30.3.3.2.1.1.1.3" xref="S1.p5.30.m30.3.3.2.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p5.30.m30.3.3.2.1.1.3" stretchy="false" xref="S1.p5.30.m30.3.3.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p5.30.m30.4.4.3.3" xref="S1.p5.30.m30.4.4.3.3.cmml">+</mo><mrow id="S1.p5.30.m30.4.4.3.2.1" xref="S1.p5.30.m30.4.4.3.2.2.cmml"><mo id="S1.p5.30.m30.4.4.3.2.1.2" stretchy="false" xref="S1.p5.30.m30.4.4.3.2.2.1.cmml">|</mo><mrow id="S1.p5.30.m30.4.4.3.2.1.1" xref="S1.p5.30.m30.4.4.3.2.1.1.cmml"><mi id="S1.p5.30.m30.4.4.3.2.1.1.2" xref="S1.p5.30.m30.4.4.3.2.1.1.2.cmml">A</mi><mo id="S1.p5.30.m30.4.4.3.2.1.1.1" xref="S1.p5.30.m30.4.4.3.2.1.1.1.cmml">∩</mo><mi id="S1.p5.30.m30.4.4.3.2.1.1.3" xref="S1.p5.30.m30.4.4.3.2.1.1.3.cmml">B</mi></mrow><mo id="S1.p5.30.m30.4.4.3.2.1.3" stretchy="false" xref="S1.p5.30.m30.4.4.3.2.2.1.cmml">|</mo></mrow></mrow><mo id="S1.p5.30.m30.4.4.6" xref="S1.p5.30.m30.4.4.6.cmml">≤</mo><mrow id="S1.p5.30.m30.4.4.7" xref="S1.p5.30.m30.4.4.7.cmml"><mrow id="S1.p5.30.m30.4.4.7.2" xref="S1.p5.30.m30.4.4.7.2.cmml"><mfrac id="S1.p5.30.m30.4.4.7.2.2" xref="S1.p5.30.m30.4.4.7.2.2.cmml"><mn id="S1.p5.30.m30.4.4.7.2.2.2" xref="S1.p5.30.m30.4.4.7.2.2.2.cmml">2</mn><mn id="S1.p5.30.m30.4.4.7.2.2.3" xref="S1.p5.30.m30.4.4.7.2.2.3.cmml">3</mn></mfrac><mo id="S1.p5.30.m30.4.4.7.2.1" xref="S1.p5.30.m30.4.4.7.2.1.cmml">⁢</mo><mrow id="S1.p5.30.m30.4.4.7.2.3.2" xref="S1.p5.30.m30.4.4.7.2.3.1.cmml"><mo id="S1.p5.30.m30.4.4.7.2.3.2.1" stretchy="false" xref="S1.p5.30.m30.4.4.7.2.3.1.1.cmml">|</mo><mi id="S1.p5.30.m30.1.1" xref="S1.p5.30.m30.1.1.cmml">W</mi><mo id="S1.p5.30.m30.4.4.7.2.3.2.2" stretchy="false" xref="S1.p5.30.m30.4.4.7.2.3.1.1.cmml">|</mo></mrow></mrow><mo id="S1.p5.30.m30.4.4.7.1" xref="S1.p5.30.m30.4.4.7.1.cmml">+</mo><mi id="S1.p5.30.m30.4.4.7.3" xref="S1.p5.30.m30.4.4.7.3.cmml">a</mi></mrow><mo id="S1.p5.30.m30.4.4.8" xref="S1.p5.30.m30.4.4.8.cmml">≤</mo><mrow id="S1.p5.30.m30.4.4.9" xref="S1.p5.30.m30.4.4.9.cmml"><mn id="S1.p5.30.m30.4.4.9.2" xref="S1.p5.30.m30.4.4.9.2.cmml">3</mn><mo id="S1.p5.30.m30.4.4.9.1" xref="S1.p5.30.m30.4.4.9.1.cmml">⁢</mo><mi id="S1.p5.30.m30.4.4.9.3" xref="S1.p5.30.m30.4.4.9.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.30.m30.4b"><apply id="S1.p5.30.m30.4.4.cmml" xref="S1.p5.30.m30.4.4"><and id="S1.p5.30.m30.4.4a.cmml" xref="S1.p5.30.m30.4.4"></and><apply id="S1.p5.30.m30.4.4b.cmml" xref="S1.p5.30.m30.4.4"><leq id="S1.p5.30.m30.4.4.5.cmml" xref="S1.p5.30.m30.4.4.5"></leq><apply id="S1.p5.30.m30.2.2.1.2.cmml" xref="S1.p5.30.m30.2.2.1.1"><abs id="S1.p5.30.m30.2.2.1.2.1.cmml" xref="S1.p5.30.m30.2.2.1.1.2"></abs><apply id="S1.p5.30.m30.2.2.1.1.1.cmml" xref="S1.p5.30.m30.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.p5.30.m30.2.2.1.1.1.1.cmml" xref="S1.p5.30.m30.2.2.1.1.1">subscript</csymbol><ci id="S1.p5.30.m30.2.2.1.1.1.2.cmml" xref="S1.p5.30.m30.2.2.1.1.1.2">𝑊</ci><ci id="S1.p5.30.m30.2.2.1.1.1.3.cmml" xref="S1.p5.30.m30.2.2.1.1.1.3">𝐴</ci></apply></apply><apply id="S1.p5.30.m30.4.4.3.cmml" xref="S1.p5.30.m30.4.4.3"><plus id="S1.p5.30.m30.4.4.3.3.cmml" xref="S1.p5.30.m30.4.4.3.3"></plus><apply id="S1.p5.30.m30.3.3.2.1.2.cmml" xref="S1.p5.30.m30.3.3.2.1.1"><abs id="S1.p5.30.m30.3.3.2.1.2.1.cmml" xref="S1.p5.30.m30.3.3.2.1.1.2"></abs><apply id="S1.p5.30.m30.3.3.2.1.1.1.cmml" xref="S1.p5.30.m30.3.3.2.1.1.1"><setdiff id="S1.p5.30.m30.3.3.2.1.1.1.1.cmml" xref="S1.p5.30.m30.3.3.2.1.1.1.1"></setdiff><ci id="S1.p5.30.m30.3.3.2.1.1.1.2.cmml" xref="S1.p5.30.m30.3.3.2.1.1.1.2">𝑊</ci><ci id="S1.p5.30.m30.3.3.2.1.1.1.3.cmml" xref="S1.p5.30.m30.3.3.2.1.1.1.3">𝐵</ci></apply></apply><apply id="S1.p5.30.m30.4.4.3.2.2.cmml" xref="S1.p5.30.m30.4.4.3.2.1"><abs id="S1.p5.30.m30.4.4.3.2.2.1.cmml" xref="S1.p5.30.m30.4.4.3.2.1.2"></abs><apply id="S1.p5.30.m30.4.4.3.2.1.1.cmml" xref="S1.p5.30.m30.4.4.3.2.1.1"><intersect id="S1.p5.30.m30.4.4.3.2.1.1.1.cmml" xref="S1.p5.30.m30.4.4.3.2.1.1.1"></intersect><ci id="S1.p5.30.m30.4.4.3.2.1.1.2.cmml" xref="S1.p5.30.m30.4.4.3.2.1.1.2">𝐴</ci><ci id="S1.p5.30.m30.4.4.3.2.1.1.3.cmml" xref="S1.p5.30.m30.4.4.3.2.1.1.3">𝐵</ci></apply></apply></apply></apply><apply id="S1.p5.30.m30.4.4c.cmml" xref="S1.p5.30.m30.4.4"><leq id="S1.p5.30.m30.4.4.6.cmml" xref="S1.p5.30.m30.4.4.6"></leq><share href="https://arxiv.org/html/2503.17112v1#S1.p5.30.m30.4.4.3.cmml" id="S1.p5.30.m30.4.4d.cmml" xref="S1.p5.30.m30.4.4"></share><apply id="S1.p5.30.m30.4.4.7.cmml" xref="S1.p5.30.m30.4.4.7"><plus id="S1.p5.30.m30.4.4.7.1.cmml" xref="S1.p5.30.m30.4.4.7.1"></plus><apply id="S1.p5.30.m30.4.4.7.2.cmml" xref="S1.p5.30.m30.4.4.7.2"><times id="S1.p5.30.m30.4.4.7.2.1.cmml" xref="S1.p5.30.m30.4.4.7.2.1"></times><apply id="S1.p5.30.m30.4.4.7.2.2.cmml" xref="S1.p5.30.m30.4.4.7.2.2"><divide id="S1.p5.30.m30.4.4.7.2.2.1.cmml" xref="S1.p5.30.m30.4.4.7.2.2"></divide><cn id="S1.p5.30.m30.4.4.7.2.2.2.cmml" type="integer" xref="S1.p5.30.m30.4.4.7.2.2.2">2</cn><cn id="S1.p5.30.m30.4.4.7.2.2.3.cmml" type="integer" xref="S1.p5.30.m30.4.4.7.2.2.3">3</cn></apply><apply id="S1.p5.30.m30.4.4.7.2.3.1.cmml" xref="S1.p5.30.m30.4.4.7.2.3.2"><abs id="S1.p5.30.m30.4.4.7.2.3.1.1.cmml" xref="S1.p5.30.m30.4.4.7.2.3.2.1"></abs><ci id="S1.p5.30.m30.1.1.cmml" xref="S1.p5.30.m30.1.1">𝑊</ci></apply></apply><ci id="S1.p5.30.m30.4.4.7.3.cmml" xref="S1.p5.30.m30.4.4.7.3">𝑎</ci></apply></apply><apply id="S1.p5.30.m30.4.4e.cmml" xref="S1.p5.30.m30.4.4"><leq id="S1.p5.30.m30.4.4.8.cmml" xref="S1.p5.30.m30.4.4.8"></leq><share href="https://arxiv.org/html/2503.17112v1#S1.p5.30.m30.4.4.7.cmml" id="S1.p5.30.m30.4.4f.cmml" xref="S1.p5.30.m30.4.4"></share><apply id="S1.p5.30.m30.4.4.9.cmml" xref="S1.p5.30.m30.4.4.9"><times id="S1.p5.30.m30.4.4.9.1.cmml" xref="S1.p5.30.m30.4.4.9.1"></times><cn id="S1.p5.30.m30.4.4.9.2.cmml" type="integer" xref="S1.p5.30.m30.4.4.9.2">3</cn><ci id="S1.p5.30.m30.4.4.9.3.cmml" xref="S1.p5.30.m30.4.4.9.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.30.m30.4c">|W_{A}|\leq|W\setminus B|+|A\cap B|\leq\tfrac{2}{3}|W|+a\leq 3a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.30.m30.4d">| italic_W start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | ≤ | italic_W ∖ italic_B | + | italic_A ∩ italic_B | ≤ divide start_ARG 2 end_ARG start_ARG 3 end_ARG | italic_W | + italic_a ≤ 3 italic_a</annotation></semantics></math> and a bag <math alttext="B_{x^{\prime}}" class="ltx_Math" display="inline" id="S1.p5.31.m31.1"><semantics id="S1.p5.31.m31.1a"><msub id="S1.p5.31.m31.1.1" xref="S1.p5.31.m31.1.1.cmml"><mi id="S1.p5.31.m31.1.1.2" xref="S1.p5.31.m31.1.1.2.cmml">B</mi><msup id="S1.p5.31.m31.1.1.3" xref="S1.p5.31.m31.1.1.3.cmml"><mi id="S1.p5.31.m31.1.1.3.2" xref="S1.p5.31.m31.1.1.3.2.cmml">x</mi><mo id="S1.p5.31.m31.1.1.3.3" xref="S1.p5.31.m31.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S1.p5.31.m31.1b"><apply id="S1.p5.31.m31.1.1.cmml" xref="S1.p5.31.m31.1.1"><csymbol cd="ambiguous" id="S1.p5.31.m31.1.1.1.cmml" xref="S1.p5.31.m31.1.1">subscript</csymbol><ci id="S1.p5.31.m31.1.1.2.cmml" xref="S1.p5.31.m31.1.1.2">𝐵</ci><apply id="S1.p5.31.m31.1.1.3.cmml" xref="S1.p5.31.m31.1.1.3"><csymbol cd="ambiguous" id="S1.p5.31.m31.1.1.3.1.cmml" xref="S1.p5.31.m31.1.1.3">superscript</csymbol><ci id="S1.p5.31.m31.1.1.3.2.cmml" xref="S1.p5.31.m31.1.1.3.2">𝑥</ci><ci id="S1.p5.31.m31.1.1.3.3.cmml" xref="S1.p5.31.m31.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.31.m31.1c">B_{x^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.31.m31.1d">italic_B start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S1.p5.32.m32.1"><semantics id="S1.p5.32.m32.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.32.m32.1.1" xref="S1.p5.32.m32.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S1.p5.32.m32.1b"><ci id="S1.p5.32.m32.1.1.cmml" xref="S1.p5.32.m32.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.32.m32.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.32.m32.1d">caligraphic_T</annotation></semantics></math> contains <math alttext="W_{A}" class="ltx_Math" display="inline" id="S1.p5.33.m33.1"><semantics id="S1.p5.33.m33.1a"><msub id="S1.p5.33.m33.1.1" xref="S1.p5.33.m33.1.1.cmml"><mi id="S1.p5.33.m33.1.1.2" xref="S1.p5.33.m33.1.1.2.cmml">W</mi><mi id="S1.p5.33.m33.1.1.3" xref="S1.p5.33.m33.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p5.33.m33.1b"><apply id="S1.p5.33.m33.1.1.cmml" xref="S1.p5.33.m33.1.1"><csymbol cd="ambiguous" id="S1.p5.33.m33.1.1.1.cmml" xref="S1.p5.33.m33.1.1">subscript</csymbol><ci id="S1.p5.33.m33.1.1.2.cmml" xref="S1.p5.33.m33.1.1.2">𝑊</ci><ci id="S1.p5.33.m33.1.1.3.cmml" xref="S1.p5.33.m33.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.33.m33.1c">W_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.33.m33.1d">italic_W start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>. The algorithm then inductively extends <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S1.p5.34.m34.1"><semantics id="S1.p5.34.m34.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.34.m34.1.1" xref="S1.p5.34.m34.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S1.p5.34.m34.1b"><ci id="S1.p5.34.m34.1.1.cmml" xref="S1.p5.34.m34.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.34.m34.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.34.m34.1d">caligraphic_T</annotation></semantics></math> to a tree decomposition of <math alttext="G_{A}" class="ltx_Math" display="inline" id="S1.p5.35.m35.1"><semantics id="S1.p5.35.m35.1a"><msub id="S1.p5.35.m35.1.1" xref="S1.p5.35.m35.1.1.cmml"><mi id="S1.p5.35.m35.1.1.2" xref="S1.p5.35.m35.1.1.2.cmml">G</mi><mi id="S1.p5.35.m35.1.1.3" xref="S1.p5.35.m35.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p5.35.m35.1b"><apply id="S1.p5.35.m35.1.1.cmml" xref="S1.p5.35.m35.1.1"><csymbol cd="ambiguous" id="S1.p5.35.m35.1.1.1.cmml" xref="S1.p5.35.m35.1.1">subscript</csymbol><ci id="S1.p5.35.m35.1.1.2.cmml" xref="S1.p5.35.m35.1.1.2">𝐺</ci><ci id="S1.p5.35.m35.1.1.3.cmml" xref="S1.p5.35.m35.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.35.m35.1c">G_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.35.m35.1d">italic_G start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> of width less than <math alttext="4a" class="ltx_Math" display="inline" id="S1.p5.36.m36.1"><semantics id="S1.p5.36.m36.1a"><mrow id="S1.p5.36.m36.1.1" xref="S1.p5.36.m36.1.1.cmml"><mn id="S1.p5.36.m36.1.1.2" xref="S1.p5.36.m36.1.1.2.cmml">4</mn><mo id="S1.p5.36.m36.1.1.1" xref="S1.p5.36.m36.1.1.1.cmml">⁢</mo><mi id="S1.p5.36.m36.1.1.3" xref="S1.p5.36.m36.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.36.m36.1b"><apply id="S1.p5.36.m36.1.1.cmml" xref="S1.p5.36.m36.1.1"><times id="S1.p5.36.m36.1.1.1.cmml" xref="S1.p5.36.m36.1.1.1"></times><cn id="S1.p5.36.m36.1.1.2.cmml" type="integer" xref="S1.p5.36.m36.1.1.2">4</cn><ci id="S1.p5.36.m36.1.1.3.cmml" xref="S1.p5.36.m36.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.36.m36.1c">4a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.36.m36.1d">4 italic_a</annotation></semantics></math>. Let <math alttext="X_{B}:=B" class="ltx_Math" display="inline" id="S1.p5.37.m37.1"><semantics id="S1.p5.37.m37.1a"><mrow id="S1.p5.37.m37.1.1" xref="S1.p5.37.m37.1.1.cmml"><msub id="S1.p5.37.m37.1.1.2" xref="S1.p5.37.m37.1.1.2.cmml"><mi id="S1.p5.37.m37.1.1.2.2" xref="S1.p5.37.m37.1.1.2.2.cmml">X</mi><mi id="S1.p5.37.m37.1.1.2.3" xref="S1.p5.37.m37.1.1.2.3.cmml">B</mi></msub><mo id="S1.p5.37.m37.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p5.37.m37.1.1.1.cmml">:=</mo><mi id="S1.p5.37.m37.1.1.3" xref="S1.p5.37.m37.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.37.m37.1b"><apply id="S1.p5.37.m37.1.1.cmml" xref="S1.p5.37.m37.1.1"><csymbol cd="latexml" id="S1.p5.37.m37.1.1.1.cmml" xref="S1.p5.37.m37.1.1.1">assign</csymbol><apply id="S1.p5.37.m37.1.1.2.cmml" xref="S1.p5.37.m37.1.1.2"><csymbol cd="ambiguous" id="S1.p5.37.m37.1.1.2.1.cmml" xref="S1.p5.37.m37.1.1.2">subscript</csymbol><ci id="S1.p5.37.m37.1.1.2.2.cmml" xref="S1.p5.37.m37.1.1.2.2">𝑋</ci><ci id="S1.p5.37.m37.1.1.2.3.cmml" xref="S1.p5.37.m37.1.1.2.3">𝐵</ci></apply><ci id="S1.p5.37.m37.1.1.3.cmml" xref="S1.p5.37.m37.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.37.m37.1c">X_{B}:=B</annotation><annotation encoding="application/x-llamapun" id="S1.p5.37.m37.1d">italic_X start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT := italic_B</annotation></semantics></math>, <math alttext="Y_{B}:=Y\cup A" class="ltx_Math" display="inline" id="S1.p5.38.m38.1"><semantics id="S1.p5.38.m38.1a"><mrow id="S1.p5.38.m38.1.1" xref="S1.p5.38.m38.1.1.cmml"><msub id="S1.p5.38.m38.1.1.2" xref="S1.p5.38.m38.1.1.2.cmml"><mi id="S1.p5.38.m38.1.1.2.2" xref="S1.p5.38.m38.1.1.2.2.cmml">Y</mi><mi id="S1.p5.38.m38.1.1.2.3" xref="S1.p5.38.m38.1.1.2.3.cmml">B</mi></msub><mo id="S1.p5.38.m38.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p5.38.m38.1.1.1.cmml">:=</mo><mrow id="S1.p5.38.m38.1.1.3" xref="S1.p5.38.m38.1.1.3.cmml"><mi id="S1.p5.38.m38.1.1.3.2" xref="S1.p5.38.m38.1.1.3.2.cmml">Y</mi><mo id="S1.p5.38.m38.1.1.3.1" xref="S1.p5.38.m38.1.1.3.1.cmml">∪</mo><mi id="S1.p5.38.m38.1.1.3.3" xref="S1.p5.38.m38.1.1.3.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.38.m38.1b"><apply id="S1.p5.38.m38.1.1.cmml" xref="S1.p5.38.m38.1.1"><csymbol cd="latexml" id="S1.p5.38.m38.1.1.1.cmml" xref="S1.p5.38.m38.1.1.1">assign</csymbol><apply id="S1.p5.38.m38.1.1.2.cmml" xref="S1.p5.38.m38.1.1.2"><csymbol cd="ambiguous" id="S1.p5.38.m38.1.1.2.1.cmml" xref="S1.p5.38.m38.1.1.2">subscript</csymbol><ci id="S1.p5.38.m38.1.1.2.2.cmml" xref="S1.p5.38.m38.1.1.2.2">𝑌</ci><ci id="S1.p5.38.m38.1.1.2.3.cmml" xref="S1.p5.38.m38.1.1.2.3">𝐵</ci></apply><apply id="S1.p5.38.m38.1.1.3.cmml" xref="S1.p5.38.m38.1.1.3"><union id="S1.p5.38.m38.1.1.3.1.cmml" xref="S1.p5.38.m38.1.1.3.1"></union><ci id="S1.p5.38.m38.1.1.3.2.cmml" xref="S1.p5.38.m38.1.1.3.2">𝑌</ci><ci id="S1.p5.38.m38.1.1.3.3.cmml" xref="S1.p5.38.m38.1.1.3.3">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.38.m38.1c">Y_{B}:=Y\cup A</annotation><annotation encoding="application/x-llamapun" id="S1.p5.38.m38.1d">italic_Y start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT := italic_Y ∪ italic_A</annotation></semantics></math>, <math alttext="G_{B}:=G" class="ltx_Math" display="inline" id="S1.p5.39.m39.1"><semantics id="S1.p5.39.m39.1a"><mrow id="S1.p5.39.m39.1.1" xref="S1.p5.39.m39.1.1.cmml"><msub id="S1.p5.39.m39.1.1.2" xref="S1.p5.39.m39.1.1.2.cmml"><mi id="S1.p5.39.m39.1.1.2.2" xref="S1.p5.39.m39.1.1.2.2.cmml">G</mi><mi id="S1.p5.39.m39.1.1.2.3" xref="S1.p5.39.m39.1.1.2.3.cmml">B</mi></msub><mo id="S1.p5.39.m39.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p5.39.m39.1.1.1.cmml">:=</mo><mi id="S1.p5.39.m39.1.1.3" xref="S1.p5.39.m39.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.39.m39.1b"><apply id="S1.p5.39.m39.1.1.cmml" xref="S1.p5.39.m39.1.1"><csymbol cd="latexml" id="S1.p5.39.m39.1.1.1.cmml" xref="S1.p5.39.m39.1.1.1">assign</csymbol><apply id="S1.p5.39.m39.1.1.2.cmml" xref="S1.p5.39.m39.1.1.2"><csymbol cd="ambiguous" id="S1.p5.39.m39.1.1.2.1.cmml" xref="S1.p5.39.m39.1.1.2">subscript</csymbol><ci id="S1.p5.39.m39.1.1.2.2.cmml" xref="S1.p5.39.m39.1.1.2.2">𝐺</ci><ci id="S1.p5.39.m39.1.1.2.3.cmml" xref="S1.p5.39.m39.1.1.2.3">𝐵</ci></apply><ci id="S1.p5.39.m39.1.1.3.cmml" xref="S1.p5.39.m39.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.39.m39.1c">G_{B}:=G</annotation><annotation encoding="application/x-llamapun" id="S1.p5.39.m39.1d">italic_G start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT := italic_G</annotation></semantics></math>, and <math alttext="W_{B}:=X_{B}\cap Y_{B}" class="ltx_Math" display="inline" id="S1.p5.40.m40.1"><semantics id="S1.p5.40.m40.1a"><mrow id="S1.p5.40.m40.1.1" xref="S1.p5.40.m40.1.1.cmml"><msub id="S1.p5.40.m40.1.1.2" xref="S1.p5.40.m40.1.1.2.cmml"><mi id="S1.p5.40.m40.1.1.2.2" xref="S1.p5.40.m40.1.1.2.2.cmml">W</mi><mi id="S1.p5.40.m40.1.1.2.3" xref="S1.p5.40.m40.1.1.2.3.cmml">B</mi></msub><mo id="S1.p5.40.m40.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p5.40.m40.1.1.1.cmml">:=</mo><mrow id="S1.p5.40.m40.1.1.3" xref="S1.p5.40.m40.1.1.3.cmml"><msub id="S1.p5.40.m40.1.1.3.2" xref="S1.p5.40.m40.1.1.3.2.cmml"><mi id="S1.p5.40.m40.1.1.3.2.2" xref="S1.p5.40.m40.1.1.3.2.2.cmml">X</mi><mi id="S1.p5.40.m40.1.1.3.2.3" xref="S1.p5.40.m40.1.1.3.2.3.cmml">B</mi></msub><mo id="S1.p5.40.m40.1.1.3.1" xref="S1.p5.40.m40.1.1.3.1.cmml">∩</mo><msub id="S1.p5.40.m40.1.1.3.3" xref="S1.p5.40.m40.1.1.3.3.cmml"><mi id="S1.p5.40.m40.1.1.3.3.2" xref="S1.p5.40.m40.1.1.3.3.2.cmml">Y</mi><mi id="S1.p5.40.m40.1.1.3.3.3" xref="S1.p5.40.m40.1.1.3.3.3.cmml">B</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.40.m40.1b"><apply id="S1.p5.40.m40.1.1.cmml" xref="S1.p5.40.m40.1.1"><csymbol cd="latexml" id="S1.p5.40.m40.1.1.1.cmml" xref="S1.p5.40.m40.1.1.1">assign</csymbol><apply id="S1.p5.40.m40.1.1.2.cmml" xref="S1.p5.40.m40.1.1.2"><csymbol cd="ambiguous" id="S1.p5.40.m40.1.1.2.1.cmml" xref="S1.p5.40.m40.1.1.2">subscript</csymbol><ci id="S1.p5.40.m40.1.1.2.2.cmml" xref="S1.p5.40.m40.1.1.2.2">𝑊</ci><ci id="S1.p5.40.m40.1.1.2.3.cmml" xref="S1.p5.40.m40.1.1.2.3">𝐵</ci></apply><apply id="S1.p5.40.m40.1.1.3.cmml" xref="S1.p5.40.m40.1.1.3"><intersect id="S1.p5.40.m40.1.1.3.1.cmml" xref="S1.p5.40.m40.1.1.3.1"></intersect><apply id="S1.p5.40.m40.1.1.3.2.cmml" xref="S1.p5.40.m40.1.1.3.2"><csymbol cd="ambiguous" id="S1.p5.40.m40.1.1.3.2.1.cmml" xref="S1.p5.40.m40.1.1.3.2">subscript</csymbol><ci id="S1.p5.40.m40.1.1.3.2.2.cmml" xref="S1.p5.40.m40.1.1.3.2.2">𝑋</ci><ci id="S1.p5.40.m40.1.1.3.2.3.cmml" xref="S1.p5.40.m40.1.1.3.2.3">𝐵</ci></apply><apply id="S1.p5.40.m40.1.1.3.3.cmml" xref="S1.p5.40.m40.1.1.3.3"><csymbol cd="ambiguous" id="S1.p5.40.m40.1.1.3.3.1.cmml" xref="S1.p5.40.m40.1.1.3.3">subscript</csymbol><ci id="S1.p5.40.m40.1.1.3.3.2.cmml" xref="S1.p5.40.m40.1.1.3.3.2">𝑌</ci><ci id="S1.p5.40.m40.1.1.3.3.3.cmml" xref="S1.p5.40.m40.1.1.3.3.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.40.m40.1c">W_{B}:=X_{B}\cap Y_{B}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.40.m40.1d">italic_W start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT := italic_X start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ∩ italic_Y start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S1.p5.41.m41.1"><semantics id="S1.p5.41.m41.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.41.m41.1.1" xref="S1.p5.41.m41.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S1.p5.41.m41.1b"><ci id="S1.p5.41.m41.1.1.cmml" xref="S1.p5.41.m41.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.41.m41.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.41.m41.1d">caligraphic_T</annotation></semantics></math> is a tree decomposition of <math alttext="Y_{B}" class="ltx_Math" display="inline" id="S1.p5.42.m42.1"><semantics id="S1.p5.42.m42.1a"><msub id="S1.p5.42.m42.1.1" xref="S1.p5.42.m42.1.1.cmml"><mi id="S1.p5.42.m42.1.1.2" xref="S1.p5.42.m42.1.1.2.cmml">Y</mi><mi id="S1.p5.42.m42.1.1.3" xref="S1.p5.42.m42.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p5.42.m42.1b"><apply id="S1.p5.42.m42.1.1.cmml" xref="S1.p5.42.m42.1.1"><csymbol cd="ambiguous" id="S1.p5.42.m42.1.1.1.cmml" xref="S1.p5.42.m42.1.1">subscript</csymbol><ci id="S1.p5.42.m42.1.1.2.cmml" xref="S1.p5.42.m42.1.1.2">𝑌</ci><ci id="S1.p5.42.m42.1.1.3.cmml" xref="S1.p5.42.m42.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.42.m42.1c">Y_{B}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.42.m42.1d">italic_Y start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> of width less than <math alttext="4a" class="ltx_Math" display="inline" id="S1.p5.43.m43.1"><semantics id="S1.p5.43.m43.1a"><mrow id="S1.p5.43.m43.1.1" xref="S1.p5.43.m43.1.1.cmml"><mn id="S1.p5.43.m43.1.1.2" xref="S1.p5.43.m43.1.1.2.cmml">4</mn><mo id="S1.p5.43.m43.1.1.1" xref="S1.p5.43.m43.1.1.1.cmml">⁢</mo><mi id="S1.p5.43.m43.1.1.3" xref="S1.p5.43.m43.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.43.m43.1b"><apply id="S1.p5.43.m43.1.1.cmml" xref="S1.p5.43.m43.1.1"><times id="S1.p5.43.m43.1.1.1.cmml" xref="S1.p5.43.m43.1.1.1"></times><cn id="S1.p5.43.m43.1.1.2.cmml" type="integer" xref="S1.p5.43.m43.1.1.2">4</cn><ci id="S1.p5.43.m43.1.1.3.cmml" xref="S1.p5.43.m43.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.43.m43.1c">4a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.43.m43.1d">4 italic_a</annotation></semantics></math> and <math alttext="(X_{B},Y_{B})" class="ltx_Math" display="inline" id="S1.p5.44.m44.2"><semantics id="S1.p5.44.m44.2a"><mrow id="S1.p5.44.m44.2.2.2" xref="S1.p5.44.m44.2.2.3.cmml"><mo id="S1.p5.44.m44.2.2.2.3" stretchy="false" xref="S1.p5.44.m44.2.2.3.cmml">(</mo><msub id="S1.p5.44.m44.1.1.1.1" xref="S1.p5.44.m44.1.1.1.1.cmml"><mi id="S1.p5.44.m44.1.1.1.1.2" xref="S1.p5.44.m44.1.1.1.1.2.cmml">X</mi><mi id="S1.p5.44.m44.1.1.1.1.3" xref="S1.p5.44.m44.1.1.1.1.3.cmml">B</mi></msub><mo id="S1.p5.44.m44.2.2.2.4" xref="S1.p5.44.m44.2.2.3.cmml">,</mo><msub id="S1.p5.44.m44.2.2.2.2" xref="S1.p5.44.m44.2.2.2.2.cmml"><mi id="S1.p5.44.m44.2.2.2.2.2" xref="S1.p5.44.m44.2.2.2.2.2.cmml">Y</mi><mi id="S1.p5.44.m44.2.2.2.2.3" xref="S1.p5.44.m44.2.2.2.2.3.cmml">B</mi></msub><mo id="S1.p5.44.m44.2.2.2.5" stretchy="false" xref="S1.p5.44.m44.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.44.m44.2b"><interval closure="open" id="S1.p5.44.m44.2.2.3.cmml" xref="S1.p5.44.m44.2.2.2"><apply id="S1.p5.44.m44.1.1.1.1.cmml" xref="S1.p5.44.m44.1.1.1.1"><csymbol cd="ambiguous" id="S1.p5.44.m44.1.1.1.1.1.cmml" xref="S1.p5.44.m44.1.1.1.1">subscript</csymbol><ci id="S1.p5.44.m44.1.1.1.1.2.cmml" xref="S1.p5.44.m44.1.1.1.1.2">𝑋</ci><ci id="S1.p5.44.m44.1.1.1.1.3.cmml" xref="S1.p5.44.m44.1.1.1.1.3">𝐵</ci></apply><apply id="S1.p5.44.m44.2.2.2.2.cmml" xref="S1.p5.44.m44.2.2.2.2"><csymbol cd="ambiguous" id="S1.p5.44.m44.2.2.2.2.1.cmml" xref="S1.p5.44.m44.2.2.2.2">subscript</csymbol><ci id="S1.p5.44.m44.2.2.2.2.2.cmml" xref="S1.p5.44.m44.2.2.2.2.2">𝑌</ci><ci id="S1.p5.44.m44.2.2.2.2.3.cmml" xref="S1.p5.44.m44.2.2.2.2.3">𝐵</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.44.m44.2c">(X_{B},Y_{B})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.44.m44.2d">( italic_X start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT )</annotation></semantics></math> is a separation of <math alttext="G_{B}" class="ltx_Math" display="inline" id="S1.p5.45.m45.1"><semantics id="S1.p5.45.m45.1a"><msub id="S1.p5.45.m45.1.1" xref="S1.p5.45.m45.1.1.cmml"><mi id="S1.p5.45.m45.1.1.2" xref="S1.p5.45.m45.1.1.2.cmml">G</mi><mi id="S1.p5.45.m45.1.1.3" xref="S1.p5.45.m45.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p5.45.m45.1b"><apply id="S1.p5.45.m45.1.1.cmml" xref="S1.p5.45.m45.1.1"><csymbol cd="ambiguous" id="S1.p5.45.m45.1.1.1.cmml" xref="S1.p5.45.m45.1.1">subscript</csymbol><ci id="S1.p5.45.m45.1.1.2.cmml" xref="S1.p5.45.m45.1.1.2">𝐺</ci><ci id="S1.p5.45.m45.1.1.3.cmml" xref="S1.p5.45.m45.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.45.m45.1c">G_{B}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.45.m45.1d">italic_G start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> of order <math alttext="|W_{B}|\leq|W\cap B|+|A\cap B|\leq\tfrac{2}{3}|W|+a\leq 3a" class="ltx_Math" display="inline" id="S1.p5.46.m46.4"><semantics id="S1.p5.46.m46.4a"><mrow id="S1.p5.46.m46.4.4" xref="S1.p5.46.m46.4.4.cmml"><mrow id="S1.p5.46.m46.2.2.1.1" xref="S1.p5.46.m46.2.2.1.2.cmml"><mo id="S1.p5.46.m46.2.2.1.1.2" stretchy="false" xref="S1.p5.46.m46.2.2.1.2.1.cmml">|</mo><msub id="S1.p5.46.m46.2.2.1.1.1" xref="S1.p5.46.m46.2.2.1.1.1.cmml"><mi id="S1.p5.46.m46.2.2.1.1.1.2" xref="S1.p5.46.m46.2.2.1.1.1.2.cmml">W</mi><mi id="S1.p5.46.m46.2.2.1.1.1.3" xref="S1.p5.46.m46.2.2.1.1.1.3.cmml">B</mi></msub><mo id="S1.p5.46.m46.2.2.1.1.3" stretchy="false" xref="S1.p5.46.m46.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p5.46.m46.4.4.5" xref="S1.p5.46.m46.4.4.5.cmml">≤</mo><mrow id="S1.p5.46.m46.4.4.3" xref="S1.p5.46.m46.4.4.3.cmml"><mrow id="S1.p5.46.m46.3.3.2.1.1" xref="S1.p5.46.m46.3.3.2.1.2.cmml"><mo id="S1.p5.46.m46.3.3.2.1.1.2" stretchy="false" xref="S1.p5.46.m46.3.3.2.1.2.1.cmml">|</mo><mrow id="S1.p5.46.m46.3.3.2.1.1.1" xref="S1.p5.46.m46.3.3.2.1.1.1.cmml"><mi id="S1.p5.46.m46.3.3.2.1.1.1.2" xref="S1.p5.46.m46.3.3.2.1.1.1.2.cmml">W</mi><mo id="S1.p5.46.m46.3.3.2.1.1.1.1" xref="S1.p5.46.m46.3.3.2.1.1.1.1.cmml">∩</mo><mi id="S1.p5.46.m46.3.3.2.1.1.1.3" xref="S1.p5.46.m46.3.3.2.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p5.46.m46.3.3.2.1.1.3" stretchy="false" xref="S1.p5.46.m46.3.3.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p5.46.m46.4.4.3.3" xref="S1.p5.46.m46.4.4.3.3.cmml">+</mo><mrow id="S1.p5.46.m46.4.4.3.2.1" xref="S1.p5.46.m46.4.4.3.2.2.cmml"><mo id="S1.p5.46.m46.4.4.3.2.1.2" stretchy="false" xref="S1.p5.46.m46.4.4.3.2.2.1.cmml">|</mo><mrow id="S1.p5.46.m46.4.4.3.2.1.1" xref="S1.p5.46.m46.4.4.3.2.1.1.cmml"><mi id="S1.p5.46.m46.4.4.3.2.1.1.2" xref="S1.p5.46.m46.4.4.3.2.1.1.2.cmml">A</mi><mo id="S1.p5.46.m46.4.4.3.2.1.1.1" xref="S1.p5.46.m46.4.4.3.2.1.1.1.cmml">∩</mo><mi id="S1.p5.46.m46.4.4.3.2.1.1.3" xref="S1.p5.46.m46.4.4.3.2.1.1.3.cmml">B</mi></mrow><mo id="S1.p5.46.m46.4.4.3.2.1.3" stretchy="false" xref="S1.p5.46.m46.4.4.3.2.2.1.cmml">|</mo></mrow></mrow><mo id="S1.p5.46.m46.4.4.6" xref="S1.p5.46.m46.4.4.6.cmml">≤</mo><mrow id="S1.p5.46.m46.4.4.7" xref="S1.p5.46.m46.4.4.7.cmml"><mrow id="S1.p5.46.m46.4.4.7.2" xref="S1.p5.46.m46.4.4.7.2.cmml"><mfrac id="S1.p5.46.m46.4.4.7.2.2" xref="S1.p5.46.m46.4.4.7.2.2.cmml"><mn id="S1.p5.46.m46.4.4.7.2.2.2" xref="S1.p5.46.m46.4.4.7.2.2.2.cmml">2</mn><mn id="S1.p5.46.m46.4.4.7.2.2.3" xref="S1.p5.46.m46.4.4.7.2.2.3.cmml">3</mn></mfrac><mo id="S1.p5.46.m46.4.4.7.2.1" xref="S1.p5.46.m46.4.4.7.2.1.cmml">⁢</mo><mrow id="S1.p5.46.m46.4.4.7.2.3.2" xref="S1.p5.46.m46.4.4.7.2.3.1.cmml"><mo id="S1.p5.46.m46.4.4.7.2.3.2.1" stretchy="false" xref="S1.p5.46.m46.4.4.7.2.3.1.1.cmml">|</mo><mi id="S1.p5.46.m46.1.1" xref="S1.p5.46.m46.1.1.cmml">W</mi><mo id="S1.p5.46.m46.4.4.7.2.3.2.2" stretchy="false" xref="S1.p5.46.m46.4.4.7.2.3.1.1.cmml">|</mo></mrow></mrow><mo id="S1.p5.46.m46.4.4.7.1" xref="S1.p5.46.m46.4.4.7.1.cmml">+</mo><mi id="S1.p5.46.m46.4.4.7.3" xref="S1.p5.46.m46.4.4.7.3.cmml">a</mi></mrow><mo id="S1.p5.46.m46.4.4.8" xref="S1.p5.46.m46.4.4.8.cmml">≤</mo><mrow id="S1.p5.46.m46.4.4.9" xref="S1.p5.46.m46.4.4.9.cmml"><mn id="S1.p5.46.m46.4.4.9.2" xref="S1.p5.46.m46.4.4.9.2.cmml">3</mn><mo id="S1.p5.46.m46.4.4.9.1" xref="S1.p5.46.m46.4.4.9.1.cmml">⁢</mo><mi id="S1.p5.46.m46.4.4.9.3" xref="S1.p5.46.m46.4.4.9.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.46.m46.4b"><apply id="S1.p5.46.m46.4.4.cmml" xref="S1.p5.46.m46.4.4"><and id="S1.p5.46.m46.4.4a.cmml" xref="S1.p5.46.m46.4.4"></and><apply id="S1.p5.46.m46.4.4b.cmml" xref="S1.p5.46.m46.4.4"><leq id="S1.p5.46.m46.4.4.5.cmml" xref="S1.p5.46.m46.4.4.5"></leq><apply id="S1.p5.46.m46.2.2.1.2.cmml" xref="S1.p5.46.m46.2.2.1.1"><abs id="S1.p5.46.m46.2.2.1.2.1.cmml" xref="S1.p5.46.m46.2.2.1.1.2"></abs><apply id="S1.p5.46.m46.2.2.1.1.1.cmml" xref="S1.p5.46.m46.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.p5.46.m46.2.2.1.1.1.1.cmml" xref="S1.p5.46.m46.2.2.1.1.1">subscript</csymbol><ci id="S1.p5.46.m46.2.2.1.1.1.2.cmml" xref="S1.p5.46.m46.2.2.1.1.1.2">𝑊</ci><ci id="S1.p5.46.m46.2.2.1.1.1.3.cmml" xref="S1.p5.46.m46.2.2.1.1.1.3">𝐵</ci></apply></apply><apply id="S1.p5.46.m46.4.4.3.cmml" xref="S1.p5.46.m46.4.4.3"><plus id="S1.p5.46.m46.4.4.3.3.cmml" xref="S1.p5.46.m46.4.4.3.3"></plus><apply id="S1.p5.46.m46.3.3.2.1.2.cmml" xref="S1.p5.46.m46.3.3.2.1.1"><abs id="S1.p5.46.m46.3.3.2.1.2.1.cmml" xref="S1.p5.46.m46.3.3.2.1.1.2"></abs><apply id="S1.p5.46.m46.3.3.2.1.1.1.cmml" xref="S1.p5.46.m46.3.3.2.1.1.1"><intersect id="S1.p5.46.m46.3.3.2.1.1.1.1.cmml" xref="S1.p5.46.m46.3.3.2.1.1.1.1"></intersect><ci id="S1.p5.46.m46.3.3.2.1.1.1.2.cmml" xref="S1.p5.46.m46.3.3.2.1.1.1.2">𝑊</ci><ci id="S1.p5.46.m46.3.3.2.1.1.1.3.cmml" xref="S1.p5.46.m46.3.3.2.1.1.1.3">𝐵</ci></apply></apply><apply id="S1.p5.46.m46.4.4.3.2.2.cmml" xref="S1.p5.46.m46.4.4.3.2.1"><abs id="S1.p5.46.m46.4.4.3.2.2.1.cmml" xref="S1.p5.46.m46.4.4.3.2.1.2"></abs><apply id="S1.p5.46.m46.4.4.3.2.1.1.cmml" xref="S1.p5.46.m46.4.4.3.2.1.1"><intersect id="S1.p5.46.m46.4.4.3.2.1.1.1.cmml" xref="S1.p5.46.m46.4.4.3.2.1.1.1"></intersect><ci id="S1.p5.46.m46.4.4.3.2.1.1.2.cmml" xref="S1.p5.46.m46.4.4.3.2.1.1.2">𝐴</ci><ci id="S1.p5.46.m46.4.4.3.2.1.1.3.cmml" xref="S1.p5.46.m46.4.4.3.2.1.1.3">𝐵</ci></apply></apply></apply></apply><apply id="S1.p5.46.m46.4.4c.cmml" xref="S1.p5.46.m46.4.4"><leq id="S1.p5.46.m46.4.4.6.cmml" xref="S1.p5.46.m46.4.4.6"></leq><share href="https://arxiv.org/html/2503.17112v1#S1.p5.46.m46.4.4.3.cmml" id="S1.p5.46.m46.4.4d.cmml" xref="S1.p5.46.m46.4.4"></share><apply id="S1.p5.46.m46.4.4.7.cmml" xref="S1.p5.46.m46.4.4.7"><plus id="S1.p5.46.m46.4.4.7.1.cmml" xref="S1.p5.46.m46.4.4.7.1"></plus><apply id="S1.p5.46.m46.4.4.7.2.cmml" xref="S1.p5.46.m46.4.4.7.2"><times id="S1.p5.46.m46.4.4.7.2.1.cmml" xref="S1.p5.46.m46.4.4.7.2.1"></times><apply id="S1.p5.46.m46.4.4.7.2.2.cmml" xref="S1.p5.46.m46.4.4.7.2.2"><divide id="S1.p5.46.m46.4.4.7.2.2.1.cmml" xref="S1.p5.46.m46.4.4.7.2.2"></divide><cn id="S1.p5.46.m46.4.4.7.2.2.2.cmml" type="integer" xref="S1.p5.46.m46.4.4.7.2.2.2">2</cn><cn id="S1.p5.46.m46.4.4.7.2.2.3.cmml" type="integer" xref="S1.p5.46.m46.4.4.7.2.2.3">3</cn></apply><apply id="S1.p5.46.m46.4.4.7.2.3.1.cmml" xref="S1.p5.46.m46.4.4.7.2.3.2"><abs id="S1.p5.46.m46.4.4.7.2.3.1.1.cmml" xref="S1.p5.46.m46.4.4.7.2.3.2.1"></abs><ci id="S1.p5.46.m46.1.1.cmml" xref="S1.p5.46.m46.1.1">𝑊</ci></apply></apply><ci id="S1.p5.46.m46.4.4.7.3.cmml" xref="S1.p5.46.m46.4.4.7.3">𝑎</ci></apply></apply><apply id="S1.p5.46.m46.4.4e.cmml" xref="S1.p5.46.m46.4.4"><leq id="S1.p5.46.m46.4.4.8.cmml" xref="S1.p5.46.m46.4.4.8"></leq><share href="https://arxiv.org/html/2503.17112v1#S1.p5.46.m46.4.4.7.cmml" id="S1.p5.46.m46.4.4f.cmml" xref="S1.p5.46.m46.4.4"></share><apply id="S1.p5.46.m46.4.4.9.cmml" xref="S1.p5.46.m46.4.4.9"><times id="S1.p5.46.m46.4.4.9.1.cmml" xref="S1.p5.46.m46.4.4.9.1"></times><cn id="S1.p5.46.m46.4.4.9.2.cmml" type="integer" xref="S1.p5.46.m46.4.4.9.2">3</cn><ci id="S1.p5.46.m46.4.4.9.3.cmml" xref="S1.p5.46.m46.4.4.9.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.46.m46.4c">|W_{B}|\leq|W\cap B|+|A\cap B|\leq\tfrac{2}{3}|W|+a\leq 3a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.46.m46.4d">| italic_W start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT | ≤ | italic_W ∩ italic_B | + | italic_A ∩ italic_B | ≤ divide start_ARG 2 end_ARG start_ARG 3 end_ARG | italic_W | + italic_a ≤ 3 italic_a</annotation></semantics></math> and a bag <math alttext="B_{x^{\prime}}" class="ltx_Math" display="inline" id="S1.p5.47.m47.1"><semantics id="S1.p5.47.m47.1a"><msub id="S1.p5.47.m47.1.1" xref="S1.p5.47.m47.1.1.cmml"><mi id="S1.p5.47.m47.1.1.2" xref="S1.p5.47.m47.1.1.2.cmml">B</mi><msup id="S1.p5.47.m47.1.1.3" xref="S1.p5.47.m47.1.1.3.cmml"><mi id="S1.p5.47.m47.1.1.3.2" xref="S1.p5.47.m47.1.1.3.2.cmml">x</mi><mo id="S1.p5.47.m47.1.1.3.3" xref="S1.p5.47.m47.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S1.p5.47.m47.1b"><apply id="S1.p5.47.m47.1.1.cmml" xref="S1.p5.47.m47.1.1"><csymbol cd="ambiguous" id="S1.p5.47.m47.1.1.1.cmml" xref="S1.p5.47.m47.1.1">subscript</csymbol><ci id="S1.p5.47.m47.1.1.2.cmml" xref="S1.p5.47.m47.1.1.2">𝐵</ci><apply id="S1.p5.47.m47.1.1.3.cmml" xref="S1.p5.47.m47.1.1.3"><csymbol cd="ambiguous" id="S1.p5.47.m47.1.1.3.1.cmml" xref="S1.p5.47.m47.1.1.3">superscript</csymbol><ci id="S1.p5.47.m47.1.1.3.2.cmml" xref="S1.p5.47.m47.1.1.3.2">𝑥</ci><ci id="S1.p5.47.m47.1.1.3.3.cmml" xref="S1.p5.47.m47.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.47.m47.1c">B_{x^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.47.m47.1d">italic_B start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S1.p5.48.m48.1"><semantics id="S1.p5.48.m48.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.48.m48.1.1" xref="S1.p5.48.m48.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S1.p5.48.m48.1b"><ci id="S1.p5.48.m48.1.1.cmml" xref="S1.p5.48.m48.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.48.m48.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.48.m48.1d">caligraphic_T</annotation></semantics></math> contains <math alttext="W_{B}" class="ltx_Math" display="inline" id="S1.p5.49.m49.1"><semantics id="S1.p5.49.m49.1a"><msub id="S1.p5.49.m49.1.1" xref="S1.p5.49.m49.1.1.cmml"><mi id="S1.p5.49.m49.1.1.2" xref="S1.p5.49.m49.1.1.2.cmml">W</mi><mi id="S1.p5.49.m49.1.1.3" xref="S1.p5.49.m49.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p5.49.m49.1b"><apply id="S1.p5.49.m49.1.1.cmml" xref="S1.p5.49.m49.1.1"><csymbol cd="ambiguous" id="S1.p5.49.m49.1.1.1.cmml" xref="S1.p5.49.m49.1.1">subscript</csymbol><ci id="S1.p5.49.m49.1.1.2.cmml" xref="S1.p5.49.m49.1.1.2">𝑊</ci><ci id="S1.p5.49.m49.1.1.3.cmml" xref="S1.p5.49.m49.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.49.m49.1c">W_{B}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.49.m49.1d">italic_W start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math>. The algorithm finishes by inductively extending <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S1.p5.50.m50.1"><semantics id="S1.p5.50.m50.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p5.50.m50.1.1" xref="S1.p5.50.m50.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S1.p5.50.m50.1b"><ci id="S1.p5.50.m50.1.1.cmml" xref="S1.p5.50.m50.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.50.m50.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.50.m50.1d">caligraphic_T</annotation></semantics></math> to a tree decomposition of <math alttext="G_{B}=G" class="ltx_Math" display="inline" id="S1.p5.51.m51.1"><semantics id="S1.p5.51.m51.1a"><mrow id="S1.p5.51.m51.1.1" xref="S1.p5.51.m51.1.1.cmml"><msub id="S1.p5.51.m51.1.1.2" xref="S1.p5.51.m51.1.1.2.cmml"><mi id="S1.p5.51.m51.1.1.2.2" xref="S1.p5.51.m51.1.1.2.2.cmml">G</mi><mi id="S1.p5.51.m51.1.1.2.3" xref="S1.p5.51.m51.1.1.2.3.cmml">B</mi></msub><mo id="S1.p5.51.m51.1.1.1" xref="S1.p5.51.m51.1.1.1.cmml">=</mo><mi id="S1.p5.51.m51.1.1.3" xref="S1.p5.51.m51.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.51.m51.1b"><apply id="S1.p5.51.m51.1.1.cmml" xref="S1.p5.51.m51.1.1"><eq id="S1.p5.51.m51.1.1.1.cmml" xref="S1.p5.51.m51.1.1.1"></eq><apply id="S1.p5.51.m51.1.1.2.cmml" xref="S1.p5.51.m51.1.1.2"><csymbol cd="ambiguous" id="S1.p5.51.m51.1.1.2.1.cmml" xref="S1.p5.51.m51.1.1.2">subscript</csymbol><ci id="S1.p5.51.m51.1.1.2.2.cmml" xref="S1.p5.51.m51.1.1.2.2">𝐺</ci><ci id="S1.p5.51.m51.1.1.2.3.cmml" xref="S1.p5.51.m51.1.1.2.3">𝐵</ci></apply><ci id="S1.p5.51.m51.1.1.3.cmml" xref="S1.p5.51.m51.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.51.m51.1c">G_{B}=G</annotation><annotation encoding="application/x-llamapun" id="S1.p5.51.m51.1d">italic_G start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_G</annotation></semantics></math> of width less than <math alttext="4a" class="ltx_Math" display="inline" id="S1.p5.52.m52.1"><semantics id="S1.p5.52.m52.1a"><mrow id="S1.p5.52.m52.1.1" xref="S1.p5.52.m52.1.1.cmml"><mn id="S1.p5.52.m52.1.1.2" xref="S1.p5.52.m52.1.1.2.cmml">4</mn><mo id="S1.p5.52.m52.1.1.1" xref="S1.p5.52.m52.1.1.1.cmml">⁢</mo><mi id="S1.p5.52.m52.1.1.3" xref="S1.p5.52.m52.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.52.m52.1b"><apply id="S1.p5.52.m52.1.1.cmml" xref="S1.p5.52.m52.1.1"><times id="S1.p5.52.m52.1.1.1.cmml" xref="S1.p5.52.m52.1.1.1"></times><cn id="S1.p5.52.m52.1.1.2.cmml" type="integer" xref="S1.p5.52.m52.1.1.2">4</cn><ci id="S1.p5.52.m52.1.1.3.cmml" xref="S1.p5.52.m52.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.52.m52.1c">4a</annotation><annotation encoding="application/x-llamapun" id="S1.p5.52.m52.1d">4 italic_a</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.7">The challenge in establishing <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm1" title="Theorem 1. ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1</span></a> is that the balanced separations in the definition of separation number are only guaranteed to balance the entire set of vertices in an arbitrary subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mi id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><ci id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">italic_G</annotation></semantics></math>, rather than separating <math alttext="G" class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><mi id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><ci id="S1.p6.2.m2.1.1.cmml" xref="S1.p6.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">italic_G</annotation></semantics></math> in such a way the vertices in a specific <math alttext="W\subseteq V(G)" class="ltx_Math" display="inline" id="S1.p6.3.m3.1"><semantics id="S1.p6.3.m3.1a"><mrow id="S1.p6.3.m3.1.2" xref="S1.p6.3.m3.1.2.cmml"><mi id="S1.p6.3.m3.1.2.2" xref="S1.p6.3.m3.1.2.2.cmml">W</mi><mo id="S1.p6.3.m3.1.2.1" xref="S1.p6.3.m3.1.2.1.cmml">⊆</mo><mrow id="S1.p6.3.m3.1.2.3" xref="S1.p6.3.m3.1.2.3.cmml"><mi id="S1.p6.3.m3.1.2.3.2" xref="S1.p6.3.m3.1.2.3.2.cmml">V</mi><mo id="S1.p6.3.m3.1.2.3.1" xref="S1.p6.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S1.p6.3.m3.1.2.3.3.2" xref="S1.p6.3.m3.1.2.3.cmml"><mo id="S1.p6.3.m3.1.2.3.3.2.1" stretchy="false" xref="S1.p6.3.m3.1.2.3.cmml">(</mo><mi id="S1.p6.3.m3.1.1" xref="S1.p6.3.m3.1.1.cmml">G</mi><mo id="S1.p6.3.m3.1.2.3.3.2.2" stretchy="false" xref="S1.p6.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.3.m3.1b"><apply id="S1.p6.3.m3.1.2.cmml" xref="S1.p6.3.m3.1.2"><subset id="S1.p6.3.m3.1.2.1.cmml" xref="S1.p6.3.m3.1.2.1"></subset><ci id="S1.p6.3.m3.1.2.2.cmml" xref="S1.p6.3.m3.1.2.2">𝑊</ci><apply id="S1.p6.3.m3.1.2.3.cmml" xref="S1.p6.3.m3.1.2.3"><times id="S1.p6.3.m3.1.2.3.1.cmml" xref="S1.p6.3.m3.1.2.3.1"></times><ci id="S1.p6.3.m3.1.2.3.2.cmml" xref="S1.p6.3.m3.1.2.3.2">𝑉</ci><ci id="S1.p6.3.m3.1.1.cmml" xref="S1.p6.3.m3.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.3.m3.1c">W\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.3.m3.1d">italic_W ⊆ italic_V ( italic_G )</annotation></semantics></math> are balanced. In the language of the previous paragraph, there is no reason that a balanced separation <math alttext="(A,B)" class="ltx_Math" display="inline" id="S1.p6.4.m4.2"><semantics id="S1.p6.4.m4.2a"><mrow id="S1.p6.4.m4.2.3.2" xref="S1.p6.4.m4.2.3.1.cmml"><mo id="S1.p6.4.m4.2.3.2.1" stretchy="false" xref="S1.p6.4.m4.2.3.1.cmml">(</mo><mi id="S1.p6.4.m4.1.1" xref="S1.p6.4.m4.1.1.cmml">A</mi><mo id="S1.p6.4.m4.2.3.2.2" xref="S1.p6.4.m4.2.3.1.cmml">,</mo><mi id="S1.p6.4.m4.2.2" xref="S1.p6.4.m4.2.2.cmml">B</mi><mo id="S1.p6.4.m4.2.3.2.3" stretchy="false" xref="S1.p6.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.4.m4.2b"><interval closure="open" id="S1.p6.4.m4.2.3.1.cmml" xref="S1.p6.4.m4.2.3.2"><ci id="S1.p6.4.m4.1.1.cmml" xref="S1.p6.4.m4.1.1">𝐴</ci><ci id="S1.p6.4.m4.2.2.cmml" xref="S1.p6.4.m4.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.4.m4.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.4.m4.2d">( italic_A , italic_B )</annotation></semantics></math> of <math alttext="G[X]" class="ltx_Math" display="inline" id="S1.p6.5.m5.1"><semantics id="S1.p6.5.m5.1a"><mrow id="S1.p6.5.m5.1.2" xref="S1.p6.5.m5.1.2.cmml"><mi id="S1.p6.5.m5.1.2.2" xref="S1.p6.5.m5.1.2.2.cmml">G</mi><mo id="S1.p6.5.m5.1.2.1" xref="S1.p6.5.m5.1.2.1.cmml">⁢</mo><mrow id="S1.p6.5.m5.1.2.3.2" xref="S1.p6.5.m5.1.2.3.1.cmml"><mo id="S1.p6.5.m5.1.2.3.2.1" stretchy="false" xref="S1.p6.5.m5.1.2.3.1.1.cmml">[</mo><mi id="S1.p6.5.m5.1.1" xref="S1.p6.5.m5.1.1.cmml">X</mi><mo id="S1.p6.5.m5.1.2.3.2.2" stretchy="false" xref="S1.p6.5.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.5.m5.1b"><apply id="S1.p6.5.m5.1.2.cmml" xref="S1.p6.5.m5.1.2"><times id="S1.p6.5.m5.1.2.1.cmml" xref="S1.p6.5.m5.1.2.1"></times><ci id="S1.p6.5.m5.1.2.2.cmml" xref="S1.p6.5.m5.1.2.2">𝐺</ci><apply id="S1.p6.5.m5.1.2.3.1.cmml" xref="S1.p6.5.m5.1.2.3.2"><csymbol cd="latexml" id="S1.p6.5.m5.1.2.3.1.1.cmml" xref="S1.p6.5.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="S1.p6.5.m5.1.1.cmml" xref="S1.p6.5.m5.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.5.m5.1c">G[X]</annotation><annotation encoding="application/x-llamapun" id="S1.p6.5.m5.1d">italic_G [ italic_X ]</annotation></semantics></math> should have <math alttext="|(A\setminus B)\cap W|<|W|" class="ltx_Math" display="inline" id="S1.p6.6.m6.2"><semantics id="S1.p6.6.m6.2a"><mrow id="S1.p6.6.m6.2.2" xref="S1.p6.6.m6.2.2.cmml"><mrow id="S1.p6.6.m6.2.2.1.1" xref="S1.p6.6.m6.2.2.1.2.cmml"><mo id="S1.p6.6.m6.2.2.1.1.2" stretchy="false" xref="S1.p6.6.m6.2.2.1.2.1.cmml">|</mo><mrow id="S1.p6.6.m6.2.2.1.1.1" xref="S1.p6.6.m6.2.2.1.1.1.cmml"><mrow id="S1.p6.6.m6.2.2.1.1.1.1.1" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.cmml"><mo id="S1.p6.6.m6.2.2.1.1.1.1.1.2" stretchy="false" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p6.6.m6.2.2.1.1.1.1.1.1" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.cmml"><mi id="S1.p6.6.m6.2.2.1.1.1.1.1.1.2" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.2.cmml">A</mi><mo id="S1.p6.6.m6.2.2.1.1.1.1.1.1.1" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S1.p6.6.m6.2.2.1.1.1.1.1.1.3" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S1.p6.6.m6.2.2.1.1.1.1.1.3" stretchy="false" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p6.6.m6.2.2.1.1.1.2" xref="S1.p6.6.m6.2.2.1.1.1.2.cmml">∩</mo><mi id="S1.p6.6.m6.2.2.1.1.1.3" xref="S1.p6.6.m6.2.2.1.1.1.3.cmml">W</mi></mrow><mo id="S1.p6.6.m6.2.2.1.1.3" stretchy="false" xref="S1.p6.6.m6.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p6.6.m6.2.2.2" xref="S1.p6.6.m6.2.2.2.cmml"><</mo><mrow id="S1.p6.6.m6.2.2.3.2" xref="S1.p6.6.m6.2.2.3.1.cmml"><mo id="S1.p6.6.m6.2.2.3.2.1" stretchy="false" xref="S1.p6.6.m6.2.2.3.1.1.cmml">|</mo><mi id="S1.p6.6.m6.1.1" xref="S1.p6.6.m6.1.1.cmml">W</mi><mo id="S1.p6.6.m6.2.2.3.2.2" stretchy="false" xref="S1.p6.6.m6.2.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.6.m6.2b"><apply id="S1.p6.6.m6.2.2.cmml" xref="S1.p6.6.m6.2.2"><lt id="S1.p6.6.m6.2.2.2.cmml" xref="S1.p6.6.m6.2.2.2"></lt><apply id="S1.p6.6.m6.2.2.1.2.cmml" xref="S1.p6.6.m6.2.2.1.1"><abs id="S1.p6.6.m6.2.2.1.2.1.cmml" xref="S1.p6.6.m6.2.2.1.1.2"></abs><apply id="S1.p6.6.m6.2.2.1.1.1.cmml" xref="S1.p6.6.m6.2.2.1.1.1"><intersect id="S1.p6.6.m6.2.2.1.1.1.2.cmml" xref="S1.p6.6.m6.2.2.1.1.1.2"></intersect><apply id="S1.p6.6.m6.2.2.1.1.1.1.1.1.cmml" xref="S1.p6.6.m6.2.2.1.1.1.1.1"><setdiff id="S1.p6.6.m6.2.2.1.1.1.1.1.1.1.cmml" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.1"></setdiff><ci id="S1.p6.6.m6.2.2.1.1.1.1.1.1.2.cmml" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.2">𝐴</ci><ci id="S1.p6.6.m6.2.2.1.1.1.1.1.1.3.cmml" xref="S1.p6.6.m6.2.2.1.1.1.1.1.1.3">𝐵</ci></apply><ci id="S1.p6.6.m6.2.2.1.1.1.3.cmml" xref="S1.p6.6.m6.2.2.1.1.1.3">𝑊</ci></apply></apply><apply id="S1.p6.6.m6.2.2.3.1.cmml" xref="S1.p6.6.m6.2.2.3.2"><abs id="S1.p6.6.m6.2.2.3.1.1.cmml" xref="S1.p6.6.m6.2.2.3.2.1"></abs><ci id="S1.p6.6.m6.1.1.cmml" xref="S1.p6.6.m6.1.1">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.6.m6.2c">|(A\setminus B)\cap W|<|W|</annotation><annotation encoding="application/x-llamapun" id="S1.p6.6.m6.2d">| ( italic_A ∖ italic_B ) ∩ italic_W | < | italic_W |</annotation></semantics></math> and <math alttext="|(B\setminus A)\cap W|<|W|" class="ltx_Math" display="inline" id="S1.p6.7.m7.2"><semantics id="S1.p6.7.m7.2a"><mrow id="S1.p6.7.m7.2.2" xref="S1.p6.7.m7.2.2.cmml"><mrow id="S1.p6.7.m7.2.2.1.1" xref="S1.p6.7.m7.2.2.1.2.cmml"><mo id="S1.p6.7.m7.2.2.1.1.2" stretchy="false" xref="S1.p6.7.m7.2.2.1.2.1.cmml">|</mo><mrow id="S1.p6.7.m7.2.2.1.1.1" xref="S1.p6.7.m7.2.2.1.1.1.cmml"><mrow id="S1.p6.7.m7.2.2.1.1.1.1.1" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.cmml"><mo id="S1.p6.7.m7.2.2.1.1.1.1.1.2" stretchy="false" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p6.7.m7.2.2.1.1.1.1.1.1" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.cmml"><mi id="S1.p6.7.m7.2.2.1.1.1.1.1.1.2" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.2.cmml">B</mi><mo id="S1.p6.7.m7.2.2.1.1.1.1.1.1.1" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S1.p6.7.m7.2.2.1.1.1.1.1.1.3" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.3.cmml">A</mi></mrow><mo id="S1.p6.7.m7.2.2.1.1.1.1.1.3" stretchy="false" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p6.7.m7.2.2.1.1.1.2" xref="S1.p6.7.m7.2.2.1.1.1.2.cmml">∩</mo><mi id="S1.p6.7.m7.2.2.1.1.1.3" xref="S1.p6.7.m7.2.2.1.1.1.3.cmml">W</mi></mrow><mo id="S1.p6.7.m7.2.2.1.1.3" stretchy="false" xref="S1.p6.7.m7.2.2.1.2.1.cmml">|</mo></mrow><mo id="S1.p6.7.m7.2.2.2" xref="S1.p6.7.m7.2.2.2.cmml"><</mo><mrow id="S1.p6.7.m7.2.2.3.2" xref="S1.p6.7.m7.2.2.3.1.cmml"><mo id="S1.p6.7.m7.2.2.3.2.1" stretchy="false" xref="S1.p6.7.m7.2.2.3.1.1.cmml">|</mo><mi id="S1.p6.7.m7.1.1" xref="S1.p6.7.m7.1.1.cmml">W</mi><mo id="S1.p6.7.m7.2.2.3.2.2" stretchy="false" xref="S1.p6.7.m7.2.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.7.m7.2b"><apply id="S1.p6.7.m7.2.2.cmml" xref="S1.p6.7.m7.2.2"><lt id="S1.p6.7.m7.2.2.2.cmml" xref="S1.p6.7.m7.2.2.2"></lt><apply id="S1.p6.7.m7.2.2.1.2.cmml" xref="S1.p6.7.m7.2.2.1.1"><abs id="S1.p6.7.m7.2.2.1.2.1.cmml" xref="S1.p6.7.m7.2.2.1.1.2"></abs><apply id="S1.p6.7.m7.2.2.1.1.1.cmml" xref="S1.p6.7.m7.2.2.1.1.1"><intersect id="S1.p6.7.m7.2.2.1.1.1.2.cmml" xref="S1.p6.7.m7.2.2.1.1.1.2"></intersect><apply id="S1.p6.7.m7.2.2.1.1.1.1.1.1.cmml" xref="S1.p6.7.m7.2.2.1.1.1.1.1"><setdiff id="S1.p6.7.m7.2.2.1.1.1.1.1.1.1.cmml" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.1"></setdiff><ci id="S1.p6.7.m7.2.2.1.1.1.1.1.1.2.cmml" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.2">𝐵</ci><ci id="S1.p6.7.m7.2.2.1.1.1.1.1.1.3.cmml" xref="S1.p6.7.m7.2.2.1.1.1.1.1.1.3">𝐴</ci></apply><ci id="S1.p6.7.m7.2.2.1.1.1.3.cmml" xref="S1.p6.7.m7.2.2.1.1.1.3">𝑊</ci></apply></apply><apply id="S1.p6.7.m7.2.2.3.1.cmml" xref="S1.p6.7.m7.2.2.3.2"><abs id="S1.p6.7.m7.2.2.3.1.1.cmml" xref="S1.p6.7.m7.2.2.3.2.1"></abs><ci id="S1.p6.7.m7.1.1.cmml" xref="S1.p6.7.m7.1.1">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.7.m7.2c">|(B\setminus A)\cap W|<|W|</annotation><annotation encoding="application/x-llamapun" id="S1.p6.7.m7.2d">| ( italic_B ∖ italic_A ) ∩ italic_W | < | italic_W |</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.11"><cite class="ltx_cite ltx_citemacro_citet">Dvořák and Norin [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib4" title="">4</a>]</cite> prove <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm1" title="Theorem 1. ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1</span></a> with the constant <math alttext="c=15" class="ltx_Math" display="inline" id="S1.p7.1.m1.1"><semantics id="S1.p7.1.m1.1a"><mrow id="S1.p7.1.m1.1.1" xref="S1.p7.1.m1.1.1.cmml"><mi id="S1.p7.1.m1.1.1.2" xref="S1.p7.1.m1.1.1.2.cmml">c</mi><mo id="S1.p7.1.m1.1.1.1" xref="S1.p7.1.m1.1.1.1.cmml">=</mo><mn id="S1.p7.1.m1.1.1.3" xref="S1.p7.1.m1.1.1.3.cmml">15</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.1.m1.1b"><apply id="S1.p7.1.m1.1.1.cmml" xref="S1.p7.1.m1.1.1"><eq id="S1.p7.1.m1.1.1.1.cmml" xref="S1.p7.1.m1.1.1.1"></eq><ci id="S1.p7.1.m1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.2">𝑐</ci><cn id="S1.p7.1.m1.1.1.3.cmml" type="integer" xref="S1.p7.1.m1.1.1.3">15</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.1.m1.1c">c=15</annotation><annotation encoding="application/x-llamapun" id="S1.p7.1.m1.1d">italic_c = 15</annotation></semantics></math>. Their proof is by contradiction and makes use of the relationship between treewidth and brambles established by <cite class="ltx_cite ltx_citemacro_citet">Seymour and Thomas [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib6" title="">6</a>]</cite>. Essentially, they show that if <math alttext="\operatorname{tw}(G)>15\operatorname{sn}(G)" class="ltx_Math" display="inline" id="S1.p7.2.m2.4"><semantics id="S1.p7.2.m2.4a"><mrow id="S1.p7.2.m2.4.5" xref="S1.p7.2.m2.4.5.cmml"><mrow id="S1.p7.2.m2.4.5.2.2" xref="S1.p7.2.m2.4.5.2.1.cmml"><mi id="S1.p7.2.m2.1.1" xref="S1.p7.2.m2.1.1.cmml">tw</mi><mo id="S1.p7.2.m2.4.5.2.2a" xref="S1.p7.2.m2.4.5.2.1.cmml">⁡</mo><mrow id="S1.p7.2.m2.4.5.2.2.1" xref="S1.p7.2.m2.4.5.2.1.cmml"><mo id="S1.p7.2.m2.4.5.2.2.1.1" stretchy="false" xref="S1.p7.2.m2.4.5.2.1.cmml">(</mo><mi id="S1.p7.2.m2.2.2" xref="S1.p7.2.m2.2.2.cmml">G</mi><mo id="S1.p7.2.m2.4.5.2.2.1.2" stretchy="false" xref="S1.p7.2.m2.4.5.2.1.cmml">)</mo></mrow></mrow><mo id="S1.p7.2.m2.4.5.1" xref="S1.p7.2.m2.4.5.1.cmml">></mo><mrow id="S1.p7.2.m2.4.5.3" xref="S1.p7.2.m2.4.5.3.cmml"><mn id="S1.p7.2.m2.4.5.3.2" xref="S1.p7.2.m2.4.5.3.2.cmml">15</mn><mo id="S1.p7.2.m2.4.5.3.1" lspace="0.167em" xref="S1.p7.2.m2.4.5.3.1.cmml">⁢</mo><mrow id="S1.p7.2.m2.4.5.3.3.2" xref="S1.p7.2.m2.4.5.3.3.1.cmml"><mi id="S1.p7.2.m2.3.3" xref="S1.p7.2.m2.3.3.cmml">sn</mi><mo id="S1.p7.2.m2.4.5.3.3.2a" xref="S1.p7.2.m2.4.5.3.3.1.cmml">⁡</mo><mrow id="S1.p7.2.m2.4.5.3.3.2.1" xref="S1.p7.2.m2.4.5.3.3.1.cmml"><mo id="S1.p7.2.m2.4.5.3.3.2.1.1" stretchy="false" xref="S1.p7.2.m2.4.5.3.3.1.cmml">(</mo><mi id="S1.p7.2.m2.4.4" xref="S1.p7.2.m2.4.4.cmml">G</mi><mo id="S1.p7.2.m2.4.5.3.3.2.1.2" stretchy="false" xref="S1.p7.2.m2.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.2.m2.4b"><apply id="S1.p7.2.m2.4.5.cmml" xref="S1.p7.2.m2.4.5"><gt id="S1.p7.2.m2.4.5.1.cmml" xref="S1.p7.2.m2.4.5.1"></gt><apply id="S1.p7.2.m2.4.5.2.1.cmml" xref="S1.p7.2.m2.4.5.2.2"><ci id="S1.p7.2.m2.1.1.cmml" xref="S1.p7.2.m2.1.1">tw</ci><ci id="S1.p7.2.m2.2.2.cmml" xref="S1.p7.2.m2.2.2">𝐺</ci></apply><apply id="S1.p7.2.m2.4.5.3.cmml" xref="S1.p7.2.m2.4.5.3"><times id="S1.p7.2.m2.4.5.3.1.cmml" xref="S1.p7.2.m2.4.5.3.1"></times><cn id="S1.p7.2.m2.4.5.3.2.cmml" type="integer" xref="S1.p7.2.m2.4.5.3.2">15</cn><apply id="S1.p7.2.m2.4.5.3.3.1.cmml" xref="S1.p7.2.m2.4.5.3.3.2"><ci id="S1.p7.2.m2.3.3.cmml" xref="S1.p7.2.m2.3.3">sn</ci><ci id="S1.p7.2.m2.4.4.cmml" xref="S1.p7.2.m2.4.4">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.2.m2.4c">\operatorname{tw}(G)>15\operatorname{sn}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p7.2.m2.4d">roman_tw ( italic_G ) > 15 roman_sn ( italic_G )</annotation></semantics></math>, then there exists an <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p7.3.m3.1"><semantics id="S1.p7.3.m3.1a"><mi id="S1.p7.3.m3.1.1" xref="S1.p7.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p7.3.m3.1b"><ci id="S1.p7.3.m3.1.1.cmml" xref="S1.p7.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p7.3.m3.1d">italic_α</annotation></semantics></math>-tame <math alttext="W" class="ltx_Math" display="inline" id="S1.p7.4.m4.1"><semantics id="S1.p7.4.m4.1a"><mi id="S1.p7.4.m4.1.1" xref="S1.p7.4.m4.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.p7.4.m4.1b"><ci id="S1.p7.4.m4.1.1.cmml" xref="S1.p7.4.m4.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.4.m4.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.p7.4.m4.1d">italic_W</annotation></semantics></math>-cloud (a special kind of network flow) which contradicts the choice of a haven (a special kind of flap assignment) that is derived from a bramble of order <math alttext="15\operatorname{sn}(G)" class="ltx_Math" display="inline" id="S1.p7.5.m5.2"><semantics id="S1.p7.5.m5.2a"><mrow id="S1.p7.5.m5.2.3" xref="S1.p7.5.m5.2.3.cmml"><mn id="S1.p7.5.m5.2.3.2" xref="S1.p7.5.m5.2.3.2.cmml">15</mn><mo id="S1.p7.5.m5.2.3.1" lspace="0.167em" xref="S1.p7.5.m5.2.3.1.cmml">⁢</mo><mrow id="S1.p7.5.m5.2.3.3.2" xref="S1.p7.5.m5.2.3.3.1.cmml"><mi id="S1.p7.5.m5.1.1" xref="S1.p7.5.m5.1.1.cmml">sn</mi><mo id="S1.p7.5.m5.2.3.3.2a" xref="S1.p7.5.m5.2.3.3.1.cmml">⁡</mo><mrow id="S1.p7.5.m5.2.3.3.2.1" xref="S1.p7.5.m5.2.3.3.1.cmml"><mo id="S1.p7.5.m5.2.3.3.2.1.1" stretchy="false" xref="S1.p7.5.m5.2.3.3.1.cmml">(</mo><mi id="S1.p7.5.m5.2.2" xref="S1.p7.5.m5.2.2.cmml">G</mi><mo id="S1.p7.5.m5.2.3.3.2.1.2" stretchy="false" xref="S1.p7.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.5.m5.2b"><apply id="S1.p7.5.m5.2.3.cmml" xref="S1.p7.5.m5.2.3"><times id="S1.p7.5.m5.2.3.1.cmml" xref="S1.p7.5.m5.2.3.1"></times><cn id="S1.p7.5.m5.2.3.2.cmml" type="integer" xref="S1.p7.5.m5.2.3.2">15</cn><apply id="S1.p7.5.m5.2.3.3.1.cmml" xref="S1.p7.5.m5.2.3.3.2"><ci id="S1.p7.5.m5.1.1.cmml" xref="S1.p7.5.m5.1.1">sn</ci><ci id="S1.p7.5.m5.2.2.cmml" xref="S1.p7.5.m5.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.5.m5.2c">15\operatorname{sn}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p7.5.m5.2d">15 roman_sn ( italic_G )</annotation></semantics></math>. The crux of their proof <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib4" title="">4</a>, Proof of Lemma 7]</cite> involves showing that, for a <em class="ltx_emph ltx_font_italic" id="S1.p7.11.1">carefully chosen</em> <math alttext="W\subseteq V(G)" class="ltx_Math" display="inline" id="S1.p7.6.m6.1"><semantics id="S1.p7.6.m6.1a"><mrow id="S1.p7.6.m6.1.2" xref="S1.p7.6.m6.1.2.cmml"><mi id="S1.p7.6.m6.1.2.2" xref="S1.p7.6.m6.1.2.2.cmml">W</mi><mo id="S1.p7.6.m6.1.2.1" xref="S1.p7.6.m6.1.2.1.cmml">⊆</mo><mrow id="S1.p7.6.m6.1.2.3" xref="S1.p7.6.m6.1.2.3.cmml"><mi id="S1.p7.6.m6.1.2.3.2" xref="S1.p7.6.m6.1.2.3.2.cmml">V</mi><mo id="S1.p7.6.m6.1.2.3.1" xref="S1.p7.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S1.p7.6.m6.1.2.3.3.2" xref="S1.p7.6.m6.1.2.3.cmml"><mo id="S1.p7.6.m6.1.2.3.3.2.1" stretchy="false" xref="S1.p7.6.m6.1.2.3.cmml">(</mo><mi id="S1.p7.6.m6.1.1" xref="S1.p7.6.m6.1.1.cmml">G</mi><mo id="S1.p7.6.m6.1.2.3.3.2.2" stretchy="false" xref="S1.p7.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.6.m6.1b"><apply id="S1.p7.6.m6.1.2.cmml" xref="S1.p7.6.m6.1.2"><subset id="S1.p7.6.m6.1.2.1.cmml" xref="S1.p7.6.m6.1.2.1"></subset><ci id="S1.p7.6.m6.1.2.2.cmml" xref="S1.p7.6.m6.1.2.2">𝑊</ci><apply id="S1.p7.6.m6.1.2.3.cmml" xref="S1.p7.6.m6.1.2.3"><times id="S1.p7.6.m6.1.2.3.1.cmml" xref="S1.p7.6.m6.1.2.3.1"></times><ci id="S1.p7.6.m6.1.2.3.2.cmml" xref="S1.p7.6.m6.1.2.3.2">𝑉</ci><ci id="S1.p7.6.m6.1.1.cmml" xref="S1.p7.6.m6.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.6.m6.1c">W\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p7.6.m6.1d">italic_W ⊆ italic_V ( italic_G )</annotation></semantics></math>, a balanced separation of the subgraph <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S1.p7.7.m7.1"><semantics id="S1.p7.7.m7.1a"><mrow id="S1.p7.7.m7.1.1" xref="S1.p7.7.m7.1.1.cmml"><mi id="S1.p7.7.m7.1.1.2" xref="S1.p7.7.m7.1.1.2.cmml">H</mi><mo id="S1.p7.7.m7.1.1.1" xref="S1.p7.7.m7.1.1.1.cmml">⊆</mo><mi id="S1.p7.7.m7.1.1.3" xref="S1.p7.7.m7.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.7.m7.1b"><apply id="S1.p7.7.m7.1.1.cmml" xref="S1.p7.7.m7.1.1"><subset id="S1.p7.7.m7.1.1.1.cmml" xref="S1.p7.7.m7.1.1.1"></subset><ci id="S1.p7.7.m7.1.1.2.cmml" xref="S1.p7.7.m7.1.1.2">𝐻</ci><ci id="S1.p7.7.m7.1.1.3.cmml" xref="S1.p7.7.m7.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.7.m7.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S1.p7.7.m7.1d">italic_H ⊆ italic_G</annotation></semantics></math> induced by the saturated and hungry vertices of an <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p7.8.m8.1"><semantics id="S1.p7.8.m8.1a"><mi id="S1.p7.8.m8.1.1" xref="S1.p7.8.m8.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p7.8.m8.1b"><ci id="S1.p7.8.m8.1.1.cmml" xref="S1.p7.8.m8.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.8.m8.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p7.8.m8.1d">italic_α</annotation></semantics></math>-tame <math alttext="W" class="ltx_Math" display="inline" id="S1.p7.9.m9.1"><semantics id="S1.p7.9.m9.1a"><mi id="S1.p7.9.m9.1.1" xref="S1.p7.9.m9.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.p7.9.m9.1b"><ci id="S1.p7.9.m9.1.1.cmml" xref="S1.p7.9.m9.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.9.m9.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.p7.9.m9.1d">italic_W</annotation></semantics></math>-cloud is, by necessity, also (rougly) <math alttext="W" class="ltx_Math" display="inline" id="S1.p7.10.m10.1"><semantics id="S1.p7.10.m10.1a"><mi id="S1.p7.10.m10.1.1" xref="S1.p7.10.m10.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.p7.10.m10.1b"><ci id="S1.p7.10.m10.1.1.cmml" xref="S1.p7.10.m10.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.10.m10.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.p7.10.m10.1d">italic_W</annotation></semantics></math>-balanced. This leads to a contradiction related to the choice of <math alttext="W" class="ltx_Math" display="inline" id="S1.p7.11.m11.1"><semantics id="S1.p7.11.m11.1a"><mi id="S1.p7.11.m11.1.1" xref="S1.p7.11.m11.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.p7.11.m11.1b"><ci id="S1.p7.11.m11.1.1.cmml" xref="S1.p7.11.m11.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.11.m11.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.p7.11.m11.1d">italic_W</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>In an earlier draft of their result, <cite class="ltx_cite ltx_citemacro_citet">Dvorák and Norin [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib3" title="">3</a>]</cite>, used tangles rather than brambles and havens, and confluent flows <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib1" title="">1</a>]</cite> rather than <math alttext="W" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">W</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_W</annotation></semantics></math>-clouds to establish <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm1" title="Theorem 1. ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1</span></a> with the constant <math alttext="c=105" class="ltx_Math" display="inline" id="footnote2.m2.1"><semantics id="footnote2.m2.1b"><mrow id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml"><mi id="footnote2.m2.1.1.2" xref="footnote2.m2.1.1.2.cmml">c</mi><mo id="footnote2.m2.1.1.1" xref="footnote2.m2.1.1.1.cmml">=</mo><mn id="footnote2.m2.1.1.3" xref="footnote2.m2.1.1.3.cmml">105</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m2.1c"><apply id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1"><eq id="footnote2.m2.1.1.1.cmml" xref="footnote2.m2.1.1.1"></eq><ci id="footnote2.m2.1.1.2.cmml" xref="footnote2.m2.1.1.2">𝑐</ci><cn id="footnote2.m2.1.1.3.cmml" type="integer" xref="footnote2.m2.1.1.3">105</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.1d">c=105</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.1e">italic_c = 105</annotation></semantics></math>. They credit an anonymous referee for help in reducing the value of <math alttext="c" class="ltx_Math" display="inline" id="footnote2.m3.1"><semantics id="footnote2.m3.1b"><mi id="footnote2.m3.1.1" xref="footnote2.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="footnote2.m3.1c"><ci id="footnote2.m3.1.1.cmml" xref="footnote2.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m3.1d">c</annotation><annotation encoding="application/x-llamapun" id="footnote2.m3.1e">italic_c</annotation></semantics></math>.</span></span></span></p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.4">In the current paper, we prove <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm1" title="Theorem 1. ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1</span></a> with the constant <math alttext="c=7915/139<56.943" class="ltx_Math" display="inline" id="S1.p8.1.m1.1"><semantics id="S1.p8.1.m1.1a"><mrow id="S1.p8.1.m1.1.1" xref="S1.p8.1.m1.1.1.cmml"><mi id="S1.p8.1.m1.1.1.2" xref="S1.p8.1.m1.1.1.2.cmml">c</mi><mo id="S1.p8.1.m1.1.1.3" xref="S1.p8.1.m1.1.1.3.cmml">=</mo><mrow id="S1.p8.1.m1.1.1.4" xref="S1.p8.1.m1.1.1.4.cmml"><mn id="S1.p8.1.m1.1.1.4.2" xref="S1.p8.1.m1.1.1.4.2.cmml">7915</mn><mo id="S1.p8.1.m1.1.1.4.1" xref="S1.p8.1.m1.1.1.4.1.cmml">/</mo><mn id="S1.p8.1.m1.1.1.4.3" xref="S1.p8.1.m1.1.1.4.3.cmml">139</mn></mrow><mo id="S1.p8.1.m1.1.1.5" xref="S1.p8.1.m1.1.1.5.cmml"><</mo><mn id="S1.p8.1.m1.1.1.6" xref="S1.p8.1.m1.1.1.6.cmml">56.943</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.1.m1.1b"><apply id="S1.p8.1.m1.1.1.cmml" xref="S1.p8.1.m1.1.1"><and id="S1.p8.1.m1.1.1a.cmml" xref="S1.p8.1.m1.1.1"></and><apply id="S1.p8.1.m1.1.1b.cmml" xref="S1.p8.1.m1.1.1"><eq id="S1.p8.1.m1.1.1.3.cmml" xref="S1.p8.1.m1.1.1.3"></eq><ci id="S1.p8.1.m1.1.1.2.cmml" xref="S1.p8.1.m1.1.1.2">𝑐</ci><apply id="S1.p8.1.m1.1.1.4.cmml" xref="S1.p8.1.m1.1.1.4"><divide id="S1.p8.1.m1.1.1.4.1.cmml" xref="S1.p8.1.m1.1.1.4.1"></divide><cn id="S1.p8.1.m1.1.1.4.2.cmml" type="integer" xref="S1.p8.1.m1.1.1.4.2">7915</cn><cn id="S1.p8.1.m1.1.1.4.3.cmml" type="integer" xref="S1.p8.1.m1.1.1.4.3">139</cn></apply></apply><apply id="S1.p8.1.m1.1.1c.cmml" xref="S1.p8.1.m1.1.1"><lt id="S1.p8.1.m1.1.1.5.cmml" xref="S1.p8.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2503.17112v1#S1.p8.1.m1.1.1.4.cmml" id="S1.p8.1.m1.1.1d.cmml" xref="S1.p8.1.m1.1.1"></share><cn id="S1.p8.1.m1.1.1.6.cmml" type="float" xref="S1.p8.1.m1.1.1.6">56.943</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.1.m1.1c">c=7915/139<56.943</annotation><annotation encoding="application/x-llamapun" id="S1.p8.1.m1.1d">italic_c = 7915 / 139 < 56.943</annotation></semantics></math>. Despite the larger constant, we believe that the proof given here has a number of advantages. The proof is constructive: It proves that <math alttext="\operatorname{tw}(G)\leq c\cdot\operatorname{sn}(G)" class="ltx_Math" display="inline" id="S1.p8.2.m2.4"><semantics id="S1.p8.2.m2.4a"><mrow id="S1.p8.2.m2.4.5" xref="S1.p8.2.m2.4.5.cmml"><mrow id="S1.p8.2.m2.4.5.2.2" xref="S1.p8.2.m2.4.5.2.1.cmml"><mi id="S1.p8.2.m2.1.1" xref="S1.p8.2.m2.1.1.cmml">tw</mi><mo id="S1.p8.2.m2.4.5.2.2a" xref="S1.p8.2.m2.4.5.2.1.cmml">⁡</mo><mrow id="S1.p8.2.m2.4.5.2.2.1" xref="S1.p8.2.m2.4.5.2.1.cmml"><mo id="S1.p8.2.m2.4.5.2.2.1.1" stretchy="false" xref="S1.p8.2.m2.4.5.2.1.cmml">(</mo><mi id="S1.p8.2.m2.2.2" xref="S1.p8.2.m2.2.2.cmml">G</mi><mo id="S1.p8.2.m2.4.5.2.2.1.2" stretchy="false" xref="S1.p8.2.m2.4.5.2.1.cmml">)</mo></mrow></mrow><mo id="S1.p8.2.m2.4.5.1" xref="S1.p8.2.m2.4.5.1.cmml">≤</mo><mrow id="S1.p8.2.m2.4.5.3" xref="S1.p8.2.m2.4.5.3.cmml"><mi id="S1.p8.2.m2.4.5.3.2" xref="S1.p8.2.m2.4.5.3.2.cmml">c</mi><mo id="S1.p8.2.m2.4.5.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p8.2.m2.4.5.3.1.cmml">⋅</mo><mrow id="S1.p8.2.m2.4.5.3.3.2" xref="S1.p8.2.m2.4.5.3.3.1.cmml"><mi id="S1.p8.2.m2.3.3" xref="S1.p8.2.m2.3.3.cmml">sn</mi><mo id="S1.p8.2.m2.4.5.3.3.2a" xref="S1.p8.2.m2.4.5.3.3.1.cmml">⁡</mo><mrow id="S1.p8.2.m2.4.5.3.3.2.1" xref="S1.p8.2.m2.4.5.3.3.1.cmml"><mo id="S1.p8.2.m2.4.5.3.3.2.1.1" stretchy="false" xref="S1.p8.2.m2.4.5.3.3.1.cmml">(</mo><mi id="S1.p8.2.m2.4.4" xref="S1.p8.2.m2.4.4.cmml">G</mi><mo id="S1.p8.2.m2.4.5.3.3.2.1.2" stretchy="false" xref="S1.p8.2.m2.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.2.m2.4b"><apply id="S1.p8.2.m2.4.5.cmml" xref="S1.p8.2.m2.4.5"><leq id="S1.p8.2.m2.4.5.1.cmml" xref="S1.p8.2.m2.4.5.1"></leq><apply id="S1.p8.2.m2.4.5.2.1.cmml" xref="S1.p8.2.m2.4.5.2.2"><ci id="S1.p8.2.m2.1.1.cmml" xref="S1.p8.2.m2.1.1">tw</ci><ci id="S1.p8.2.m2.2.2.cmml" xref="S1.p8.2.m2.2.2">𝐺</ci></apply><apply id="S1.p8.2.m2.4.5.3.cmml" xref="S1.p8.2.m2.4.5.3"><ci id="S1.p8.2.m2.4.5.3.1.cmml" xref="S1.p8.2.m2.4.5.3.1">⋅</ci><ci id="S1.p8.2.m2.4.5.3.2.cmml" xref="S1.p8.2.m2.4.5.3.2">𝑐</ci><apply id="S1.p8.2.m2.4.5.3.3.1.cmml" xref="S1.p8.2.m2.4.5.3.3.2"><ci id="S1.p8.2.m2.3.3.cmml" xref="S1.p8.2.m2.3.3">sn</ci><ci id="S1.p8.2.m2.4.4.cmml" xref="S1.p8.2.m2.4.4">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.2.m2.4c">\operatorname{tw}(G)\leq c\cdot\operatorname{sn}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p8.2.m2.4d">roman_tw ( italic_G ) ≤ italic_c ⋅ roman_sn ( italic_G )</annotation></semantics></math> by constructing a tree decomposition of <math alttext="G" class="ltx_Math" display="inline" id="S1.p8.3.m3.1"><semantics id="S1.p8.3.m3.1a"><mi id="S1.p8.3.m3.1.1" xref="S1.p8.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p8.3.m3.1b"><ci id="S1.p8.3.m3.1.1.cmml" xref="S1.p8.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p8.3.m3.1d">italic_G</annotation></semantics></math> having width less than <math alttext="c\cdot\operatorname{sn}(G)" class="ltx_Math" display="inline" id="S1.p8.4.m4.2"><semantics id="S1.p8.4.m4.2a"><mrow id="S1.p8.4.m4.2.3" xref="S1.p8.4.m4.2.3.cmml"><mi id="S1.p8.4.m4.2.3.2" xref="S1.p8.4.m4.2.3.2.cmml">c</mi><mo id="S1.p8.4.m4.2.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p8.4.m4.2.3.1.cmml">⋅</mo><mrow id="S1.p8.4.m4.2.3.3.2" xref="S1.p8.4.m4.2.3.3.1.cmml"><mi id="S1.p8.4.m4.1.1" xref="S1.p8.4.m4.1.1.cmml">sn</mi><mo id="S1.p8.4.m4.2.3.3.2a" xref="S1.p8.4.m4.2.3.3.1.cmml">⁡</mo><mrow id="S1.p8.4.m4.2.3.3.2.1" xref="S1.p8.4.m4.2.3.3.1.cmml"><mo id="S1.p8.4.m4.2.3.3.2.1.1" stretchy="false" xref="S1.p8.4.m4.2.3.3.1.cmml">(</mo><mi id="S1.p8.4.m4.2.2" xref="S1.p8.4.m4.2.2.cmml">G</mi><mo id="S1.p8.4.m4.2.3.3.2.1.2" stretchy="false" xref="S1.p8.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.4.m4.2b"><apply id="S1.p8.4.m4.2.3.cmml" xref="S1.p8.4.m4.2.3"><ci id="S1.p8.4.m4.2.3.1.cmml" xref="S1.p8.4.m4.2.3.1">⋅</ci><ci id="S1.p8.4.m4.2.3.2.cmml" xref="S1.p8.4.m4.2.3.2">𝑐</ci><apply id="S1.p8.4.m4.2.3.3.1.cmml" xref="S1.p8.4.m4.2.3.3.2"><ci id="S1.p8.4.m4.1.1.cmml" xref="S1.p8.4.m4.1.1">sn</ci><ci id="S1.p8.4.m4.2.2.cmml" xref="S1.p8.4.m4.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.4.m4.2c">c\cdot\operatorname{sn}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p8.4.m4.2d">italic_c ⋅ roman_sn ( italic_G )</annotation></semantics></math>. The proof requires fewer definitions and previous results: It does not use brambles, havens, or network flows. Brambles and havens are avoided entirely. The use of network flows is replaced by a collection of paths obtained from repeated applications of the simplest version of Menger’s Theorem on vertex-disjoint paths in (unweighted undirected) graphs.</p> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.30">Most importantly, the proof given here is built around a generalization of <math alttext="W" class="ltx_Math" display="inline" id="S1.p9.1.m1.1"><semantics id="S1.p9.1.m1.1a"><mi id="S1.p9.1.m1.1.1" xref="S1.p9.1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.p9.1.m1.1b"><ci id="S1.p9.1.m1.1.1.cmml" xref="S1.p9.1.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.1.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.p9.1.m1.1d">italic_W</annotation></semantics></math>-balanced separations: For a sufficiently large <math alttext="t>0" class="ltx_Math" display="inline" id="S1.p9.2.m2.1"><semantics id="S1.p9.2.m2.1a"><mrow id="S1.p9.2.m2.1.1" xref="S1.p9.2.m2.1.1.cmml"><mi id="S1.p9.2.m2.1.1.2" xref="S1.p9.2.m2.1.1.2.cmml">t</mi><mo id="S1.p9.2.m2.1.1.1" xref="S1.p9.2.m2.1.1.1.cmml">></mo><mn id="S1.p9.2.m2.1.1.3" xref="S1.p9.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.2.m2.1b"><apply id="S1.p9.2.m2.1.1.cmml" xref="S1.p9.2.m2.1.1"><gt id="S1.p9.2.m2.1.1.1.cmml" xref="S1.p9.2.m2.1.1.1"></gt><ci id="S1.p9.2.m2.1.1.2.cmml" xref="S1.p9.2.m2.1.1.2">𝑡</ci><cn id="S1.p9.2.m2.1.1.3.cmml" type="integer" xref="S1.p9.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.2.m2.1c">t>0</annotation><annotation encoding="application/x-llamapun" id="S1.p9.2.m2.1d">italic_t > 0</annotation></semantics></math> and an <em class="ltx_emph ltx_font_italic" id="S1.p9.30.1">arbitrary</em> <math alttext="W\subseteq V(G)" class="ltx_Math" display="inline" id="S1.p9.3.m3.1"><semantics id="S1.p9.3.m3.1a"><mrow id="S1.p9.3.m3.1.2" xref="S1.p9.3.m3.1.2.cmml"><mi id="S1.p9.3.m3.1.2.2" xref="S1.p9.3.m3.1.2.2.cmml">W</mi><mo id="S1.p9.3.m3.1.2.1" xref="S1.p9.3.m3.1.2.1.cmml">⊆</mo><mrow id="S1.p9.3.m3.1.2.3" xref="S1.p9.3.m3.1.2.3.cmml"><mi id="S1.p9.3.m3.1.2.3.2" xref="S1.p9.3.m3.1.2.3.2.cmml">V</mi><mo id="S1.p9.3.m3.1.2.3.1" xref="S1.p9.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S1.p9.3.m3.1.2.3.3.2" xref="S1.p9.3.m3.1.2.3.cmml"><mo id="S1.p9.3.m3.1.2.3.3.2.1" stretchy="false" xref="S1.p9.3.m3.1.2.3.cmml">(</mo><mi id="S1.p9.3.m3.1.1" xref="S1.p9.3.m3.1.1.cmml">G</mi><mo id="S1.p9.3.m3.1.2.3.3.2.2" stretchy="false" xref="S1.p9.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.3.m3.1b"><apply id="S1.p9.3.m3.1.2.cmml" xref="S1.p9.3.m3.1.2"><subset id="S1.p9.3.m3.1.2.1.cmml" xref="S1.p9.3.m3.1.2.1"></subset><ci id="S1.p9.3.m3.1.2.2.cmml" xref="S1.p9.3.m3.1.2.2">𝑊</ci><apply id="S1.p9.3.m3.1.2.3.cmml" xref="S1.p9.3.m3.1.2.3"><times id="S1.p9.3.m3.1.2.3.1.cmml" xref="S1.p9.3.m3.1.2.3.1"></times><ci id="S1.p9.3.m3.1.2.3.2.cmml" xref="S1.p9.3.m3.1.2.3.2">𝑉</ci><ci id="S1.p9.3.m3.1.1.cmml" xref="S1.p9.3.m3.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.3.m3.1c">W\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p9.3.m3.1d">italic_W ⊆ italic_V ( italic_G )</annotation></semantics></math> of size at least <math alttext="t\cdot\operatorname{sn}(G)" class="ltx_Math" display="inline" id="S1.p9.4.m4.2"><semantics id="S1.p9.4.m4.2a"><mrow id="S1.p9.4.m4.2.3" xref="S1.p9.4.m4.2.3.cmml"><mi id="S1.p9.4.m4.2.3.2" xref="S1.p9.4.m4.2.3.2.cmml">t</mi><mo id="S1.p9.4.m4.2.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p9.4.m4.2.3.1.cmml">⋅</mo><mrow id="S1.p9.4.m4.2.3.3.2" xref="S1.p9.4.m4.2.3.3.1.cmml"><mi id="S1.p9.4.m4.1.1" xref="S1.p9.4.m4.1.1.cmml">sn</mi><mo id="S1.p9.4.m4.2.3.3.2a" xref="S1.p9.4.m4.2.3.3.1.cmml">⁡</mo><mrow id="S1.p9.4.m4.2.3.3.2.1" xref="S1.p9.4.m4.2.3.3.1.cmml"><mo id="S1.p9.4.m4.2.3.3.2.1.1" stretchy="false" xref="S1.p9.4.m4.2.3.3.1.cmml">(</mo><mi id="S1.p9.4.m4.2.2" xref="S1.p9.4.m4.2.2.cmml">G</mi><mo id="S1.p9.4.m4.2.3.3.2.1.2" stretchy="false" xref="S1.p9.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.4.m4.2b"><apply id="S1.p9.4.m4.2.3.cmml" xref="S1.p9.4.m4.2.3"><ci id="S1.p9.4.m4.2.3.1.cmml" xref="S1.p9.4.m4.2.3.1">⋅</ci><ci id="S1.p9.4.m4.2.3.2.cmml" xref="S1.p9.4.m4.2.3.2">𝑡</ci><apply id="S1.p9.4.m4.2.3.3.1.cmml" xref="S1.p9.4.m4.2.3.3.2"><ci id="S1.p9.4.m4.1.1.cmml" xref="S1.p9.4.m4.1.1">sn</ci><ci id="S1.p9.4.m4.2.2.cmml" xref="S1.p9.4.m4.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.4.m4.2c">t\cdot\operatorname{sn}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p9.4.m4.2d">italic_t ⋅ roman_sn ( italic_G )</annotation></semantics></math>, there exists a subgraph <math alttext="H" class="ltx_Math" display="inline" id="S1.p9.5.m5.1"><semantics id="S1.p9.5.m5.1a"><mi id="S1.p9.5.m5.1.1" xref="S1.p9.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p9.5.m5.1b"><ci id="S1.p9.5.m5.1.1.cmml" xref="S1.p9.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p9.5.m5.1d">italic_H</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S1.p9.6.m6.1"><semantics id="S1.p9.6.m6.1a"><mi id="S1.p9.6.m6.1.1" xref="S1.p9.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p9.6.m6.1b"><ci id="S1.p9.6.m6.1.1.cmml" xref="S1.p9.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p9.6.m6.1d">italic_G</annotation></semantics></math> with <math alttext="W\subseteq V(H)" class="ltx_Math" display="inline" id="S1.p9.7.m7.1"><semantics id="S1.p9.7.m7.1a"><mrow id="S1.p9.7.m7.1.2" xref="S1.p9.7.m7.1.2.cmml"><mi id="S1.p9.7.m7.1.2.2" xref="S1.p9.7.m7.1.2.2.cmml">W</mi><mo id="S1.p9.7.m7.1.2.1" xref="S1.p9.7.m7.1.2.1.cmml">⊆</mo><mrow id="S1.p9.7.m7.1.2.3" xref="S1.p9.7.m7.1.2.3.cmml"><mi id="S1.p9.7.m7.1.2.3.2" xref="S1.p9.7.m7.1.2.3.2.cmml">V</mi><mo id="S1.p9.7.m7.1.2.3.1" xref="S1.p9.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S1.p9.7.m7.1.2.3.3.2" xref="S1.p9.7.m7.1.2.3.cmml"><mo id="S1.p9.7.m7.1.2.3.3.2.1" stretchy="false" xref="S1.p9.7.m7.1.2.3.cmml">(</mo><mi id="S1.p9.7.m7.1.1" xref="S1.p9.7.m7.1.1.cmml">H</mi><mo id="S1.p9.7.m7.1.2.3.3.2.2" stretchy="false" xref="S1.p9.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.7.m7.1b"><apply id="S1.p9.7.m7.1.2.cmml" xref="S1.p9.7.m7.1.2"><subset id="S1.p9.7.m7.1.2.1.cmml" xref="S1.p9.7.m7.1.2.1"></subset><ci id="S1.p9.7.m7.1.2.2.cmml" xref="S1.p9.7.m7.1.2.2">𝑊</ci><apply id="S1.p9.7.m7.1.2.3.cmml" xref="S1.p9.7.m7.1.2.3"><times id="S1.p9.7.m7.1.2.3.1.cmml" xref="S1.p9.7.m7.1.2.3.1"></times><ci id="S1.p9.7.m7.1.2.3.2.cmml" xref="S1.p9.7.m7.1.2.3.2">𝑉</ci><ci id="S1.p9.7.m7.1.1.cmml" xref="S1.p9.7.m7.1.1">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.7.m7.1c">W\subseteq V(H)</annotation><annotation encoding="application/x-llamapun" id="S1.p9.7.m7.1d">italic_W ⊆ italic_V ( italic_H )</annotation></semantics></math> and a separation <math alttext="(X,Y)" class="ltx_Math" display="inline" id="S1.p9.8.m8.2"><semantics id="S1.p9.8.m8.2a"><mrow id="S1.p9.8.m8.2.3.2" xref="S1.p9.8.m8.2.3.1.cmml"><mo id="S1.p9.8.m8.2.3.2.1" stretchy="false" xref="S1.p9.8.m8.2.3.1.cmml">(</mo><mi id="S1.p9.8.m8.1.1" xref="S1.p9.8.m8.1.1.cmml">X</mi><mo id="S1.p9.8.m8.2.3.2.2" xref="S1.p9.8.m8.2.3.1.cmml">,</mo><mi id="S1.p9.8.m8.2.2" xref="S1.p9.8.m8.2.2.cmml">Y</mi><mo id="S1.p9.8.m8.2.3.2.3" stretchy="false" xref="S1.p9.8.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.8.m8.2b"><interval closure="open" id="S1.p9.8.m8.2.3.1.cmml" xref="S1.p9.8.m8.2.3.2"><ci id="S1.p9.8.m8.1.1.cmml" xref="S1.p9.8.m8.1.1">𝑋</ci><ci id="S1.p9.8.m8.2.2.cmml" xref="S1.p9.8.m8.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.8.m8.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S1.p9.8.m8.2d">( italic_X , italic_Y )</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S1.p9.9.m9.1"><semantics id="S1.p9.9.m9.1a"><mi id="S1.p9.9.m9.1.1" xref="S1.p9.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p9.9.m9.1b"><ci id="S1.p9.9.m9.1.1.cmml" xref="S1.p9.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p9.9.m9.1d">italic_G</annotation></semantics></math> with <math alttext="W\subseteq Y" class="ltx_Math" display="inline" id="S1.p9.10.m10.1"><semantics id="S1.p9.10.m10.1a"><mrow id="S1.p9.10.m10.1.1" xref="S1.p9.10.m10.1.1.cmml"><mi id="S1.p9.10.m10.1.1.2" xref="S1.p9.10.m10.1.1.2.cmml">W</mi><mo id="S1.p9.10.m10.1.1.1" xref="S1.p9.10.m10.1.1.1.cmml">⊆</mo><mi id="S1.p9.10.m10.1.1.3" xref="S1.p9.10.m10.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.10.m10.1b"><apply id="S1.p9.10.m10.1.1.cmml" xref="S1.p9.10.m10.1.1"><subset id="S1.p9.10.m10.1.1.1.cmml" xref="S1.p9.10.m10.1.1.1"></subset><ci id="S1.p9.10.m10.1.1.2.cmml" xref="S1.p9.10.m10.1.1.2">𝑊</ci><ci id="S1.p9.10.m10.1.1.3.cmml" xref="S1.p9.10.m10.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.10.m10.1c">W\subseteq Y</annotation><annotation encoding="application/x-llamapun" id="S1.p9.10.m10.1d">italic_W ⊆ italic_Y</annotation></semantics></math> and <math alttext="V(G)\setminus V(H)\subseteq X\setminus Y" class="ltx_Math" display="inline" id="S1.p9.11.m11.2"><semantics id="S1.p9.11.m11.2a"><mrow id="S1.p9.11.m11.2.3" xref="S1.p9.11.m11.2.3.cmml"><mrow id="S1.p9.11.m11.2.3.2" xref="S1.p9.11.m11.2.3.2.cmml"><mrow id="S1.p9.11.m11.2.3.2.2" xref="S1.p9.11.m11.2.3.2.2.cmml"><mi id="S1.p9.11.m11.2.3.2.2.2" xref="S1.p9.11.m11.2.3.2.2.2.cmml">V</mi><mo id="S1.p9.11.m11.2.3.2.2.1" xref="S1.p9.11.m11.2.3.2.2.1.cmml">⁢</mo><mrow id="S1.p9.11.m11.2.3.2.2.3.2" xref="S1.p9.11.m11.2.3.2.2.cmml"><mo id="S1.p9.11.m11.2.3.2.2.3.2.1" stretchy="false" xref="S1.p9.11.m11.2.3.2.2.cmml">(</mo><mi id="S1.p9.11.m11.1.1" xref="S1.p9.11.m11.1.1.cmml">G</mi><mo id="S1.p9.11.m11.2.3.2.2.3.2.2" stretchy="false" xref="S1.p9.11.m11.2.3.2.2.cmml">)</mo></mrow></mrow><mo id="S1.p9.11.m11.2.3.2.1" xref="S1.p9.11.m11.2.3.2.1.cmml">∖</mo><mrow id="S1.p9.11.m11.2.3.2.3" xref="S1.p9.11.m11.2.3.2.3.cmml"><mi id="S1.p9.11.m11.2.3.2.3.2" xref="S1.p9.11.m11.2.3.2.3.2.cmml">V</mi><mo id="S1.p9.11.m11.2.3.2.3.1" xref="S1.p9.11.m11.2.3.2.3.1.cmml">⁢</mo><mrow id="S1.p9.11.m11.2.3.2.3.3.2" xref="S1.p9.11.m11.2.3.2.3.cmml"><mo id="S1.p9.11.m11.2.3.2.3.3.2.1" stretchy="false" xref="S1.p9.11.m11.2.3.2.3.cmml">(</mo><mi id="S1.p9.11.m11.2.2" xref="S1.p9.11.m11.2.2.cmml">H</mi><mo id="S1.p9.11.m11.2.3.2.3.3.2.2" stretchy="false" xref="S1.p9.11.m11.2.3.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S1.p9.11.m11.2.3.1" xref="S1.p9.11.m11.2.3.1.cmml">⊆</mo><mrow id="S1.p9.11.m11.2.3.3" xref="S1.p9.11.m11.2.3.3.cmml"><mi id="S1.p9.11.m11.2.3.3.2" xref="S1.p9.11.m11.2.3.3.2.cmml">X</mi><mo id="S1.p9.11.m11.2.3.3.1" xref="S1.p9.11.m11.2.3.3.1.cmml">∖</mo><mi id="S1.p9.11.m11.2.3.3.3" xref="S1.p9.11.m11.2.3.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.11.m11.2b"><apply id="S1.p9.11.m11.2.3.cmml" xref="S1.p9.11.m11.2.3"><subset id="S1.p9.11.m11.2.3.1.cmml" xref="S1.p9.11.m11.2.3.1"></subset><apply id="S1.p9.11.m11.2.3.2.cmml" xref="S1.p9.11.m11.2.3.2"><setdiff id="S1.p9.11.m11.2.3.2.1.cmml" xref="S1.p9.11.m11.2.3.2.1"></setdiff><apply id="S1.p9.11.m11.2.3.2.2.cmml" xref="S1.p9.11.m11.2.3.2.2"><times id="S1.p9.11.m11.2.3.2.2.1.cmml" xref="S1.p9.11.m11.2.3.2.2.1"></times><ci id="S1.p9.11.m11.2.3.2.2.2.cmml" xref="S1.p9.11.m11.2.3.2.2.2">𝑉</ci><ci id="S1.p9.11.m11.1.1.cmml" xref="S1.p9.11.m11.1.1">𝐺</ci></apply><apply id="S1.p9.11.m11.2.3.2.3.cmml" xref="S1.p9.11.m11.2.3.2.3"><times id="S1.p9.11.m11.2.3.2.3.1.cmml" xref="S1.p9.11.m11.2.3.2.3.1"></times><ci id="S1.p9.11.m11.2.3.2.3.2.cmml" xref="S1.p9.11.m11.2.3.2.3.2">𝑉</ci><ci id="S1.p9.11.m11.2.2.cmml" xref="S1.p9.11.m11.2.2">𝐻</ci></apply></apply><apply id="S1.p9.11.m11.2.3.3.cmml" xref="S1.p9.11.m11.2.3.3"><setdiff id="S1.p9.11.m11.2.3.3.1.cmml" xref="S1.p9.11.m11.2.3.3.1"></setdiff><ci id="S1.p9.11.m11.2.3.3.2.cmml" xref="S1.p9.11.m11.2.3.3.2">𝑋</ci><ci id="S1.p9.11.m11.2.3.3.3.cmml" xref="S1.p9.11.m11.2.3.3.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.11.m11.2c">V(G)\setminus V(H)\subseteq X\setminus Y</annotation><annotation encoding="application/x-llamapun" id="S1.p9.11.m11.2d">italic_V ( italic_G ) ∖ italic_V ( italic_H ) ⊆ italic_X ∖ italic_Y</annotation></semantics></math> and having order less than <math alttext="|W|" class="ltx_Math" display="inline" id="S1.p9.12.m12.1"><semantics id="S1.p9.12.m12.1a"><mrow id="S1.p9.12.m12.1.2.2" xref="S1.p9.12.m12.1.2.1.cmml"><mo id="S1.p9.12.m12.1.2.2.1" stretchy="false" xref="S1.p9.12.m12.1.2.1.1.cmml">|</mo><mi id="S1.p9.12.m12.1.1" xref="S1.p9.12.m12.1.1.cmml">W</mi><mo id="S1.p9.12.m12.1.2.2.2" stretchy="false" xref="S1.p9.12.m12.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.12.m12.1b"><apply id="S1.p9.12.m12.1.2.1.cmml" xref="S1.p9.12.m12.1.2.2"><abs id="S1.p9.12.m12.1.2.1.1.cmml" xref="S1.p9.12.m12.1.2.2.1"></abs><ci id="S1.p9.12.m12.1.1.cmml" xref="S1.p9.12.m12.1.1">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.12.m12.1c">|W|</annotation><annotation encoding="application/x-llamapun" id="S1.p9.12.m12.1d">| italic_W |</annotation></semantics></math> and such that any balanced separation of <math alttext="H" class="ltx_Math" display="inline" id="S1.p9.13.m13.1"><semantics id="S1.p9.13.m13.1a"><mi id="S1.p9.13.m13.1.1" xref="S1.p9.13.m13.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p9.13.m13.1b"><ci id="S1.p9.13.m13.1.1.cmml" xref="S1.p9.13.m13.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.13.m13.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p9.13.m13.1d">italic_H</annotation></semantics></math> must necessarily balance <math alttext="W\cup(X\cap Y)" class="ltx_Math" display="inline" id="S1.p9.14.m14.1"><semantics id="S1.p9.14.m14.1a"><mrow id="S1.p9.14.m14.1.1" xref="S1.p9.14.m14.1.1.cmml"><mi id="S1.p9.14.m14.1.1.3" xref="S1.p9.14.m14.1.1.3.cmml">W</mi><mo id="S1.p9.14.m14.1.1.2" xref="S1.p9.14.m14.1.1.2.cmml">∪</mo><mrow id="S1.p9.14.m14.1.1.1.1" xref="S1.p9.14.m14.1.1.1.1.1.cmml"><mo id="S1.p9.14.m14.1.1.1.1.2" stretchy="false" xref="S1.p9.14.m14.1.1.1.1.1.cmml">(</mo><mrow id="S1.p9.14.m14.1.1.1.1.1" xref="S1.p9.14.m14.1.1.1.1.1.cmml"><mi id="S1.p9.14.m14.1.1.1.1.1.2" xref="S1.p9.14.m14.1.1.1.1.1.2.cmml">X</mi><mo id="S1.p9.14.m14.1.1.1.1.1.1" xref="S1.p9.14.m14.1.1.1.1.1.1.cmml">∩</mo><mi id="S1.p9.14.m14.1.1.1.1.1.3" xref="S1.p9.14.m14.1.1.1.1.1.3.cmml">Y</mi></mrow><mo id="S1.p9.14.m14.1.1.1.1.3" stretchy="false" xref="S1.p9.14.m14.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.14.m14.1b"><apply id="S1.p9.14.m14.1.1.cmml" xref="S1.p9.14.m14.1.1"><union id="S1.p9.14.m14.1.1.2.cmml" xref="S1.p9.14.m14.1.1.2"></union><ci id="S1.p9.14.m14.1.1.3.cmml" xref="S1.p9.14.m14.1.1.3">𝑊</ci><apply id="S1.p9.14.m14.1.1.1.1.1.cmml" xref="S1.p9.14.m14.1.1.1.1"><intersect id="S1.p9.14.m14.1.1.1.1.1.1.cmml" xref="S1.p9.14.m14.1.1.1.1.1.1"></intersect><ci id="S1.p9.14.m14.1.1.1.1.1.2.cmml" xref="S1.p9.14.m14.1.1.1.1.1.2">𝑋</ci><ci id="S1.p9.14.m14.1.1.1.1.1.3.cmml" xref="S1.p9.14.m14.1.1.1.1.1.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.14.m14.1c">W\cup(X\cap Y)</annotation><annotation encoding="application/x-llamapun" id="S1.p9.14.m14.1d">italic_W ∪ ( italic_X ∩ italic_Y )</annotation></semantics></math>. This leads to an algorithm for constructing a tree decomposition of <math alttext="G" class="ltx_Math" display="inline" id="S1.p9.15.m15.1"><semantics id="S1.p9.15.m15.1a"><mi id="S1.p9.15.m15.1.1" xref="S1.p9.15.m15.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p9.15.m15.1b"><ci id="S1.p9.15.m15.1.1.cmml" xref="S1.p9.15.m15.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.15.m15.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p9.15.m15.1d">italic_G</annotation></semantics></math> similar in spirit to the algorithm outlined above. In particular, recursively taking balanced separations of subgraphs of <math alttext="H" class="ltx_Math" display="inline" id="S1.p9.16.m16.1"><semantics id="S1.p9.16.m16.1a"><mi id="S1.p9.16.m16.1.1" xref="S1.p9.16.m16.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p9.16.m16.1b"><ci id="S1.p9.16.m16.1.1.cmml" xref="S1.p9.16.m16.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.16.m16.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p9.16.m16.1d">italic_H</annotation></semantics></math> gives an algorithm for constructing a tree decomposition <math alttext="\mathcal{T}_{Y}:=(B_{x}:x\in V(T_{Y}))" class="ltx_math_unparsed" display="inline" id="S1.p9.17.m17.1"><semantics id="S1.p9.17.m17.1a"><mrow id="S1.p9.17.m17.1b"><msub id="S1.p9.17.m17.1.1"><mi class="ltx_font_mathcaligraphic" id="S1.p9.17.m17.1.1.2">𝒯</mi><mi id="S1.p9.17.m17.1.1.3">Y</mi></msub><mo id="S1.p9.17.m17.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S1.p9.17.m17.1.3"><mo id="S1.p9.17.m17.1.3.1" stretchy="false">(</mo><msub id="S1.p9.17.m17.1.3.2"><mi id="S1.p9.17.m17.1.3.2.2">B</mi><mi id="S1.p9.17.m17.1.3.2.3">x</mi></msub><mo id="S1.p9.17.m17.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S1.p9.17.m17.1.3.4">x</mi><mo id="S1.p9.17.m17.1.3.5">∈</mo><mi id="S1.p9.17.m17.1.3.6">V</mi><mrow id="S1.p9.17.m17.1.3.7"><mo id="S1.p9.17.m17.1.3.7.1" stretchy="false">(</mo><msub id="S1.p9.17.m17.1.3.7.2"><mi id="S1.p9.17.m17.1.3.7.2.2">T</mi><mi id="S1.p9.17.m17.1.3.7.2.3">Y</mi></msub><mo id="S1.p9.17.m17.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S1.p9.17.m17.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p9.17.m17.1c">\mathcal{T}_{Y}:=(B_{x}:x\in V(T_{Y}))</annotation><annotation encoding="application/x-llamapun" id="S1.p9.17.m17.1d">caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) )</annotation></semantics></math> of <math alttext="G[Y]" class="ltx_Math" display="inline" id="S1.p9.18.m18.1"><semantics id="S1.p9.18.m18.1a"><mrow id="S1.p9.18.m18.1.2" xref="S1.p9.18.m18.1.2.cmml"><mi id="S1.p9.18.m18.1.2.2" xref="S1.p9.18.m18.1.2.2.cmml">G</mi><mo id="S1.p9.18.m18.1.2.1" xref="S1.p9.18.m18.1.2.1.cmml">⁢</mo><mrow id="S1.p9.18.m18.1.2.3.2" xref="S1.p9.18.m18.1.2.3.1.cmml"><mo id="S1.p9.18.m18.1.2.3.2.1" stretchy="false" xref="S1.p9.18.m18.1.2.3.1.1.cmml">[</mo><mi id="S1.p9.18.m18.1.1" xref="S1.p9.18.m18.1.1.cmml">Y</mi><mo id="S1.p9.18.m18.1.2.3.2.2" stretchy="false" xref="S1.p9.18.m18.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.18.m18.1b"><apply id="S1.p9.18.m18.1.2.cmml" xref="S1.p9.18.m18.1.2"><times id="S1.p9.18.m18.1.2.1.cmml" xref="S1.p9.18.m18.1.2.1"></times><ci id="S1.p9.18.m18.1.2.2.cmml" xref="S1.p9.18.m18.1.2.2">𝐺</ci><apply id="S1.p9.18.m18.1.2.3.1.cmml" xref="S1.p9.18.m18.1.2.3.2"><csymbol cd="latexml" id="S1.p9.18.m18.1.2.3.1.1.cmml" xref="S1.p9.18.m18.1.2.3.2.1">delimited-[]</csymbol><ci id="S1.p9.18.m18.1.1.cmml" xref="S1.p9.18.m18.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.18.m18.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="S1.p9.18.m18.1d">italic_G [ italic_Y ]</annotation></semantics></math> in which some bag <math alttext="B_{y}" class="ltx_Math" display="inline" id="S1.p9.19.m19.1"><semantics id="S1.p9.19.m19.1a"><msub id="S1.p9.19.m19.1.1" xref="S1.p9.19.m19.1.1.cmml"><mi id="S1.p9.19.m19.1.1.2" xref="S1.p9.19.m19.1.1.2.cmml">B</mi><mi id="S1.p9.19.m19.1.1.3" xref="S1.p9.19.m19.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p9.19.m19.1b"><apply id="S1.p9.19.m19.1.1.cmml" xref="S1.p9.19.m19.1.1"><csymbol cd="ambiguous" id="S1.p9.19.m19.1.1.1.cmml" xref="S1.p9.19.m19.1.1">subscript</csymbol><ci id="S1.p9.19.m19.1.1.2.cmml" xref="S1.p9.19.m19.1.1.2">𝐵</ci><ci id="S1.p9.19.m19.1.1.3.cmml" xref="S1.p9.19.m19.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.19.m19.1c">B_{y}</annotation><annotation encoding="application/x-llamapun" id="S1.p9.19.m19.1d">italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> contains <math alttext="W\cup(X\cap Y)" class="ltx_Math" display="inline" id="S1.p9.20.m20.1"><semantics id="S1.p9.20.m20.1a"><mrow id="S1.p9.20.m20.1.1" xref="S1.p9.20.m20.1.1.cmml"><mi id="S1.p9.20.m20.1.1.3" xref="S1.p9.20.m20.1.1.3.cmml">W</mi><mo id="S1.p9.20.m20.1.1.2" xref="S1.p9.20.m20.1.1.2.cmml">∪</mo><mrow id="S1.p9.20.m20.1.1.1.1" xref="S1.p9.20.m20.1.1.1.1.1.cmml"><mo id="S1.p9.20.m20.1.1.1.1.2" stretchy="false" xref="S1.p9.20.m20.1.1.1.1.1.cmml">(</mo><mrow id="S1.p9.20.m20.1.1.1.1.1" xref="S1.p9.20.m20.1.1.1.1.1.cmml"><mi id="S1.p9.20.m20.1.1.1.1.1.2" xref="S1.p9.20.m20.1.1.1.1.1.2.cmml">X</mi><mo id="S1.p9.20.m20.1.1.1.1.1.1" xref="S1.p9.20.m20.1.1.1.1.1.1.cmml">∩</mo><mi id="S1.p9.20.m20.1.1.1.1.1.3" xref="S1.p9.20.m20.1.1.1.1.1.3.cmml">Y</mi></mrow><mo id="S1.p9.20.m20.1.1.1.1.3" stretchy="false" xref="S1.p9.20.m20.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.20.m20.1b"><apply id="S1.p9.20.m20.1.1.cmml" xref="S1.p9.20.m20.1.1"><union id="S1.p9.20.m20.1.1.2.cmml" xref="S1.p9.20.m20.1.1.2"></union><ci id="S1.p9.20.m20.1.1.3.cmml" xref="S1.p9.20.m20.1.1.3">𝑊</ci><apply id="S1.p9.20.m20.1.1.1.1.1.cmml" xref="S1.p9.20.m20.1.1.1.1"><intersect id="S1.p9.20.m20.1.1.1.1.1.1.cmml" xref="S1.p9.20.m20.1.1.1.1.1.1"></intersect><ci id="S1.p9.20.m20.1.1.1.1.1.2.cmml" xref="S1.p9.20.m20.1.1.1.1.1.2">𝑋</ci><ci id="S1.p9.20.m20.1.1.1.1.1.3.cmml" xref="S1.p9.20.m20.1.1.1.1.1.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.20.m20.1c">W\cup(X\cap Y)</annotation><annotation encoding="application/x-llamapun" id="S1.p9.20.m20.1d">italic_W ∪ ( italic_X ∩ italic_Y )</annotation></semantics></math>. Then, recursion/induction is used to find a tree decomposition <math alttext="\mathcal{T}_{X}:=(B_{x}:x\in V(T_{X}))" class="ltx_math_unparsed" display="inline" id="S1.p9.21.m21.1"><semantics id="S1.p9.21.m21.1a"><mrow id="S1.p9.21.m21.1b"><msub id="S1.p9.21.m21.1.1"><mi class="ltx_font_mathcaligraphic" id="S1.p9.21.m21.1.1.2">𝒯</mi><mi id="S1.p9.21.m21.1.1.3">X</mi></msub><mo id="S1.p9.21.m21.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S1.p9.21.m21.1.3"><mo id="S1.p9.21.m21.1.3.1" stretchy="false">(</mo><msub id="S1.p9.21.m21.1.3.2"><mi id="S1.p9.21.m21.1.3.2.2">B</mi><mi id="S1.p9.21.m21.1.3.2.3">x</mi></msub><mo id="S1.p9.21.m21.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S1.p9.21.m21.1.3.4">x</mi><mo id="S1.p9.21.m21.1.3.5">∈</mo><mi id="S1.p9.21.m21.1.3.6">V</mi><mrow id="S1.p9.21.m21.1.3.7"><mo id="S1.p9.21.m21.1.3.7.1" stretchy="false">(</mo><msub id="S1.p9.21.m21.1.3.7.2"><mi id="S1.p9.21.m21.1.3.7.2.2">T</mi><mi id="S1.p9.21.m21.1.3.7.2.3">X</mi></msub><mo id="S1.p9.21.m21.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S1.p9.21.m21.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p9.21.m21.1c">\mathcal{T}_{X}:=(B_{x}:x\in V(T_{X}))</annotation><annotation encoding="application/x-llamapun" id="S1.p9.21.m21.1d">caligraphic_T start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) )</annotation></semantics></math> of <math alttext="G^{\prime}:=G[X]" class="ltx_Math" display="inline" id="S1.p9.22.m22.1"><semantics id="S1.p9.22.m22.1a"><mrow id="S1.p9.22.m22.1.2" xref="S1.p9.22.m22.1.2.cmml"><msup id="S1.p9.22.m22.1.2.2" xref="S1.p9.22.m22.1.2.2.cmml"><mi id="S1.p9.22.m22.1.2.2.2" xref="S1.p9.22.m22.1.2.2.2.cmml">G</mi><mo id="S1.p9.22.m22.1.2.2.3" xref="S1.p9.22.m22.1.2.2.3.cmml">′</mo></msup><mo id="S1.p9.22.m22.1.2.1" lspace="0.278em" rspace="0.278em" xref="S1.p9.22.m22.1.2.1.cmml">:=</mo><mrow id="S1.p9.22.m22.1.2.3" xref="S1.p9.22.m22.1.2.3.cmml"><mi id="S1.p9.22.m22.1.2.3.2" xref="S1.p9.22.m22.1.2.3.2.cmml">G</mi><mo id="S1.p9.22.m22.1.2.3.1" xref="S1.p9.22.m22.1.2.3.1.cmml">⁢</mo><mrow id="S1.p9.22.m22.1.2.3.3.2" xref="S1.p9.22.m22.1.2.3.3.1.cmml"><mo id="S1.p9.22.m22.1.2.3.3.2.1" stretchy="false" xref="S1.p9.22.m22.1.2.3.3.1.1.cmml">[</mo><mi id="S1.p9.22.m22.1.1" xref="S1.p9.22.m22.1.1.cmml">X</mi><mo id="S1.p9.22.m22.1.2.3.3.2.2" stretchy="false" xref="S1.p9.22.m22.1.2.3.3.1.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.22.m22.1b"><apply id="S1.p9.22.m22.1.2.cmml" xref="S1.p9.22.m22.1.2"><csymbol cd="latexml" id="S1.p9.22.m22.1.2.1.cmml" xref="S1.p9.22.m22.1.2.1">assign</csymbol><apply id="S1.p9.22.m22.1.2.2.cmml" xref="S1.p9.22.m22.1.2.2"><csymbol cd="ambiguous" id="S1.p9.22.m22.1.2.2.1.cmml" xref="S1.p9.22.m22.1.2.2">superscript</csymbol><ci id="S1.p9.22.m22.1.2.2.2.cmml" xref="S1.p9.22.m22.1.2.2.2">𝐺</ci><ci id="S1.p9.22.m22.1.2.2.3.cmml" xref="S1.p9.22.m22.1.2.2.3">′</ci></apply><apply id="S1.p9.22.m22.1.2.3.cmml" xref="S1.p9.22.m22.1.2.3"><times id="S1.p9.22.m22.1.2.3.1.cmml" xref="S1.p9.22.m22.1.2.3.1"></times><ci id="S1.p9.22.m22.1.2.3.2.cmml" xref="S1.p9.22.m22.1.2.3.2">𝐺</ci><apply id="S1.p9.22.m22.1.2.3.3.1.cmml" xref="S1.p9.22.m22.1.2.3.3.2"><csymbol cd="latexml" id="S1.p9.22.m22.1.2.3.3.1.1.cmml" xref="S1.p9.22.m22.1.2.3.3.2.1">delimited-[]</csymbol><ci id="S1.p9.22.m22.1.1.cmml" xref="S1.p9.22.m22.1.1">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.22.m22.1c">G^{\prime}:=G[X]</annotation><annotation encoding="application/x-llamapun" id="S1.p9.22.m22.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_G [ italic_X ]</annotation></semantics></math> in which some bag <math alttext="B_{x}" class="ltx_Math" display="inline" id="S1.p9.23.m23.1"><semantics id="S1.p9.23.m23.1a"><msub id="S1.p9.23.m23.1.1" xref="S1.p9.23.m23.1.1.cmml"><mi id="S1.p9.23.m23.1.1.2" xref="S1.p9.23.m23.1.1.2.cmml">B</mi><mi id="S1.p9.23.m23.1.1.3" xref="S1.p9.23.m23.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p9.23.m23.1b"><apply id="S1.p9.23.m23.1.1.cmml" xref="S1.p9.23.m23.1.1"><csymbol cd="ambiguous" id="S1.p9.23.m23.1.1.1.cmml" xref="S1.p9.23.m23.1.1">subscript</csymbol><ci id="S1.p9.23.m23.1.1.2.cmml" xref="S1.p9.23.m23.1.1.2">𝐵</ci><ci id="S1.p9.23.m23.1.1.3.cmml" xref="S1.p9.23.m23.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.23.m23.1c">B_{x}</annotation><annotation encoding="application/x-llamapun" id="S1.p9.23.m23.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> contains <math alttext="W^{\prime}:=X\cap Y" class="ltx_Math" display="inline" id="S1.p9.24.m24.1"><semantics id="S1.p9.24.m24.1a"><mrow id="S1.p9.24.m24.1.1" xref="S1.p9.24.m24.1.1.cmml"><msup id="S1.p9.24.m24.1.1.2" xref="S1.p9.24.m24.1.1.2.cmml"><mi id="S1.p9.24.m24.1.1.2.2" xref="S1.p9.24.m24.1.1.2.2.cmml">W</mi><mo id="S1.p9.24.m24.1.1.2.3" xref="S1.p9.24.m24.1.1.2.3.cmml">′</mo></msup><mo id="S1.p9.24.m24.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p9.24.m24.1.1.1.cmml">:=</mo><mrow id="S1.p9.24.m24.1.1.3" xref="S1.p9.24.m24.1.1.3.cmml"><mi id="S1.p9.24.m24.1.1.3.2" xref="S1.p9.24.m24.1.1.3.2.cmml">X</mi><mo id="S1.p9.24.m24.1.1.3.1" xref="S1.p9.24.m24.1.1.3.1.cmml">∩</mo><mi id="S1.p9.24.m24.1.1.3.3" xref="S1.p9.24.m24.1.1.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.24.m24.1b"><apply id="S1.p9.24.m24.1.1.cmml" xref="S1.p9.24.m24.1.1"><csymbol cd="latexml" id="S1.p9.24.m24.1.1.1.cmml" xref="S1.p9.24.m24.1.1.1">assign</csymbol><apply id="S1.p9.24.m24.1.1.2.cmml" xref="S1.p9.24.m24.1.1.2"><csymbol cd="ambiguous" id="S1.p9.24.m24.1.1.2.1.cmml" xref="S1.p9.24.m24.1.1.2">superscript</csymbol><ci id="S1.p9.24.m24.1.1.2.2.cmml" xref="S1.p9.24.m24.1.1.2.2">𝑊</ci><ci id="S1.p9.24.m24.1.1.2.3.cmml" xref="S1.p9.24.m24.1.1.2.3">′</ci></apply><apply id="S1.p9.24.m24.1.1.3.cmml" xref="S1.p9.24.m24.1.1.3"><intersect id="S1.p9.24.m24.1.1.3.1.cmml" xref="S1.p9.24.m24.1.1.3.1"></intersect><ci id="S1.p9.24.m24.1.1.3.2.cmml" xref="S1.p9.24.m24.1.1.3.2">𝑋</ci><ci id="S1.p9.24.m24.1.1.3.3.cmml" xref="S1.p9.24.m24.1.1.3.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.24.m24.1c">W^{\prime}:=X\cap Y</annotation><annotation encoding="application/x-llamapun" id="S1.p9.24.m24.1d">italic_W start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_X ∩ italic_Y</annotation></semantics></math>. Joining <math alttext="T_{X}" class="ltx_Math" display="inline" id="S1.p9.25.m25.1"><semantics id="S1.p9.25.m25.1a"><msub id="S1.p9.25.m25.1.1" xref="S1.p9.25.m25.1.1.cmml"><mi id="S1.p9.25.m25.1.1.2" xref="S1.p9.25.m25.1.1.2.cmml">T</mi><mi id="S1.p9.25.m25.1.1.3" xref="S1.p9.25.m25.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p9.25.m25.1b"><apply id="S1.p9.25.m25.1.1.cmml" xref="S1.p9.25.m25.1.1"><csymbol cd="ambiguous" id="S1.p9.25.m25.1.1.1.cmml" xref="S1.p9.25.m25.1.1">subscript</csymbol><ci id="S1.p9.25.m25.1.1.2.cmml" xref="S1.p9.25.m25.1.1.2">𝑇</ci><ci id="S1.p9.25.m25.1.1.3.cmml" xref="S1.p9.25.m25.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.25.m25.1c">T_{X}</annotation><annotation encoding="application/x-llamapun" id="S1.p9.25.m25.1d">italic_T start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="T_{Y}" class="ltx_Math" display="inline" id="S1.p9.26.m26.1"><semantics id="S1.p9.26.m26.1a"><msub id="S1.p9.26.m26.1.1" xref="S1.p9.26.m26.1.1.cmml"><mi id="S1.p9.26.m26.1.1.2" xref="S1.p9.26.m26.1.1.2.cmml">T</mi><mi id="S1.p9.26.m26.1.1.3" xref="S1.p9.26.m26.1.1.3.cmml">Y</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p9.26.m26.1b"><apply id="S1.p9.26.m26.1.1.cmml" xref="S1.p9.26.m26.1.1"><csymbol cd="ambiguous" id="S1.p9.26.m26.1.1.1.cmml" xref="S1.p9.26.m26.1.1">subscript</csymbol><ci id="S1.p9.26.m26.1.1.2.cmml" xref="S1.p9.26.m26.1.1.2">𝑇</ci><ci id="S1.p9.26.m26.1.1.3.cmml" xref="S1.p9.26.m26.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.26.m26.1c">T_{Y}</annotation><annotation encoding="application/x-llamapun" id="S1.p9.26.m26.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math> with the edge <math alttext="xy" class="ltx_Math" display="inline" id="S1.p9.27.m27.1"><semantics id="S1.p9.27.m27.1a"><mrow id="S1.p9.27.m27.1.1" xref="S1.p9.27.m27.1.1.cmml"><mi id="S1.p9.27.m27.1.1.2" xref="S1.p9.27.m27.1.1.2.cmml">x</mi><mo id="S1.p9.27.m27.1.1.1" xref="S1.p9.27.m27.1.1.1.cmml">⁢</mo><mi id="S1.p9.27.m27.1.1.3" xref="S1.p9.27.m27.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.27.m27.1b"><apply id="S1.p9.27.m27.1.1.cmml" xref="S1.p9.27.m27.1.1"><times id="S1.p9.27.m27.1.1.1.cmml" xref="S1.p9.27.m27.1.1.1"></times><ci id="S1.p9.27.m27.1.1.2.cmml" xref="S1.p9.27.m27.1.1.2">𝑥</ci><ci id="S1.p9.27.m27.1.1.3.cmml" xref="S1.p9.27.m27.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.27.m27.1c">xy</annotation><annotation encoding="application/x-llamapun" id="S1.p9.27.m27.1d">italic_x italic_y</annotation></semantics></math> gives a tree <math alttext="T" class="ltx_Math" display="inline" id="S1.p9.28.m28.1"><semantics id="S1.p9.28.m28.1a"><mi id="S1.p9.28.m28.1.1" xref="S1.p9.28.m28.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.p9.28.m28.1b"><ci id="S1.p9.28.m28.1.1.cmml" xref="S1.p9.28.m28.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.28.m28.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.p9.28.m28.1d">italic_T</annotation></semantics></math> such that <math alttext="\mathcal{T}:=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="S1.p9.29.m29.1"><semantics id="S1.p9.29.m29.1a"><mrow id="S1.p9.29.m29.1b"><mi class="ltx_font_mathcaligraphic" id="S1.p9.29.m29.1.1">𝒯</mi><mo id="S1.p9.29.m29.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S1.p9.29.m29.1.3"><mo id="S1.p9.29.m29.1.3.1" stretchy="false">(</mo><msub id="S1.p9.29.m29.1.3.2"><mi id="S1.p9.29.m29.1.3.2.2">B</mi><mi id="S1.p9.29.m29.1.3.2.3">x</mi></msub><mo id="S1.p9.29.m29.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S1.p9.29.m29.1.3.4">x</mi><mo id="S1.p9.29.m29.1.3.5">∈</mo><mi id="S1.p9.29.m29.1.3.6">V</mi><mrow id="S1.p9.29.m29.1.3.7"><mo id="S1.p9.29.m29.1.3.7.1" stretchy="false">(</mo><mi id="S1.p9.29.m29.1.3.7.2">T</mi><mo id="S1.p9.29.m29.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S1.p9.29.m29.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.p9.29.m29.1c">\mathcal{T}:=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S1.p9.29.m29.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> is the desired tree decomposition of <math alttext="G" class="ltx_Math" display="inline" id="S1.p9.30.m30.1"><semantics id="S1.p9.30.m30.1a"><mi id="S1.p9.30.m30.1.1" xref="S1.p9.30.m30.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p9.30.m30.1b"><ci id="S1.p9.30.m30.1.1.cmml" xref="S1.p9.30.m30.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.30.m30.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p9.30.m30.1d">italic_G</annotation></semantics></math>.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Preliminaries</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.38">For standard graph theoretic terminology and notations, see <cite class="ltx_cite ltx_citemacro_citet">Diestel [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib2" title="">2</a>]</cite>. Let <math alttext="G" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_G</annotation></semantics></math> be a graph and let <math alttext="S,T\subseteq V(G)" class="ltx_Math" display="inline" id="S2.p1.2.m2.3"><semantics id="S2.p1.2.m2.3a"><mrow id="S2.p1.2.m2.3.4" xref="S2.p1.2.m2.3.4.cmml"><mrow id="S2.p1.2.m2.3.4.2.2" xref="S2.p1.2.m2.3.4.2.1.cmml"><mi id="S2.p1.2.m2.2.2" xref="S2.p1.2.m2.2.2.cmml">S</mi><mo id="S2.p1.2.m2.3.4.2.2.1" xref="S2.p1.2.m2.3.4.2.1.cmml">,</mo><mi id="S2.p1.2.m2.3.3" xref="S2.p1.2.m2.3.3.cmml">T</mi></mrow><mo id="S2.p1.2.m2.3.4.1" xref="S2.p1.2.m2.3.4.1.cmml">⊆</mo><mrow id="S2.p1.2.m2.3.4.3" xref="S2.p1.2.m2.3.4.3.cmml"><mi id="S2.p1.2.m2.3.4.3.2" xref="S2.p1.2.m2.3.4.3.2.cmml">V</mi><mo id="S2.p1.2.m2.3.4.3.1" xref="S2.p1.2.m2.3.4.3.1.cmml">⁢</mo><mrow id="S2.p1.2.m2.3.4.3.3.2" xref="S2.p1.2.m2.3.4.3.cmml"><mo id="S2.p1.2.m2.3.4.3.3.2.1" stretchy="false" xref="S2.p1.2.m2.3.4.3.cmml">(</mo><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">G</mi><mo id="S2.p1.2.m2.3.4.3.3.2.2" stretchy="false" xref="S2.p1.2.m2.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.3b"><apply id="S2.p1.2.m2.3.4.cmml" xref="S2.p1.2.m2.3.4"><subset id="S2.p1.2.m2.3.4.1.cmml" xref="S2.p1.2.m2.3.4.1"></subset><list id="S2.p1.2.m2.3.4.2.1.cmml" xref="S2.p1.2.m2.3.4.2.2"><ci id="S2.p1.2.m2.2.2.cmml" xref="S2.p1.2.m2.2.2">𝑆</ci><ci id="S2.p1.2.m2.3.3.cmml" xref="S2.p1.2.m2.3.3">𝑇</ci></list><apply id="S2.p1.2.m2.3.4.3.cmml" xref="S2.p1.2.m2.3.4.3"><times id="S2.p1.2.m2.3.4.3.1.cmml" xref="S2.p1.2.m2.3.4.3.1"></times><ci id="S2.p1.2.m2.3.4.3.2.cmml" xref="S2.p1.2.m2.3.4.3.2">𝑉</ci><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.3c">S,T\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.3d">italic_S , italic_T ⊆ italic_V ( italic_G )</annotation></semantics></math>. An <em class="ltx_emph ltx_font_italic" id="S2.p1.4.2"><math alttext="S" class="ltx_Math" display="inline" id="S2.p1.3.1.m1.1"><semantics id="S2.p1.3.1.m1.1a"><mi id="S2.p1.3.1.m1.1.1" mathcolor="#C22147" xref="S2.p1.3.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.3.1.m1.1b"><ci id="S2.p1.3.1.m1.1.1.cmml" xref="S2.p1.3.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.1.m1.1d">italic_S</annotation></semantics></math><span class="ltx_text" id="S2.p1.4.2.1" style="color:#C22147;">-<math alttext="T" class="ltx_Math" display="inline" id="S2.p1.4.2.1.m1.1"><semantics id="S2.p1.4.2.1.m1.1a"><mi id="S2.p1.4.2.1.m1.1.1" mathcolor="#C22147" xref="S2.p1.4.2.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p1.4.2.1.m1.1b"><ci id="S2.p1.4.2.1.m1.1.1.cmml" xref="S2.p1.4.2.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.2.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.2.1.m1.1d">italic_T</annotation></semantics></math> path</span></em> in <math alttext="G" class="ltx_Math" display="inline" id="S2.p1.5.m3.1"><semantics id="S2.p1.5.m3.1a"><mi id="S2.p1.5.m3.1.1" xref="S2.p1.5.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p1.5.m3.1b"><ci id="S2.p1.5.m3.1.1.cmml" xref="S2.p1.5.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m3.1d">italic_G</annotation></semantics></math> is a path in <math alttext="G" class="ltx_Math" display="inline" id="S2.p1.6.m4.1"><semantics id="S2.p1.6.m4.1a"><mi id="S2.p1.6.m4.1.1" xref="S2.p1.6.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p1.6.m4.1b"><ci id="S2.p1.6.m4.1.1.cmml" xref="S2.p1.6.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m4.1d">italic_G</annotation></semantics></math> whose first vertex is in <math alttext="S" class="ltx_Math" display="inline" id="S2.p1.7.m5.1"><semantics id="S2.p1.7.m5.1a"><mi id="S2.p1.7.m5.1.1" xref="S2.p1.7.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.7.m5.1b"><ci id="S2.p1.7.m5.1.1.cmml" xref="S2.p1.7.m5.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m5.1d">italic_S</annotation></semantics></math> and whose last vertex is in <math alttext="T" class="ltx_Math" display="inline" id="S2.p1.8.m6.1"><semantics id="S2.p1.8.m6.1a"><mi id="S2.p1.8.m6.1.1" xref="S2.p1.8.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p1.8.m6.1b"><ci id="S2.p1.8.m6.1.1.cmml" xref="S2.p1.8.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m6.1d">italic_T</annotation></semantics></math>. We say that a set <math alttext="Z\subseteq V(G)" class="ltx_Math" display="inline" id="S2.p1.9.m7.1"><semantics id="S2.p1.9.m7.1a"><mrow id="S2.p1.9.m7.1.2" xref="S2.p1.9.m7.1.2.cmml"><mi id="S2.p1.9.m7.1.2.2" xref="S2.p1.9.m7.1.2.2.cmml">Z</mi><mo id="S2.p1.9.m7.1.2.1" xref="S2.p1.9.m7.1.2.1.cmml">⊆</mo><mrow id="S2.p1.9.m7.1.2.3" xref="S2.p1.9.m7.1.2.3.cmml"><mi id="S2.p1.9.m7.1.2.3.2" xref="S2.p1.9.m7.1.2.3.2.cmml">V</mi><mo id="S2.p1.9.m7.1.2.3.1" xref="S2.p1.9.m7.1.2.3.1.cmml">⁢</mo><mrow id="S2.p1.9.m7.1.2.3.3.2" xref="S2.p1.9.m7.1.2.3.cmml"><mo id="S2.p1.9.m7.1.2.3.3.2.1" stretchy="false" xref="S2.p1.9.m7.1.2.3.cmml">(</mo><mi id="S2.p1.9.m7.1.1" xref="S2.p1.9.m7.1.1.cmml">G</mi><mo id="S2.p1.9.m7.1.2.3.3.2.2" stretchy="false" xref="S2.p1.9.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m7.1b"><apply id="S2.p1.9.m7.1.2.cmml" xref="S2.p1.9.m7.1.2"><subset id="S2.p1.9.m7.1.2.1.cmml" xref="S2.p1.9.m7.1.2.1"></subset><ci id="S2.p1.9.m7.1.2.2.cmml" xref="S2.p1.9.m7.1.2.2">𝑍</ci><apply id="S2.p1.9.m7.1.2.3.cmml" xref="S2.p1.9.m7.1.2.3"><times id="S2.p1.9.m7.1.2.3.1.cmml" xref="S2.p1.9.m7.1.2.3.1"></times><ci id="S2.p1.9.m7.1.2.3.2.cmml" xref="S2.p1.9.m7.1.2.3.2">𝑉</ci><ci id="S2.p1.9.m7.1.1.cmml" xref="S2.p1.9.m7.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m7.1c">Z\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m7.1d">italic_Z ⊆ italic_V ( italic_G )</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S2.p1.38.4" style="color:#C22147;">separates</em> <math alttext="S" class="ltx_Math" display="inline" id="S2.p1.10.m8.1"><semantics id="S2.p1.10.m8.1a"><mi id="S2.p1.10.m8.1.1" xref="S2.p1.10.m8.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.10.m8.1b"><ci id="S2.p1.10.m8.1.1.cmml" xref="S2.p1.10.m8.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m8.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m8.1d">italic_S</annotation></semantics></math> and <math alttext="T" class="ltx_Math" display="inline" id="S2.p1.11.m9.1"><semantics id="S2.p1.11.m9.1a"><mi id="S2.p1.11.m9.1.1" xref="S2.p1.11.m9.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p1.11.m9.1b"><ci id="S2.p1.11.m9.1.1.cmml" xref="S2.p1.11.m9.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m9.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m9.1d">italic_T</annotation></semantics></math> if <math alttext="G-X" class="ltx_Math" display="inline" id="S2.p1.12.m10.1"><semantics id="S2.p1.12.m10.1a"><mrow id="S2.p1.12.m10.1.1" xref="S2.p1.12.m10.1.1.cmml"><mi id="S2.p1.12.m10.1.1.2" xref="S2.p1.12.m10.1.1.2.cmml">G</mi><mo id="S2.p1.12.m10.1.1.1" xref="S2.p1.12.m10.1.1.1.cmml">−</mo><mi id="S2.p1.12.m10.1.1.3" xref="S2.p1.12.m10.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.12.m10.1b"><apply id="S2.p1.12.m10.1.1.cmml" xref="S2.p1.12.m10.1.1"><minus id="S2.p1.12.m10.1.1.1.cmml" xref="S2.p1.12.m10.1.1.1"></minus><ci id="S2.p1.12.m10.1.1.2.cmml" xref="S2.p1.12.m10.1.1.2">𝐺</ci><ci id="S2.p1.12.m10.1.1.3.cmml" xref="S2.p1.12.m10.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m10.1c">G-X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m10.1d">italic_G - italic_X</annotation></semantics></math> has no <math alttext="S" class="ltx_Math" display="inline" id="S2.p1.13.m11.1"><semantics id="S2.p1.13.m11.1a"><mi id="S2.p1.13.m11.1.1" xref="S2.p1.13.m11.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.13.m11.1b"><ci id="S2.p1.13.m11.1.1.cmml" xref="S2.p1.13.m11.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m11.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m11.1d">italic_S</annotation></semantics></math>-<math alttext="T" class="ltx_Math" display="inline" id="S2.p1.14.m12.1"><semantics id="S2.p1.14.m12.1a"><mi id="S2.p1.14.m12.1.1" xref="S2.p1.14.m12.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p1.14.m12.1b"><ci id="S2.p1.14.m12.1.1.cmml" xref="S2.p1.14.m12.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m12.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m12.1d">italic_T</annotation></semantics></math> path. When <math alttext="Z" class="ltx_Math" display="inline" id="S2.p1.15.m13.1"><semantics id="S2.p1.15.m13.1a"><mi id="S2.p1.15.m13.1.1" xref="S2.p1.15.m13.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S2.p1.15.m13.1b"><ci id="S2.p1.15.m13.1.1.cmml" xref="S2.p1.15.m13.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m13.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m13.1d">italic_Z</annotation></semantics></math> separates <math alttext="S" class="ltx_Math" display="inline" id="S2.p1.16.m14.1"><semantics id="S2.p1.16.m14.1a"><mi id="S2.p1.16.m14.1.1" xref="S2.p1.16.m14.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.16.m14.1b"><ci id="S2.p1.16.m14.1.1.cmml" xref="S2.p1.16.m14.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.16.m14.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.16.m14.1d">italic_S</annotation></semantics></math> and <math alttext="T" class="ltx_Math" display="inline" id="S2.p1.17.m15.1"><semantics id="S2.p1.17.m15.1a"><mi id="S2.p1.17.m15.1.1" xref="S2.p1.17.m15.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p1.17.m15.1b"><ci id="S2.p1.17.m15.1.1.cmml" xref="S2.p1.17.m15.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.17.m15.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p1.17.m15.1d">italic_T</annotation></semantics></math>, any separation <math alttext="(X,Y)" class="ltx_Math" display="inline" id="S2.p1.18.m16.2"><semantics id="S2.p1.18.m16.2a"><mrow id="S2.p1.18.m16.2.3.2" xref="S2.p1.18.m16.2.3.1.cmml"><mo id="S2.p1.18.m16.2.3.2.1" stretchy="false" xref="S2.p1.18.m16.2.3.1.cmml">(</mo><mi id="S2.p1.18.m16.1.1" xref="S2.p1.18.m16.1.1.cmml">X</mi><mo id="S2.p1.18.m16.2.3.2.2" xref="S2.p1.18.m16.2.3.1.cmml">,</mo><mi id="S2.p1.18.m16.2.2" xref="S2.p1.18.m16.2.2.cmml">Y</mi><mo id="S2.p1.18.m16.2.3.2.3" stretchy="false" xref="S2.p1.18.m16.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.18.m16.2b"><interval closure="open" id="S2.p1.18.m16.2.3.1.cmml" xref="S2.p1.18.m16.2.3.2"><ci id="S2.p1.18.m16.1.1.cmml" xref="S2.p1.18.m16.1.1">𝑋</ci><ci id="S2.p1.18.m16.2.2.cmml" xref="S2.p1.18.m16.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.18.m16.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.18.m16.2d">( italic_X , italic_Y )</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S2.p1.19.m17.1"><semantics id="S2.p1.19.m17.1a"><mi id="S2.p1.19.m17.1.1" xref="S2.p1.19.m17.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p1.19.m17.1b"><ci id="S2.p1.19.m17.1.1.cmml" xref="S2.p1.19.m17.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.19.m17.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p1.19.m17.1d">italic_G</annotation></semantics></math> with <math alttext="S\subseteq X" class="ltx_Math" display="inline" id="S2.p1.20.m18.1"><semantics id="S2.p1.20.m18.1a"><mrow id="S2.p1.20.m18.1.1" xref="S2.p1.20.m18.1.1.cmml"><mi id="S2.p1.20.m18.1.1.2" xref="S2.p1.20.m18.1.1.2.cmml">S</mi><mo id="S2.p1.20.m18.1.1.1" xref="S2.p1.20.m18.1.1.1.cmml">⊆</mo><mi id="S2.p1.20.m18.1.1.3" xref="S2.p1.20.m18.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.20.m18.1b"><apply id="S2.p1.20.m18.1.1.cmml" xref="S2.p1.20.m18.1.1"><subset id="S2.p1.20.m18.1.1.1.cmml" xref="S2.p1.20.m18.1.1.1"></subset><ci id="S2.p1.20.m18.1.1.2.cmml" xref="S2.p1.20.m18.1.1.2">𝑆</ci><ci id="S2.p1.20.m18.1.1.3.cmml" xref="S2.p1.20.m18.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.20.m18.1c">S\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.20.m18.1d">italic_S ⊆ italic_X</annotation></semantics></math>, <math alttext="T\subseteq Y" class="ltx_Math" display="inline" id="S2.p1.21.m19.1"><semantics id="S2.p1.21.m19.1a"><mrow id="S2.p1.21.m19.1.1" xref="S2.p1.21.m19.1.1.cmml"><mi id="S2.p1.21.m19.1.1.2" xref="S2.p1.21.m19.1.1.2.cmml">T</mi><mo id="S2.p1.21.m19.1.1.1" xref="S2.p1.21.m19.1.1.1.cmml">⊆</mo><mi id="S2.p1.21.m19.1.1.3" xref="S2.p1.21.m19.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.21.m19.1b"><apply id="S2.p1.21.m19.1.1.cmml" xref="S2.p1.21.m19.1.1"><subset id="S2.p1.21.m19.1.1.1.cmml" xref="S2.p1.21.m19.1.1.1"></subset><ci id="S2.p1.21.m19.1.1.2.cmml" xref="S2.p1.21.m19.1.1.2">𝑇</ci><ci id="S2.p1.21.m19.1.1.3.cmml" xref="S2.p1.21.m19.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.21.m19.1c">T\subseteq Y</annotation><annotation encoding="application/x-llamapun" id="S2.p1.21.m19.1d">italic_T ⊆ italic_Y</annotation></semantics></math> and <math alttext="X\cap Y=Z" class="ltx_Math" display="inline" id="S2.p1.22.m20.1"><semantics id="S2.p1.22.m20.1a"><mrow id="S2.p1.22.m20.1.1" xref="S2.p1.22.m20.1.1.cmml"><mrow id="S2.p1.22.m20.1.1.2" xref="S2.p1.22.m20.1.1.2.cmml"><mi id="S2.p1.22.m20.1.1.2.2" xref="S2.p1.22.m20.1.1.2.2.cmml">X</mi><mo id="S2.p1.22.m20.1.1.2.1" xref="S2.p1.22.m20.1.1.2.1.cmml">∩</mo><mi id="S2.p1.22.m20.1.1.2.3" xref="S2.p1.22.m20.1.1.2.3.cmml">Y</mi></mrow><mo id="S2.p1.22.m20.1.1.1" xref="S2.p1.22.m20.1.1.1.cmml">=</mo><mi id="S2.p1.22.m20.1.1.3" xref="S2.p1.22.m20.1.1.3.cmml">Z</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.22.m20.1b"><apply id="S2.p1.22.m20.1.1.cmml" xref="S2.p1.22.m20.1.1"><eq id="S2.p1.22.m20.1.1.1.cmml" xref="S2.p1.22.m20.1.1.1"></eq><apply id="S2.p1.22.m20.1.1.2.cmml" xref="S2.p1.22.m20.1.1.2"><intersect id="S2.p1.22.m20.1.1.2.1.cmml" xref="S2.p1.22.m20.1.1.2.1"></intersect><ci id="S2.p1.22.m20.1.1.2.2.cmml" xref="S2.p1.22.m20.1.1.2.2">𝑋</ci><ci id="S2.p1.22.m20.1.1.2.3.cmml" xref="S2.p1.22.m20.1.1.2.3">𝑌</ci></apply><ci id="S2.p1.22.m20.1.1.3.cmml" xref="S2.p1.22.m20.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.22.m20.1c">X\cap Y=Z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.22.m20.1d">italic_X ∩ italic_Y = italic_Z</annotation></semantics></math> is called an <em class="ltx_emph ltx_font_italic" id="S2.p1.23.3"><math alttext="(S,Z,T)" class="ltx_Math" display="inline" id="S2.p1.23.3.m1.3"><semantics id="S2.p1.23.3.m1.3a"><mrow id="S2.p1.23.3.m1.3.4.2" xref="S2.p1.23.3.m1.3.4.1.cmml"><mo id="S2.p1.23.3.m1.3.4.2.1" mathcolor="#C22147" stretchy="false" xref="S2.p1.23.3.m1.3.4.1.cmml">(</mo><mi id="S2.p1.23.3.m1.1.1" mathcolor="#C22147" xref="S2.p1.23.3.m1.1.1.cmml">S</mi><mo id="S2.p1.23.3.m1.3.4.2.2" mathcolor="#C22147" xref="S2.p1.23.3.m1.3.4.1.cmml">,</mo><mi id="S2.p1.23.3.m1.2.2" mathcolor="#C22147" xref="S2.p1.23.3.m1.2.2.cmml">Z</mi><mo id="S2.p1.23.3.m1.3.4.2.3" mathcolor="#C22147" xref="S2.p1.23.3.m1.3.4.1.cmml">,</mo><mi id="S2.p1.23.3.m1.3.3" mathcolor="#C22147" xref="S2.p1.23.3.m1.3.3.cmml">T</mi><mo id="S2.p1.23.3.m1.3.4.2.4" mathcolor="#C22147" stretchy="false" xref="S2.p1.23.3.m1.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.23.3.m1.3b"><vector id="S2.p1.23.3.m1.3.4.1.cmml" xref="S2.p1.23.3.m1.3.4.2"><ci id="S2.p1.23.3.m1.1.1.cmml" xref="S2.p1.23.3.m1.1.1">𝑆</ci><ci id="S2.p1.23.3.m1.2.2.cmml" xref="S2.p1.23.3.m1.2.2">𝑍</ci><ci id="S2.p1.23.3.m1.3.3.cmml" xref="S2.p1.23.3.m1.3.3">𝑇</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.23.3.m1.3c">(S,Z,T)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.23.3.m1.3d">( italic_S , italic_Z , italic_T )</annotation></semantics></math><span class="ltx_text" id="S2.p1.23.3.1" style="color:#C22147;">-separation</span></em>. To see that an <math alttext="(S,Z,T)" class="ltx_Math" display="inline" id="S2.p1.24.m21.3"><semantics id="S2.p1.24.m21.3a"><mrow id="S2.p1.24.m21.3.4.2" xref="S2.p1.24.m21.3.4.1.cmml"><mo id="S2.p1.24.m21.3.4.2.1" stretchy="false" xref="S2.p1.24.m21.3.4.1.cmml">(</mo><mi id="S2.p1.24.m21.1.1" xref="S2.p1.24.m21.1.1.cmml">S</mi><mo id="S2.p1.24.m21.3.4.2.2" xref="S2.p1.24.m21.3.4.1.cmml">,</mo><mi id="S2.p1.24.m21.2.2" xref="S2.p1.24.m21.2.2.cmml">Z</mi><mo id="S2.p1.24.m21.3.4.2.3" xref="S2.p1.24.m21.3.4.1.cmml">,</mo><mi id="S2.p1.24.m21.3.3" xref="S2.p1.24.m21.3.3.cmml">T</mi><mo id="S2.p1.24.m21.3.4.2.4" stretchy="false" xref="S2.p1.24.m21.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.24.m21.3b"><vector id="S2.p1.24.m21.3.4.1.cmml" xref="S2.p1.24.m21.3.4.2"><ci id="S2.p1.24.m21.1.1.cmml" xref="S2.p1.24.m21.1.1">𝑆</ci><ci id="S2.p1.24.m21.2.2.cmml" xref="S2.p1.24.m21.2.2">𝑍</ci><ci id="S2.p1.24.m21.3.3.cmml" xref="S2.p1.24.m21.3.3">𝑇</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.24.m21.3c">(S,Z,T)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.24.m21.3d">( italic_S , italic_Z , italic_T )</annotation></semantics></math>-separation always exists, let <math alttext="G_{X}" class="ltx_Math" display="inline" id="S2.p1.25.m22.1"><semantics id="S2.p1.25.m22.1a"><msub id="S2.p1.25.m22.1.1" xref="S2.p1.25.m22.1.1.cmml"><mi id="S2.p1.25.m22.1.1.2" xref="S2.p1.25.m22.1.1.2.cmml">G</mi><mi id="S2.p1.25.m22.1.1.3" xref="S2.p1.25.m22.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.25.m22.1b"><apply id="S2.p1.25.m22.1.1.cmml" xref="S2.p1.25.m22.1.1"><csymbol cd="ambiguous" id="S2.p1.25.m22.1.1.1.cmml" xref="S2.p1.25.m22.1.1">subscript</csymbol><ci id="S2.p1.25.m22.1.1.2.cmml" xref="S2.p1.25.m22.1.1.2">𝐺</ci><ci id="S2.p1.25.m22.1.1.3.cmml" xref="S2.p1.25.m22.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.25.m22.1c">G_{X}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.25.m22.1d">italic_G start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> be the union of all components of <math alttext="G-Z" class="ltx_Math" display="inline" id="S2.p1.26.m23.1"><semantics id="S2.p1.26.m23.1a"><mrow id="S2.p1.26.m23.1.1" xref="S2.p1.26.m23.1.1.cmml"><mi id="S2.p1.26.m23.1.1.2" xref="S2.p1.26.m23.1.1.2.cmml">G</mi><mo id="S2.p1.26.m23.1.1.1" xref="S2.p1.26.m23.1.1.1.cmml">−</mo><mi id="S2.p1.26.m23.1.1.3" xref="S2.p1.26.m23.1.1.3.cmml">Z</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.26.m23.1b"><apply id="S2.p1.26.m23.1.1.cmml" xref="S2.p1.26.m23.1.1"><minus id="S2.p1.26.m23.1.1.1.cmml" xref="S2.p1.26.m23.1.1.1"></minus><ci id="S2.p1.26.m23.1.1.2.cmml" xref="S2.p1.26.m23.1.1.2">𝐺</ci><ci id="S2.p1.26.m23.1.1.3.cmml" xref="S2.p1.26.m23.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.26.m23.1c">G-Z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.26.m23.1d">italic_G - italic_Z</annotation></semantics></math> that contain a vertex of <math alttext="S" class="ltx_Math" display="inline" id="S2.p1.27.m24.1"><semantics id="S2.p1.27.m24.1a"><mi id="S2.p1.27.m24.1.1" xref="S2.p1.27.m24.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.27.m24.1b"><ci id="S2.p1.27.m24.1.1.cmml" xref="S2.p1.27.m24.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.27.m24.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.27.m24.1d">italic_S</annotation></semantics></math>. Then <math alttext="S\subseteq V(G_{X})\cup Z" class="ltx_Math" display="inline" id="S2.p1.28.m25.1"><semantics id="S2.p1.28.m25.1a"><mrow id="S2.p1.28.m25.1.1" xref="S2.p1.28.m25.1.1.cmml"><mi id="S2.p1.28.m25.1.1.3" xref="S2.p1.28.m25.1.1.3.cmml">S</mi><mo id="S2.p1.28.m25.1.1.2" xref="S2.p1.28.m25.1.1.2.cmml">⊆</mo><mrow id="S2.p1.28.m25.1.1.1" xref="S2.p1.28.m25.1.1.1.cmml"><mrow id="S2.p1.28.m25.1.1.1.1" xref="S2.p1.28.m25.1.1.1.1.cmml"><mi id="S2.p1.28.m25.1.1.1.1.3" xref="S2.p1.28.m25.1.1.1.1.3.cmml">V</mi><mo id="S2.p1.28.m25.1.1.1.1.2" xref="S2.p1.28.m25.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.p1.28.m25.1.1.1.1.1.1" xref="S2.p1.28.m25.1.1.1.1.1.1.1.cmml"><mo id="S2.p1.28.m25.1.1.1.1.1.1.2" stretchy="false" xref="S2.p1.28.m25.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.p1.28.m25.1.1.1.1.1.1.1" xref="S2.p1.28.m25.1.1.1.1.1.1.1.cmml"><mi id="S2.p1.28.m25.1.1.1.1.1.1.1.2" xref="S2.p1.28.m25.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="S2.p1.28.m25.1.1.1.1.1.1.1.3" xref="S2.p1.28.m25.1.1.1.1.1.1.1.3.cmml">X</mi></msub><mo id="S2.p1.28.m25.1.1.1.1.1.1.3" stretchy="false" xref="S2.p1.28.m25.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.28.m25.1.1.1.2" xref="S2.p1.28.m25.1.1.1.2.cmml">∪</mo><mi id="S2.p1.28.m25.1.1.1.3" xref="S2.p1.28.m25.1.1.1.3.cmml">Z</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.28.m25.1b"><apply id="S2.p1.28.m25.1.1.cmml" xref="S2.p1.28.m25.1.1"><subset id="S2.p1.28.m25.1.1.2.cmml" xref="S2.p1.28.m25.1.1.2"></subset><ci id="S2.p1.28.m25.1.1.3.cmml" xref="S2.p1.28.m25.1.1.3">𝑆</ci><apply id="S2.p1.28.m25.1.1.1.cmml" xref="S2.p1.28.m25.1.1.1"><union id="S2.p1.28.m25.1.1.1.2.cmml" xref="S2.p1.28.m25.1.1.1.2"></union><apply id="S2.p1.28.m25.1.1.1.1.cmml" xref="S2.p1.28.m25.1.1.1.1"><times id="S2.p1.28.m25.1.1.1.1.2.cmml" xref="S2.p1.28.m25.1.1.1.1.2"></times><ci id="S2.p1.28.m25.1.1.1.1.3.cmml" xref="S2.p1.28.m25.1.1.1.1.3">𝑉</ci><apply id="S2.p1.28.m25.1.1.1.1.1.1.1.cmml" xref="S2.p1.28.m25.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.28.m25.1.1.1.1.1.1.1.1.cmml" xref="S2.p1.28.m25.1.1.1.1.1.1">subscript</csymbol><ci id="S2.p1.28.m25.1.1.1.1.1.1.1.2.cmml" xref="S2.p1.28.m25.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S2.p1.28.m25.1.1.1.1.1.1.1.3.cmml" xref="S2.p1.28.m25.1.1.1.1.1.1.1.3">𝑋</ci></apply></apply><ci id="S2.p1.28.m25.1.1.1.3.cmml" xref="S2.p1.28.m25.1.1.1.3">𝑍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.28.m25.1c">S\subseteq V(G_{X})\cup Z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.28.m25.1d">italic_S ⊆ italic_V ( italic_G start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) ∪ italic_Z</annotation></semantics></math>. Take <math alttext="G_{Y}" class="ltx_Math" display="inline" id="S2.p1.29.m26.1"><semantics id="S2.p1.29.m26.1a"><msub id="S2.p1.29.m26.1.1" xref="S2.p1.29.m26.1.1.cmml"><mi id="S2.p1.29.m26.1.1.2" xref="S2.p1.29.m26.1.1.2.cmml">G</mi><mi id="S2.p1.29.m26.1.1.3" xref="S2.p1.29.m26.1.1.3.cmml">Y</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.29.m26.1b"><apply id="S2.p1.29.m26.1.1.cmml" xref="S2.p1.29.m26.1.1"><csymbol cd="ambiguous" id="S2.p1.29.m26.1.1.1.cmml" xref="S2.p1.29.m26.1.1">subscript</csymbol><ci id="S2.p1.29.m26.1.1.2.cmml" xref="S2.p1.29.m26.1.1.2">𝐺</ci><ci id="S2.p1.29.m26.1.1.3.cmml" xref="S2.p1.29.m26.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.29.m26.1c">G_{Y}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.29.m26.1d">italic_G start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math> to be the union of all components of <math alttext="G-Z" class="ltx_Math" display="inline" id="S2.p1.30.m27.1"><semantics id="S2.p1.30.m27.1a"><mrow id="S2.p1.30.m27.1.1" xref="S2.p1.30.m27.1.1.cmml"><mi id="S2.p1.30.m27.1.1.2" xref="S2.p1.30.m27.1.1.2.cmml">G</mi><mo id="S2.p1.30.m27.1.1.1" xref="S2.p1.30.m27.1.1.1.cmml">−</mo><mi id="S2.p1.30.m27.1.1.3" xref="S2.p1.30.m27.1.1.3.cmml">Z</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.30.m27.1b"><apply id="S2.p1.30.m27.1.1.cmml" xref="S2.p1.30.m27.1.1"><minus id="S2.p1.30.m27.1.1.1.cmml" xref="S2.p1.30.m27.1.1.1"></minus><ci id="S2.p1.30.m27.1.1.2.cmml" xref="S2.p1.30.m27.1.1.2">𝐺</ci><ci id="S2.p1.30.m27.1.1.3.cmml" xref="S2.p1.30.m27.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.30.m27.1c">G-Z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.30.m27.1d">italic_G - italic_Z</annotation></semantics></math> not included in <math alttext="G_{X}" class="ltx_Math" display="inline" id="S2.p1.31.m28.1"><semantics id="S2.p1.31.m28.1a"><msub id="S2.p1.31.m28.1.1" xref="S2.p1.31.m28.1.1.cmml"><mi id="S2.p1.31.m28.1.1.2" xref="S2.p1.31.m28.1.1.2.cmml">G</mi><mi id="S2.p1.31.m28.1.1.3" xref="S2.p1.31.m28.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.31.m28.1b"><apply id="S2.p1.31.m28.1.1.cmml" xref="S2.p1.31.m28.1.1"><csymbol cd="ambiguous" id="S2.p1.31.m28.1.1.1.cmml" xref="S2.p1.31.m28.1.1">subscript</csymbol><ci id="S2.p1.31.m28.1.1.2.cmml" xref="S2.p1.31.m28.1.1.2">𝐺</ci><ci id="S2.p1.31.m28.1.1.3.cmml" xref="S2.p1.31.m28.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.31.m28.1c">G_{X}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.31.m28.1d">italic_G start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math>. Since every component of <math alttext="G_{X}" class="ltx_Math" display="inline" id="S2.p1.32.m29.1"><semantics id="S2.p1.32.m29.1a"><msub id="S2.p1.32.m29.1.1" xref="S2.p1.32.m29.1.1.cmml"><mi id="S2.p1.32.m29.1.1.2" xref="S2.p1.32.m29.1.1.2.cmml">G</mi><mi id="S2.p1.32.m29.1.1.3" xref="S2.p1.32.m29.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.32.m29.1b"><apply id="S2.p1.32.m29.1.1.cmml" xref="S2.p1.32.m29.1.1"><csymbol cd="ambiguous" id="S2.p1.32.m29.1.1.1.cmml" xref="S2.p1.32.m29.1.1">subscript</csymbol><ci id="S2.p1.32.m29.1.1.2.cmml" xref="S2.p1.32.m29.1.1.2">𝐺</ci><ci id="S2.p1.32.m29.1.1.3.cmml" xref="S2.p1.32.m29.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.32.m29.1c">G_{X}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.32.m29.1d">italic_G start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> contains a vertex in <math alttext="S" class="ltx_Math" display="inline" id="S2.p1.33.m30.1"><semantics id="S2.p1.33.m30.1a"><mi id="S2.p1.33.m30.1.1" xref="S2.p1.33.m30.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.33.m30.1b"><ci id="S2.p1.33.m30.1.1.cmml" xref="S2.p1.33.m30.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.33.m30.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.33.m30.1d">italic_S</annotation></semantics></math>, no component of <math alttext="G_{X}" class="ltx_Math" display="inline" id="S2.p1.34.m31.1"><semantics id="S2.p1.34.m31.1a"><msub id="S2.p1.34.m31.1.1" xref="S2.p1.34.m31.1.1.cmml"><mi id="S2.p1.34.m31.1.1.2" xref="S2.p1.34.m31.1.1.2.cmml">G</mi><mi id="S2.p1.34.m31.1.1.3" xref="S2.p1.34.m31.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.34.m31.1b"><apply id="S2.p1.34.m31.1.1.cmml" xref="S2.p1.34.m31.1.1"><csymbol cd="ambiguous" id="S2.p1.34.m31.1.1.1.cmml" xref="S2.p1.34.m31.1.1">subscript</csymbol><ci id="S2.p1.34.m31.1.1.2.cmml" xref="S2.p1.34.m31.1.1.2">𝐺</ci><ci id="S2.p1.34.m31.1.1.3.cmml" xref="S2.p1.34.m31.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.34.m31.1c">G_{X}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.34.m31.1d">italic_G start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> contains a vertex in <math alttext="T" class="ltx_Math" display="inline" id="S2.p1.35.m32.1"><semantics id="S2.p1.35.m32.1a"><mi id="S2.p1.35.m32.1.1" xref="S2.p1.35.m32.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p1.35.m32.1b"><ci id="S2.p1.35.m32.1.1.cmml" xref="S2.p1.35.m32.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.35.m32.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p1.35.m32.1d">italic_T</annotation></semantics></math>. Therefore, <math alttext="T\subseteq V(G_{Y})\cup Z" class="ltx_Math" display="inline" id="S2.p1.36.m33.1"><semantics id="S2.p1.36.m33.1a"><mrow id="S2.p1.36.m33.1.1" xref="S2.p1.36.m33.1.1.cmml"><mi id="S2.p1.36.m33.1.1.3" xref="S2.p1.36.m33.1.1.3.cmml">T</mi><mo id="S2.p1.36.m33.1.1.2" xref="S2.p1.36.m33.1.1.2.cmml">⊆</mo><mrow id="S2.p1.36.m33.1.1.1" xref="S2.p1.36.m33.1.1.1.cmml"><mrow id="S2.p1.36.m33.1.1.1.1" xref="S2.p1.36.m33.1.1.1.1.cmml"><mi id="S2.p1.36.m33.1.1.1.1.3" xref="S2.p1.36.m33.1.1.1.1.3.cmml">V</mi><mo id="S2.p1.36.m33.1.1.1.1.2" xref="S2.p1.36.m33.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.p1.36.m33.1.1.1.1.1.1" xref="S2.p1.36.m33.1.1.1.1.1.1.1.cmml"><mo id="S2.p1.36.m33.1.1.1.1.1.1.2" stretchy="false" xref="S2.p1.36.m33.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.p1.36.m33.1.1.1.1.1.1.1" xref="S2.p1.36.m33.1.1.1.1.1.1.1.cmml"><mi id="S2.p1.36.m33.1.1.1.1.1.1.1.2" xref="S2.p1.36.m33.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="S2.p1.36.m33.1.1.1.1.1.1.1.3" xref="S2.p1.36.m33.1.1.1.1.1.1.1.3.cmml">Y</mi></msub><mo id="S2.p1.36.m33.1.1.1.1.1.1.3" stretchy="false" xref="S2.p1.36.m33.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.36.m33.1.1.1.2" xref="S2.p1.36.m33.1.1.1.2.cmml">∪</mo><mi id="S2.p1.36.m33.1.1.1.3" xref="S2.p1.36.m33.1.1.1.3.cmml">Z</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.36.m33.1b"><apply id="S2.p1.36.m33.1.1.cmml" xref="S2.p1.36.m33.1.1"><subset id="S2.p1.36.m33.1.1.2.cmml" xref="S2.p1.36.m33.1.1.2"></subset><ci id="S2.p1.36.m33.1.1.3.cmml" xref="S2.p1.36.m33.1.1.3">𝑇</ci><apply id="S2.p1.36.m33.1.1.1.cmml" xref="S2.p1.36.m33.1.1.1"><union id="S2.p1.36.m33.1.1.1.2.cmml" xref="S2.p1.36.m33.1.1.1.2"></union><apply id="S2.p1.36.m33.1.1.1.1.cmml" xref="S2.p1.36.m33.1.1.1.1"><times id="S2.p1.36.m33.1.1.1.1.2.cmml" xref="S2.p1.36.m33.1.1.1.1.2"></times><ci id="S2.p1.36.m33.1.1.1.1.3.cmml" xref="S2.p1.36.m33.1.1.1.1.3">𝑉</ci><apply id="S2.p1.36.m33.1.1.1.1.1.1.1.cmml" xref="S2.p1.36.m33.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.36.m33.1.1.1.1.1.1.1.1.cmml" xref="S2.p1.36.m33.1.1.1.1.1.1">subscript</csymbol><ci id="S2.p1.36.m33.1.1.1.1.1.1.1.2.cmml" xref="S2.p1.36.m33.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S2.p1.36.m33.1.1.1.1.1.1.1.3.cmml" xref="S2.p1.36.m33.1.1.1.1.1.1.1.3">𝑌</ci></apply></apply><ci id="S2.p1.36.m33.1.1.1.3.cmml" xref="S2.p1.36.m33.1.1.1.3">𝑍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.36.m33.1c">T\subseteq V(G_{Y})\cup Z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.36.m33.1d">italic_T ⊆ italic_V ( italic_G start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) ∪ italic_Z</annotation></semantics></math>. Then the separation <math alttext="(X,Y):=(V(G_{X})\cup Z,V(G_{Y})\cup Z)" class="ltx_Math" display="inline" id="S2.p1.37.m34.4"><semantics id="S2.p1.37.m34.4a"><mrow id="S2.p1.37.m34.4.4" xref="S2.p1.37.m34.4.4.cmml"><mrow id="S2.p1.37.m34.4.4.4.2" xref="S2.p1.37.m34.4.4.4.1.cmml"><mo id="S2.p1.37.m34.4.4.4.2.1" stretchy="false" xref="S2.p1.37.m34.4.4.4.1.cmml">(</mo><mi id="S2.p1.37.m34.1.1" xref="S2.p1.37.m34.1.1.cmml">X</mi><mo id="S2.p1.37.m34.4.4.4.2.2" xref="S2.p1.37.m34.4.4.4.1.cmml">,</mo><mi id="S2.p1.37.m34.2.2" xref="S2.p1.37.m34.2.2.cmml">Y</mi><mo id="S2.p1.37.m34.4.4.4.2.3" rspace="0.278em" stretchy="false" xref="S2.p1.37.m34.4.4.4.1.cmml">)</mo></mrow><mo id="S2.p1.37.m34.4.4.3" rspace="0.278em" xref="S2.p1.37.m34.4.4.3.cmml">:=</mo><mrow id="S2.p1.37.m34.4.4.2.2" xref="S2.p1.37.m34.4.4.2.3.cmml"><mo id="S2.p1.37.m34.4.4.2.2.3" stretchy="false" xref="S2.p1.37.m34.4.4.2.3.cmml">(</mo><mrow id="S2.p1.37.m34.3.3.1.1.1" xref="S2.p1.37.m34.3.3.1.1.1.cmml"><mrow id="S2.p1.37.m34.3.3.1.1.1.1" xref="S2.p1.37.m34.3.3.1.1.1.1.cmml"><mi id="S2.p1.37.m34.3.3.1.1.1.1.3" xref="S2.p1.37.m34.3.3.1.1.1.1.3.cmml">V</mi><mo id="S2.p1.37.m34.3.3.1.1.1.1.2" xref="S2.p1.37.m34.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.p1.37.m34.3.3.1.1.1.1.1.1" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.cmml"><mo id="S2.p1.37.m34.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.p1.37.m34.3.3.1.1.1.1.1.1.1" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.cmml"><mi id="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.2" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.3" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.3.cmml">X</mi></msub><mo id="S2.p1.37.m34.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.37.m34.3.3.1.1.1.2" xref="S2.p1.37.m34.3.3.1.1.1.2.cmml">∪</mo><mi id="S2.p1.37.m34.3.3.1.1.1.3" xref="S2.p1.37.m34.3.3.1.1.1.3.cmml">Z</mi></mrow><mo id="S2.p1.37.m34.4.4.2.2.4" xref="S2.p1.37.m34.4.4.2.3.cmml">,</mo><mrow id="S2.p1.37.m34.4.4.2.2.2" xref="S2.p1.37.m34.4.4.2.2.2.cmml"><mrow id="S2.p1.37.m34.4.4.2.2.2.1" xref="S2.p1.37.m34.4.4.2.2.2.1.cmml"><mi id="S2.p1.37.m34.4.4.2.2.2.1.3" xref="S2.p1.37.m34.4.4.2.2.2.1.3.cmml">V</mi><mo id="S2.p1.37.m34.4.4.2.2.2.1.2" xref="S2.p1.37.m34.4.4.2.2.2.1.2.cmml">⁢</mo><mrow id="S2.p1.37.m34.4.4.2.2.2.1.1.1" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.cmml"><mo id="S2.p1.37.m34.4.4.2.2.2.1.1.1.2" stretchy="false" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.cmml">(</mo><msub id="S2.p1.37.m34.4.4.2.2.2.1.1.1.1" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.cmml"><mi id="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.2" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.2.cmml">G</mi><mi id="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.3" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.3.cmml">Y</mi></msub><mo id="S2.p1.37.m34.4.4.2.2.2.1.1.1.3" stretchy="false" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.37.m34.4.4.2.2.2.2" xref="S2.p1.37.m34.4.4.2.2.2.2.cmml">∪</mo><mi id="S2.p1.37.m34.4.4.2.2.2.3" xref="S2.p1.37.m34.4.4.2.2.2.3.cmml">Z</mi></mrow><mo id="S2.p1.37.m34.4.4.2.2.5" stretchy="false" xref="S2.p1.37.m34.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.37.m34.4b"><apply id="S2.p1.37.m34.4.4.cmml" xref="S2.p1.37.m34.4.4"><csymbol cd="latexml" id="S2.p1.37.m34.4.4.3.cmml" xref="S2.p1.37.m34.4.4.3">assign</csymbol><interval closure="open" id="S2.p1.37.m34.4.4.4.1.cmml" xref="S2.p1.37.m34.4.4.4.2"><ci id="S2.p1.37.m34.1.1.cmml" xref="S2.p1.37.m34.1.1">𝑋</ci><ci id="S2.p1.37.m34.2.2.cmml" xref="S2.p1.37.m34.2.2">𝑌</ci></interval><interval closure="open" id="S2.p1.37.m34.4.4.2.3.cmml" xref="S2.p1.37.m34.4.4.2.2"><apply id="S2.p1.37.m34.3.3.1.1.1.cmml" xref="S2.p1.37.m34.3.3.1.1.1"><union id="S2.p1.37.m34.3.3.1.1.1.2.cmml" xref="S2.p1.37.m34.3.3.1.1.1.2"></union><apply id="S2.p1.37.m34.3.3.1.1.1.1.cmml" xref="S2.p1.37.m34.3.3.1.1.1.1"><times id="S2.p1.37.m34.3.3.1.1.1.1.2.cmml" xref="S2.p1.37.m34.3.3.1.1.1.1.2"></times><ci id="S2.p1.37.m34.3.3.1.1.1.1.3.cmml" xref="S2.p1.37.m34.3.3.1.1.1.1.3">𝑉</ci><apply id="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.cmml" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.2.cmml" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.3.cmml" xref="S2.p1.37.m34.3.3.1.1.1.1.1.1.1.3">𝑋</ci></apply></apply><ci id="S2.p1.37.m34.3.3.1.1.1.3.cmml" xref="S2.p1.37.m34.3.3.1.1.1.3">𝑍</ci></apply><apply id="S2.p1.37.m34.4.4.2.2.2.cmml" xref="S2.p1.37.m34.4.4.2.2.2"><union id="S2.p1.37.m34.4.4.2.2.2.2.cmml" xref="S2.p1.37.m34.4.4.2.2.2.2"></union><apply id="S2.p1.37.m34.4.4.2.2.2.1.cmml" xref="S2.p1.37.m34.4.4.2.2.2.1"><times id="S2.p1.37.m34.4.4.2.2.2.1.2.cmml" xref="S2.p1.37.m34.4.4.2.2.2.1.2"></times><ci id="S2.p1.37.m34.4.4.2.2.2.1.3.cmml" xref="S2.p1.37.m34.4.4.2.2.2.1.3">𝑉</ci><apply id="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.cmml" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.1.cmml" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1">subscript</csymbol><ci id="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.2.cmml" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.2">𝐺</ci><ci id="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.3.cmml" xref="S2.p1.37.m34.4.4.2.2.2.1.1.1.1.3">𝑌</ci></apply></apply><ci id="S2.p1.37.m34.4.4.2.2.2.3.cmml" xref="S2.p1.37.m34.4.4.2.2.2.3">𝑍</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.37.m34.4c">(X,Y):=(V(G_{X})\cup Z,V(G_{Y})\cup Z)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.37.m34.4d">( italic_X , italic_Y ) := ( italic_V ( italic_G start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) ∪ italic_Z , italic_V ( italic_G start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) ∪ italic_Z )</annotation></semantics></math> is an <math alttext="(S,Z,T)" class="ltx_Math" display="inline" id="S2.p1.38.m35.3"><semantics id="S2.p1.38.m35.3a"><mrow id="S2.p1.38.m35.3.4.2" xref="S2.p1.38.m35.3.4.1.cmml"><mo id="S2.p1.38.m35.3.4.2.1" stretchy="false" xref="S2.p1.38.m35.3.4.1.cmml">(</mo><mi id="S2.p1.38.m35.1.1" xref="S2.p1.38.m35.1.1.cmml">S</mi><mo id="S2.p1.38.m35.3.4.2.2" xref="S2.p1.38.m35.3.4.1.cmml">,</mo><mi id="S2.p1.38.m35.2.2" xref="S2.p1.38.m35.2.2.cmml">Z</mi><mo id="S2.p1.38.m35.3.4.2.3" xref="S2.p1.38.m35.3.4.1.cmml">,</mo><mi id="S2.p1.38.m35.3.3" xref="S2.p1.38.m35.3.3.cmml">T</mi><mo id="S2.p1.38.m35.3.4.2.4" stretchy="false" xref="S2.p1.38.m35.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.38.m35.3b"><vector id="S2.p1.38.m35.3.4.1.cmml" xref="S2.p1.38.m35.3.4.2"><ci id="S2.p1.38.m35.1.1.cmml" xref="S2.p1.38.m35.1.1">𝑆</ci><ci id="S2.p1.38.m35.2.2.cmml" xref="S2.p1.38.m35.2.2">𝑍</ci><ci id="S2.p1.38.m35.3.3.cmml" xref="S2.p1.38.m35.3.3">𝑇</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.38.m35.3c">(S,Z,T)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.38.m35.3d">( italic_S , italic_Z , italic_T )</annotation></semantics></math>-separation.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">We make use of the following vertex connectivity version of Menger’s Theorem (see, for example, <cite class="ltx_cite ltx_citemacro_citet">Diestel [<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib2" title="">2</a>, Theorem 3.3.1]</cite>):</p> </div> <div class="ltx_theorem ltx_theorem_thm" id="Thmthm3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm3.1.1.1">Theorem 3</span></span><span class="ltx_text ltx_font_bold" id="Thmthm3.2.2"> </span>(Menger’s Theorem)<span class="ltx_text ltx_font_bold" id="Thmthm3.3.3">.</span> </h6> <div class="ltx_para" id="Thmthm3.p1"> <p class="ltx_p" id="Thmthm3.p1.5"><span class="ltx_text ltx_font_italic" id="Thmthm3.p1.5.5">Let <math alttext="G" class="ltx_Math" display="inline" id="Thmthm3.p1.1.1.m1.1"><semantics id="Thmthm3.p1.1.1.m1.1a"><mi id="Thmthm3.p1.1.1.m1.1.1" xref="Thmthm3.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm3.p1.1.1.m1.1b"><ci id="Thmthm3.p1.1.1.m1.1.1.cmml" xref="Thmthm3.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm3.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm3.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be a graph and let <math alttext="S" class="ltx_Math" display="inline" id="Thmthm3.p1.2.2.m2.1"><semantics id="Thmthm3.p1.2.2.m2.1a"><mi id="Thmthm3.p1.2.2.m2.1.1" xref="Thmthm3.p1.2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="Thmthm3.p1.2.2.m2.1b"><ci id="Thmthm3.p1.2.2.m2.1.1.cmml" xref="Thmthm3.p1.2.2.m2.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm3.p1.2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="Thmthm3.p1.2.2.m2.1d">italic_S</annotation></semantics></math> and <math alttext="T" class="ltx_Math" display="inline" id="Thmthm3.p1.3.3.m3.1"><semantics id="Thmthm3.p1.3.3.m3.1a"><mi id="Thmthm3.p1.3.3.m3.1.1" xref="Thmthm3.p1.3.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="Thmthm3.p1.3.3.m3.1b"><ci id="Thmthm3.p1.3.3.m3.1.1.cmml" xref="Thmthm3.p1.3.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm3.p1.3.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="Thmthm3.p1.3.3.m3.1d">italic_T</annotation></semantics></math> be subsets of <math alttext="V(G)" class="ltx_Math" display="inline" id="Thmthm3.p1.4.4.m4.1"><semantics id="Thmthm3.p1.4.4.m4.1a"><mrow id="Thmthm3.p1.4.4.m4.1.2" xref="Thmthm3.p1.4.4.m4.1.2.cmml"><mi id="Thmthm3.p1.4.4.m4.1.2.2" xref="Thmthm3.p1.4.4.m4.1.2.2.cmml">V</mi><mo id="Thmthm3.p1.4.4.m4.1.2.1" xref="Thmthm3.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="Thmthm3.p1.4.4.m4.1.2.3.2" xref="Thmthm3.p1.4.4.m4.1.2.cmml"><mo id="Thmthm3.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="Thmthm3.p1.4.4.m4.1.2.cmml">(</mo><mi id="Thmthm3.p1.4.4.m4.1.1" xref="Thmthm3.p1.4.4.m4.1.1.cmml">G</mi><mo id="Thmthm3.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="Thmthm3.p1.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm3.p1.4.4.m4.1b"><apply id="Thmthm3.p1.4.4.m4.1.2.cmml" xref="Thmthm3.p1.4.4.m4.1.2"><times id="Thmthm3.p1.4.4.m4.1.2.1.cmml" xref="Thmthm3.p1.4.4.m4.1.2.1"></times><ci id="Thmthm3.p1.4.4.m4.1.2.2.cmml" xref="Thmthm3.p1.4.4.m4.1.2.2">𝑉</ci><ci id="Thmthm3.p1.4.4.m4.1.1.cmml" xref="Thmthm3.p1.4.4.m4.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm3.p1.4.4.m4.1c">V(G)</annotation><annotation encoding="application/x-llamapun" id="Thmthm3.p1.4.4.m4.1d">italic_V ( italic_G )</annotation></semantics></math>. For each <math alttext="k\in\mathbb{N}" class="ltx_Math" display="inline" id="Thmthm3.p1.5.5.m5.1"><semantics id="Thmthm3.p1.5.5.m5.1a"><mrow id="Thmthm3.p1.5.5.m5.1.1" xref="Thmthm3.p1.5.5.m5.1.1.cmml"><mi id="Thmthm3.p1.5.5.m5.1.1.2" xref="Thmthm3.p1.5.5.m5.1.1.2.cmml">k</mi><mo id="Thmthm3.p1.5.5.m5.1.1.1" xref="Thmthm3.p1.5.5.m5.1.1.1.cmml">∈</mo><mi id="Thmthm3.p1.5.5.m5.1.1.3" xref="Thmthm3.p1.5.5.m5.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmthm3.p1.5.5.m5.1b"><apply id="Thmthm3.p1.5.5.m5.1.1.cmml" xref="Thmthm3.p1.5.5.m5.1.1"><in id="Thmthm3.p1.5.5.m5.1.1.1.cmml" xref="Thmthm3.p1.5.5.m5.1.1.1"></in><ci id="Thmthm3.p1.5.5.m5.1.1.2.cmml" xref="Thmthm3.p1.5.5.m5.1.1.2">𝑘</ci><ci id="Thmthm3.p1.5.5.m5.1.1.3.cmml" xref="Thmthm3.p1.5.5.m5.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm3.p1.5.5.m5.1c">k\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="Thmthm3.p1.5.5.m5.1d">italic_k ∈ blackboard_N</annotation></semantics></math>, exactly one of the following is true:</span></p> <ol class="ltx_enumerate" id="S2.I1"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.4"><math alttext="G" class="ltx_Math" display="inline" id="S2.I1.i1.p1.1.m1.1"><semantics id="S2.I1.i1.p1.1.m1.1a"><mi id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><ci id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.1"> contains </span><math alttext="k" class="ltx_Math" display="inline" id="S2.I1.i1.p1.2.m2.1"><semantics id="S2.I1.i1.p1.2.m2.1a"><mi id="S2.I1.i1.p1.2.m2.1.1" xref="S2.I1.i1.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.2.m2.1b"><ci id="S2.I1.i1.p1.2.m2.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.2.m2.1d">italic_k</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.2"> pairwise vertex-disjoint </span><math alttext="S" class="ltx_Math" display="inline" id="S2.I1.i1.p1.3.m3.1"><semantics id="S2.I1.i1.p1.3.m3.1a"><mi id="S2.I1.i1.p1.3.m3.1.1" xref="S2.I1.i1.p1.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.3.m3.1b"><ci id="S2.I1.i1.p1.3.m3.1.1.cmml" xref="S2.I1.i1.p1.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.3.m3.1d">italic_S</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.3">-</span><math alttext="T" class="ltx_Math" display="inline" id="S2.I1.i1.p1.4.m4.1"><semantics id="S2.I1.i1.p1.4.m4.1a"><mi id="S2.I1.i1.p1.4.m4.1.1" xref="S2.I1.i1.p1.4.m4.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.4.m4.1b"><ci id="S2.I1.i1.p1.4.m4.1.1.cmml" xref="S2.I1.i1.p1.4.m4.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.4.m4.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.4.m4.1d">italic_T</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.4.4"> paths; 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">(ii)</span> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.5"><math alttext="G" class="ltx_Math" display="inline" id="S2.I1.i2.p1.1.m1.1"><semantics id="S2.I1.i2.p1.1.m1.1a"><mi id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.1b"><ci id="S2.I1.i2.p1.1.m1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.1.m1.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.5.1"> has a vertex subset </span><math alttext="Z" class="ltx_Math" display="inline" id="S2.I1.i2.p1.2.m2.1"><semantics id="S2.I1.i2.p1.2.m2.1a"><mi id="S2.I1.i2.p1.2.m2.1.1" xref="S2.I1.i2.p1.2.m2.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.1b"><ci id="S2.I1.i2.p1.2.m2.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.2.m2.1d">italic_Z</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.5.2"> of size less than </span><math alttext="k" class="ltx_Math" display="inline" id="S2.I1.i2.p1.3.m3.1"><semantics id="S2.I1.i2.p1.3.m3.1a"><mi id="S2.I1.i2.p1.3.m3.1.1" xref="S2.I1.i2.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.3.m3.1b"><ci id="S2.I1.i2.p1.3.m3.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.3.m3.1d">italic_k</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.5.3"> that separates </span><math alttext="S" class="ltx_Math" display="inline" id="S2.I1.i2.p1.4.m4.1"><semantics id="S2.I1.i2.p1.4.m4.1a"><mi id="S2.I1.i2.p1.4.m4.1.1" xref="S2.I1.i2.p1.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.4.m4.1b"><ci id="S2.I1.i2.p1.4.m4.1.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.4.m4.1d">italic_S</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.5.4"> and </span><math alttext="T" class="ltx_Math" display="inline" id="S2.I1.i2.p1.5.m5.1"><semantics id="S2.I1.i2.p1.5.m5.1a"><mi id="S2.I1.i2.p1.5.m5.1.1" xref="S2.I1.i2.p1.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.5.m5.1b"><ci id="S2.I1.i2.p1.5.m5.1.1.cmml" xref="S2.I1.i2.p1.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.5.m5.1d">italic_T</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.5.5">.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.13">The <em class="ltx_emph ltx_font_italic" id="S2.p3.13.1" style="color:#C22147;">depth</em>, <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{depth}_{T}(x)" class="ltx_Math" display="inline" id="S2.p3.1.m1.2"><semantics id="S2.p3.1.m1.2a"><mrow id="S2.p3.1.m1.2.2.1" xref="S2.p3.1.m1.2.2.2.cmml"><msub id="S2.p3.1.m1.2.2.1.1" xref="S2.p3.1.m1.2.2.1.1.cmml"><mi id="S2.p3.1.m1.2.2.1.1.2" mathcolor="#C22147" xref="S2.p3.1.m1.2.2.1.1.2.cmml">depth</mi><mi id="S2.p3.1.m1.2.2.1.1.3" mathcolor="#C22147" xref="S2.p3.1.m1.2.2.1.1.3.cmml">T</mi></msub><mo id="S2.p3.1.m1.2.2.1a" xref="S2.p3.1.m1.2.2.2.cmml">⁡</mo><mrow id="S2.p3.1.m1.2.2.1.2" xref="S2.p3.1.m1.2.2.2.cmml"><mo id="S2.p3.1.m1.2.2.1.2.1" mathcolor="#C22147" stretchy="false" xref="S2.p3.1.m1.2.2.2.cmml">(</mo><mi id="S2.p3.1.m1.1.1" mathcolor="#C22147" xref="S2.p3.1.m1.1.1.cmml">x</mi><mo id="S2.p3.1.m1.2.2.1.2.2" mathcolor="#C22147" stretchy="false" xref="S2.p3.1.m1.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.2b"><apply id="S2.p3.1.m1.2.2.2.cmml" xref="S2.p3.1.m1.2.2.1"><apply id="S2.p3.1.m1.2.2.1.1.cmml" xref="S2.p3.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.p3.1.m1.2.2.1.1.1.cmml" xref="S2.p3.1.m1.2.2.1.1">subscript</csymbol><ci id="S2.p3.1.m1.2.2.1.1.2.cmml" xref="S2.p3.1.m1.2.2.1.1.2">depth</ci><ci id="S2.p3.1.m1.2.2.1.1.3.cmml" xref="S2.p3.1.m1.2.2.1.1.3">𝑇</ci></apply><ci id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.2c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{depth}_{T}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.2d">roman_depth start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> of a node <math alttext="x" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><mi id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><ci id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_x</annotation></semantics></math> in a rooted tree <math alttext="T" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><mi id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.1b"><ci id="S2.p3.3.m3.1.1.cmml" xref="S2.p3.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">italic_T</annotation></semantics></math> is the number of edges on the path from <math alttext="x" class="ltx_Math" display="inline" id="S2.p3.4.m4.1"><semantics id="S2.p3.4.m4.1a"><mi id="S2.p3.4.m4.1.1" xref="S2.p3.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p3.4.m4.1b"><ci id="S2.p3.4.m4.1.1.cmml" xref="S2.p3.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m4.1d">italic_x</annotation></semantics></math> to the root of <math alttext="T" class="ltx_Math" display="inline" id="S2.p3.5.m5.1"><semantics id="S2.p3.5.m5.1a"><mi id="S2.p3.5.m5.1.1" xref="S2.p3.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p3.5.m5.1b"><ci id="S2.p3.5.m5.1.1.cmml" xref="S2.p3.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m5.1d">italic_T</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.p3.13.2" style="color:#C22147;">height</em> of a rooted tree <math alttext="T" class="ltx_Math" display="inline" id="S2.p3.6.m6.1"><semantics id="S2.p3.6.m6.1a"><mi id="S2.p3.6.m6.1.1" xref="S2.p3.6.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p3.6.m6.1b"><ci id="S2.p3.6.m6.1.1.cmml" xref="S2.p3.6.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.6.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p3.6.m6.1d">italic_T</annotation></semantics></math> is <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{height}(T):=\max\{\operatorname{depth}_{T}(x):x% \in V(T)\}" class="ltx_Math" display="inline" id="S2.p3.7.m7.6"><semantics id="S2.p3.7.m7.6a"><mrow id="S2.p3.7.m7.6.6" xref="S2.p3.7.m7.6.6.cmml"><mrow id="S2.p3.7.m7.6.6.3.2" xref="S2.p3.7.m7.6.6.3.1.cmml"><mi id="S2.p3.7.m7.1.1" mathcolor="#C22147" xref="S2.p3.7.m7.1.1.cmml">height</mi><mo id="S2.p3.7.m7.6.6.3.2a" xref="S2.p3.7.m7.6.6.3.1.cmml">⁡</mo><mrow id="S2.p3.7.m7.6.6.3.2.1" xref="S2.p3.7.m7.6.6.3.1.cmml"><mo id="S2.p3.7.m7.6.6.3.2.1.1" mathcolor="#C22147" stretchy="false" xref="S2.p3.7.m7.6.6.3.1.cmml">(</mo><mi id="S2.p3.7.m7.2.2" mathcolor="#C22147" xref="S2.p3.7.m7.2.2.cmml">T</mi><mo id="S2.p3.7.m7.6.6.3.2.1.2" mathcolor="#C22147" rspace="0.278em" stretchy="false" xref="S2.p3.7.m7.6.6.3.1.cmml">)</mo></mrow></mrow><mo id="S2.p3.7.m7.6.6.2" rspace="0.278em" xref="S2.p3.7.m7.6.6.2.cmml">:=</mo><mrow id="S2.p3.7.m7.6.6.1.1" xref="S2.p3.7.m7.6.6.1.2.cmml"><mi id="S2.p3.7.m7.5.5" xref="S2.p3.7.m7.5.5.cmml">max</mi><mo id="S2.p3.7.m7.6.6.1.1a" xref="S2.p3.7.m7.6.6.1.2.cmml">⁡</mo><mrow id="S2.p3.7.m7.6.6.1.1.1" xref="S2.p3.7.m7.6.6.1.2.cmml"><mo id="S2.p3.7.m7.6.6.1.1.1.2" stretchy="false" xref="S2.p3.7.m7.6.6.1.2.cmml">{</mo><mrow id="S2.p3.7.m7.6.6.1.1.1.1" xref="S2.p3.7.m7.6.6.1.1.1.1.cmml"><mrow id="S2.p3.7.m7.6.6.1.1.1.1.1.1" xref="S2.p3.7.m7.6.6.1.1.1.1.1.2.cmml"><msub id="S2.p3.7.m7.6.6.1.1.1.1.1.1.1" xref="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.cmml"><mi id="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.2" xref="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.2.cmml">depth</mi><mi id="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.3" xref="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.3.cmml">T</mi></msub><mo id="S2.p3.7.m7.6.6.1.1.1.1.1.1a" xref="S2.p3.7.m7.6.6.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S2.p3.7.m7.6.6.1.1.1.1.1.1.2" xref="S2.p3.7.m7.6.6.1.1.1.1.1.2.cmml"><mo id="S2.p3.7.m7.6.6.1.1.1.1.1.1.2.1" stretchy="false" xref="S2.p3.7.m7.6.6.1.1.1.1.1.2.cmml">(</mo><mi id="S2.p3.7.m7.3.3" xref="S2.p3.7.m7.3.3.cmml">x</mi><mo id="S2.p3.7.m7.6.6.1.1.1.1.1.1.2.2" rspace="0.278em" stretchy="false" xref="S2.p3.7.m7.6.6.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p3.7.m7.6.6.1.1.1.1.2" rspace="0.278em" xref="S2.p3.7.m7.6.6.1.1.1.1.2.cmml">:</mo><mrow id="S2.p3.7.m7.6.6.1.1.1.1.3" xref="S2.p3.7.m7.6.6.1.1.1.1.3.cmml"><mi id="S2.p3.7.m7.6.6.1.1.1.1.3.2" xref="S2.p3.7.m7.6.6.1.1.1.1.3.2.cmml">x</mi><mo id="S2.p3.7.m7.6.6.1.1.1.1.3.1" xref="S2.p3.7.m7.6.6.1.1.1.1.3.1.cmml">∈</mo><mrow id="S2.p3.7.m7.6.6.1.1.1.1.3.3" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3.cmml"><mi id="S2.p3.7.m7.6.6.1.1.1.1.3.3.2" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3.2.cmml">V</mi><mo id="S2.p3.7.m7.6.6.1.1.1.1.3.3.1" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3.1.cmml">⁢</mo><mrow id="S2.p3.7.m7.6.6.1.1.1.1.3.3.3.2" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3.cmml"><mo id="S2.p3.7.m7.6.6.1.1.1.1.3.3.3.2.1" stretchy="false" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3.cmml">(</mo><mi id="S2.p3.7.m7.4.4" xref="S2.p3.7.m7.4.4.cmml">T</mi><mo id="S2.p3.7.m7.6.6.1.1.1.1.3.3.3.2.2" stretchy="false" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.p3.7.m7.6.6.1.1.1.3" stretchy="false" xref="S2.p3.7.m7.6.6.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.7.m7.6b"><apply id="S2.p3.7.m7.6.6.cmml" xref="S2.p3.7.m7.6.6"><csymbol cd="latexml" id="S2.p3.7.m7.6.6.2.cmml" xref="S2.p3.7.m7.6.6.2">assign</csymbol><apply id="S2.p3.7.m7.6.6.3.1.cmml" xref="S2.p3.7.m7.6.6.3.2"><ci id="S2.p3.7.m7.1.1.cmml" xref="S2.p3.7.m7.1.1">height</ci><ci id="S2.p3.7.m7.2.2.cmml" xref="S2.p3.7.m7.2.2">𝑇</ci></apply><apply id="S2.p3.7.m7.6.6.1.2.cmml" xref="S2.p3.7.m7.6.6.1.1"><max id="S2.p3.7.m7.5.5.cmml" xref="S2.p3.7.m7.5.5"></max><apply id="S2.p3.7.m7.6.6.1.1.1.1.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1"><ci id="S2.p3.7.m7.6.6.1.1.1.1.2.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.2">:</ci><apply id="S2.p3.7.m7.6.6.1.1.1.1.1.2.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.1.1"><apply id="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.1.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.2.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.2">depth</ci><ci id="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.3.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.1.1.1.3">𝑇</ci></apply><ci id="S2.p3.7.m7.3.3.cmml" xref="S2.p3.7.m7.3.3">𝑥</ci></apply><apply id="S2.p3.7.m7.6.6.1.1.1.1.3.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.3"><in id="S2.p3.7.m7.6.6.1.1.1.1.3.1.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.3.1"></in><ci id="S2.p3.7.m7.6.6.1.1.1.1.3.2.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.3.2">𝑥</ci><apply id="S2.p3.7.m7.6.6.1.1.1.1.3.3.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3"><times id="S2.p3.7.m7.6.6.1.1.1.1.3.3.1.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3.1"></times><ci id="S2.p3.7.m7.6.6.1.1.1.1.3.3.2.cmml" xref="S2.p3.7.m7.6.6.1.1.1.1.3.3.2">𝑉</ci><ci id="S2.p3.7.m7.4.4.cmml" xref="S2.p3.7.m7.4.4">𝑇</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.7.m7.6c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{height}(T):=\max\{\operatorname{depth}_{T}(x):x% \in V(T)\}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.7.m7.6d">roman_height ( italic_T ) := roman_max { roman_depth start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x ) : italic_x ∈ italic_V ( italic_T ) }</annotation></semantics></math>. For a node <math alttext="x" class="ltx_Math" display="inline" id="S2.p3.8.m8.1"><semantics id="S2.p3.8.m8.1a"><mi id="S2.p3.8.m8.1.1" xref="S2.p3.8.m8.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p3.8.m8.1b"><ci id="S2.p3.8.m8.1.1.cmml" xref="S2.p3.8.m8.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.8.m8.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p3.8.m8.1d">italic_x</annotation></semantics></math> in a rooted tree <math alttext="T" class="ltx_Math" display="inline" id="S2.p3.9.m9.1"><semantics id="S2.p3.9.m9.1a"><mi id="S2.p3.9.m9.1.1" xref="S2.p3.9.m9.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p3.9.m9.1b"><ci id="S2.p3.9.m9.1.1.cmml" xref="S2.p3.9.m9.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.9.m9.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p3.9.m9.1d">italic_T</annotation></semantics></math>, we let <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}T_{x}" class="ltx_Math" display="inline" id="S2.p3.10.m10.1"><semantics id="S2.p3.10.m10.1a"><msub id="S2.p3.10.m10.1.1" xref="S2.p3.10.m10.1.1.cmml"><mi id="S2.p3.10.m10.1.1.2" mathcolor="#C22147" xref="S2.p3.10.m10.1.1.2.cmml">T</mi><mi id="S2.p3.10.m10.1.1.3" mathcolor="#C22147" xref="S2.p3.10.m10.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p3.10.m10.1b"><apply id="S2.p3.10.m10.1.1.cmml" xref="S2.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S2.p3.10.m10.1.1.1.cmml" xref="S2.p3.10.m10.1.1">subscript</csymbol><ci id="S2.p3.10.m10.1.1.2.cmml" xref="S2.p3.10.m10.1.1.2">𝑇</ci><ci id="S2.p3.10.m10.1.1.3.cmml" xref="S2.p3.10.m10.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.10.m10.1c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}T_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.10.m10.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> denote the subtree of <math alttext="T" class="ltx_Math" display="inline" id="S2.p3.11.m11.1"><semantics id="S2.p3.11.m11.1a"><mi id="S2.p3.11.m11.1.1" xref="S2.p3.11.m11.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p3.11.m11.1b"><ci id="S2.p3.11.m11.1.1.cmml" xref="S2.p3.11.m11.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.11.m11.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p3.11.m11.1d">italic_T</annotation></semantics></math> induced by all the descendants of <math alttext="x" class="ltx_Math" display="inline" id="S2.p3.12.m12.1"><semantics id="S2.p3.12.m12.1a"><mi id="S2.p3.12.m12.1.1" xref="S2.p3.12.m12.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p3.12.m12.1b"><ci id="S2.p3.12.m12.1.1.cmml" xref="S2.p3.12.m12.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.12.m12.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p3.12.m12.1d">italic_x</annotation></semantics></math>, including <math alttext="x" class="ltx_Math" display="inline" id="S2.p3.13.m13.1"><semantics id="S2.p3.13.m13.1a"><mi id="S2.p3.13.m13.1.1" xref="S2.p3.13.m13.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p3.13.m13.1b"><ci id="S2.p3.13.m13.1.1.cmml" xref="S2.p3.13.m13.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.13.m13.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p3.13.m13.1d">italic_x</annotation></semantics></math> itself.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.29">A tree decomposition <math alttext="\mathcal{T}:=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="S2.p4.1.m1.1"><semantics id="S2.p4.1.m1.1a"><mrow id="S2.p4.1.m1.1b"><mi class="ltx_font_mathcaligraphic" id="S2.p4.1.m1.1.1">𝒯</mi><mo id="S2.p4.1.m1.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S2.p4.1.m1.1.3"><mo id="S2.p4.1.m1.1.3.1" stretchy="false">(</mo><msub id="S2.p4.1.m1.1.3.2"><mi id="S2.p4.1.m1.1.3.2.2">B</mi><mi id="S2.p4.1.m1.1.3.2.3">x</mi></msub><mo id="S2.p4.1.m1.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.p4.1.m1.1.3.4">x</mi><mo id="S2.p4.1.m1.1.3.5">∈</mo><mi id="S2.p4.1.m1.1.3.6">V</mi><mrow id="S2.p4.1.m1.1.3.7"><mo id="S2.p4.1.m1.1.3.7.1" stretchy="false">(</mo><mi id="S2.p4.1.m1.1.3.7.2">T</mi><mo id="S2.p4.1.m1.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S2.p4.1.m1.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.p4.1.m1.1c">\mathcal{T}:=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> of a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.p4.2.m2.1"><semantics id="S2.p4.2.m2.1a"><mi id="S2.p4.2.m2.1.1" xref="S2.p4.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p4.2.m2.1b"><ci id="S2.p4.2.m2.1.1.cmml" xref="S2.p4.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p4.2.m2.1d">italic_G</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.p4.29.1" style="color:#C22147;">rooted</em> if the tree <math alttext="T" class="ltx_Math" display="inline" id="S2.p4.3.m3.1"><semantics id="S2.p4.3.m3.1a"><mi id="S2.p4.3.m3.1.1" xref="S2.p4.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p4.3.m3.1b"><ci id="S2.p4.3.m3.1.1.cmml" xref="S2.p4.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p4.3.m3.1d">italic_T</annotation></semantics></math> is rooted. If <math alttext="x_{0}" class="ltx_Math" display="inline" id="S2.p4.4.m4.1"><semantics id="S2.p4.4.m4.1a"><msub id="S2.p4.4.m4.1.1" xref="S2.p4.4.m4.1.1.cmml"><mi id="S2.p4.4.m4.1.1.2" xref="S2.p4.4.m4.1.1.2.cmml">x</mi><mn id="S2.p4.4.m4.1.1.3" xref="S2.p4.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p4.4.m4.1b"><apply id="S2.p4.4.m4.1.1.cmml" xref="S2.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p4.4.m4.1.1.1.cmml" xref="S2.p4.4.m4.1.1">subscript</csymbol><ci id="S2.p4.4.m4.1.1.2.cmml" xref="S2.p4.4.m4.1.1.2">𝑥</ci><cn id="S2.p4.4.m4.1.1.3.cmml" type="integer" xref="S2.p4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.4.m4.1c">x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.4.m4.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the root of <math alttext="T" class="ltx_Math" display="inline" id="S2.p4.5.m5.1"><semantics id="S2.p4.5.m5.1a"><mi id="S2.p4.5.m5.1.1" xref="S2.p4.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p4.5.m5.1b"><ci id="S2.p4.5.m5.1.1.cmml" xref="S2.p4.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p4.5.m5.1d">italic_T</annotation></semantics></math>, then <math alttext="B_{x_{0}}" class="ltx_Math" display="inline" id="S2.p4.6.m6.1"><semantics id="S2.p4.6.m6.1a"><msub id="S2.p4.6.m6.1.1" xref="S2.p4.6.m6.1.1.cmml"><mi id="S2.p4.6.m6.1.1.2" xref="S2.p4.6.m6.1.1.2.cmml">B</mi><msub id="S2.p4.6.m6.1.1.3" xref="S2.p4.6.m6.1.1.3.cmml"><mi id="S2.p4.6.m6.1.1.3.2" xref="S2.p4.6.m6.1.1.3.2.cmml">x</mi><mn id="S2.p4.6.m6.1.1.3.3" xref="S2.p4.6.m6.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.p4.6.m6.1b"><apply id="S2.p4.6.m6.1.1.cmml" xref="S2.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S2.p4.6.m6.1.1.1.cmml" xref="S2.p4.6.m6.1.1">subscript</csymbol><ci id="S2.p4.6.m6.1.1.2.cmml" xref="S2.p4.6.m6.1.1.2">𝐵</ci><apply id="S2.p4.6.m6.1.1.3.cmml" xref="S2.p4.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.p4.6.m6.1.1.3.1.cmml" xref="S2.p4.6.m6.1.1.3">subscript</csymbol><ci id="S2.p4.6.m6.1.1.3.2.cmml" xref="S2.p4.6.m6.1.1.3.2">𝑥</ci><cn id="S2.p4.6.m6.1.1.3.3.cmml" type="integer" xref="S2.p4.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.6.m6.1c">B_{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.6.m6.1d">italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is called the <em class="ltx_emph ltx_font_italic" id="S2.p4.29.2" style="color:#C22147;">root bag</em> of <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S2.p4.7.m7.1"><semantics id="S2.p4.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S2.p4.7.m7.1.1" xref="S2.p4.7.m7.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S2.p4.7.m7.1b"><ci id="S2.p4.7.m7.1.1.cmml" xref="S2.p4.7.m7.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.7.m7.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.7.m7.1d">caligraphic_T</annotation></semantics></math>. Let <math alttext="\mathcal{T}:=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="S2.p4.8.m8.1"><semantics id="S2.p4.8.m8.1a"><mrow id="S2.p4.8.m8.1b"><mi class="ltx_font_mathcaligraphic" id="S2.p4.8.m8.1.1">𝒯</mi><mo id="S2.p4.8.m8.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S2.p4.8.m8.1.3"><mo id="S2.p4.8.m8.1.3.1" stretchy="false">(</mo><msub id="S2.p4.8.m8.1.3.2"><mi id="S2.p4.8.m8.1.3.2.2">B</mi><mi id="S2.p4.8.m8.1.3.2.3">x</mi></msub><mo id="S2.p4.8.m8.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.p4.8.m8.1.3.4">x</mi><mo id="S2.p4.8.m8.1.3.5">∈</mo><mi id="S2.p4.8.m8.1.3.6">V</mi><mrow id="S2.p4.8.m8.1.3.7"><mo id="S2.p4.8.m8.1.3.7.1" stretchy="false">(</mo><mi id="S2.p4.8.m8.1.3.7.2">T</mi><mo id="S2.p4.8.m8.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S2.p4.8.m8.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.p4.8.m8.1c">\mathcal{T}:=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S2.p4.8.m8.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> be a rooted tree decomposition of a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.p4.9.m9.1"><semantics id="S2.p4.9.m9.1a"><mi id="S2.p4.9.m9.1.1" xref="S2.p4.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p4.9.m9.1b"><ci id="S2.p4.9.m9.1.1.cmml" xref="S2.p4.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p4.9.m9.1d">italic_G</annotation></semantics></math> where <math alttext="x_{0}" class="ltx_Math" display="inline" id="S2.p4.10.m10.1"><semantics id="S2.p4.10.m10.1a"><msub id="S2.p4.10.m10.1.1" xref="S2.p4.10.m10.1.1.cmml"><mi id="S2.p4.10.m10.1.1.2" xref="S2.p4.10.m10.1.1.2.cmml">x</mi><mn id="S2.p4.10.m10.1.1.3" xref="S2.p4.10.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p4.10.m10.1b"><apply id="S2.p4.10.m10.1.1.cmml" xref="S2.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S2.p4.10.m10.1.1.1.cmml" xref="S2.p4.10.m10.1.1">subscript</csymbol><ci id="S2.p4.10.m10.1.1.2.cmml" xref="S2.p4.10.m10.1.1.2">𝑥</ci><cn id="S2.p4.10.m10.1.1.3.cmml" type="integer" xref="S2.p4.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.10.m10.1c">x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.10.m10.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the root of <math alttext="T" class="ltx_Math" display="inline" id="S2.p4.11.m11.1"><semantics id="S2.p4.11.m11.1a"><mi id="S2.p4.11.m11.1.1" xref="S2.p4.11.m11.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p4.11.m11.1b"><ci id="S2.p4.11.m11.1.1.cmml" xref="S2.p4.11.m11.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.11.m11.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p4.11.m11.1d">italic_T</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.p4.29.3" style="color:#C22147;">boundary</em> of <math alttext="x_{0}" class="ltx_Math" display="inline" id="S2.p4.12.m12.1"><semantics id="S2.p4.12.m12.1a"><msub id="S2.p4.12.m12.1.1" xref="S2.p4.12.m12.1.1.cmml"><mi id="S2.p4.12.m12.1.1.2" xref="S2.p4.12.m12.1.1.2.cmml">x</mi><mn id="S2.p4.12.m12.1.1.3" xref="S2.p4.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p4.12.m12.1b"><apply id="S2.p4.12.m12.1.1.cmml" xref="S2.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S2.p4.12.m12.1.1.1.cmml" xref="S2.p4.12.m12.1.1">subscript</csymbol><ci id="S2.p4.12.m12.1.1.2.cmml" xref="S2.p4.12.m12.1.1.2">𝑥</ci><cn id="S2.p4.12.m12.1.1.3.cmml" type="integer" xref="S2.p4.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.12.m12.1c">x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.12.m12.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{\partial}_{\mathcal{T}}(x_{0}):=\emptyset" class="ltx_Math" display="inline" id="S2.p4.13.m13.2"><semantics id="S2.p4.13.m13.2a"><mrow id="S2.p4.13.m13.2.2" xref="S2.p4.13.m13.2.2.cmml"><mrow id="S2.p4.13.m13.2.2.2.2" xref="S2.p4.13.m13.2.2.2.3.cmml"><msub id="S2.p4.13.m13.1.1.1.1.1" xref="S2.p4.13.m13.1.1.1.1.1.cmml"><mi id="S2.p4.13.m13.1.1.1.1.1.2" mathcolor="#C22147" mathvariant="normal" xref="S2.p4.13.m13.1.1.1.1.1.2.cmml">∂</mi><mi class="ltx_font_mathcaligraphic" id="S2.p4.13.m13.1.1.1.1.1.3" mathcolor="#C22147" xref="S2.p4.13.m13.1.1.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.p4.13.m13.2.2.2.2a" xref="S2.p4.13.m13.2.2.2.3.cmml">⁡</mo><mrow id="S2.p4.13.m13.2.2.2.2.2" xref="S2.p4.13.m13.2.2.2.3.cmml"><mo id="S2.p4.13.m13.2.2.2.2.2.2" mathcolor="#C22147" stretchy="false" xref="S2.p4.13.m13.2.2.2.3.cmml">(</mo><msub id="S2.p4.13.m13.2.2.2.2.2.1" xref="S2.p4.13.m13.2.2.2.2.2.1.cmml"><mi id="S2.p4.13.m13.2.2.2.2.2.1.2" mathcolor="#C22147" xref="S2.p4.13.m13.2.2.2.2.2.1.2.cmml">x</mi><mn id="S2.p4.13.m13.2.2.2.2.2.1.3" mathcolor="#C22147" xref="S2.p4.13.m13.2.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S2.p4.13.m13.2.2.2.2.2.3" mathcolor="#C22147" rspace="0.278em" stretchy="false" xref="S2.p4.13.m13.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.p4.13.m13.2.2.3" rspace="0.278em" xref="S2.p4.13.m13.2.2.3.cmml">:=</mo><mi id="S2.p4.13.m13.2.2.4" mathvariant="normal" xref="S2.p4.13.m13.2.2.4.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.13.m13.2b"><apply id="S2.p4.13.m13.2.2.cmml" xref="S2.p4.13.m13.2.2"><csymbol cd="latexml" id="S2.p4.13.m13.2.2.3.cmml" xref="S2.p4.13.m13.2.2.3">assign</csymbol><apply id="S2.p4.13.m13.2.2.2.3.cmml" xref="S2.p4.13.m13.2.2.2.2"><apply id="S2.p4.13.m13.1.1.1.1.1.cmml" xref="S2.p4.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p4.13.m13.1.1.1.1.1.1.cmml" xref="S2.p4.13.m13.1.1.1.1.1">subscript</csymbol><partialdiff id="S2.p4.13.m13.1.1.1.1.1.2.cmml" xref="S2.p4.13.m13.1.1.1.1.1.2"></partialdiff><ci id="S2.p4.13.m13.1.1.1.1.1.3.cmml" xref="S2.p4.13.m13.1.1.1.1.1.3">𝒯</ci></apply><apply id="S2.p4.13.m13.2.2.2.2.2.1.cmml" xref="S2.p4.13.m13.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S2.p4.13.m13.2.2.2.2.2.1.1.cmml" xref="S2.p4.13.m13.2.2.2.2.2.1">subscript</csymbol><ci id="S2.p4.13.m13.2.2.2.2.2.1.2.cmml" xref="S2.p4.13.m13.2.2.2.2.2.1.2">𝑥</ci><cn id="S2.p4.13.m13.2.2.2.2.2.1.3.cmml" type="integer" xref="S2.p4.13.m13.2.2.2.2.2.1.3">0</cn></apply></apply><emptyset id="S2.p4.13.m13.2.2.4.cmml" xref="S2.p4.13.m13.2.2.4"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.13.m13.2c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{\partial}_{\mathcal{T}}(x_{0}):=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S2.p4.13.m13.2d">∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) := ∅</annotation></semantics></math>. For a node <math alttext="x" class="ltx_Math" display="inline" id="S2.p4.14.m14.1"><semantics id="S2.p4.14.m14.1a"><mi id="S2.p4.14.m14.1.1" xref="S2.p4.14.m14.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p4.14.m14.1b"><ci id="S2.p4.14.m14.1.1.cmml" xref="S2.p4.14.m14.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.14.m14.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p4.14.m14.1d">italic_x</annotation></semantics></math> of <math alttext="T" class="ltx_Math" display="inline" id="S2.p4.15.m15.1"><semantics id="S2.p4.15.m15.1a"><mi id="S2.p4.15.m15.1.1" xref="S2.p4.15.m15.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p4.15.m15.1b"><ci id="S2.p4.15.m15.1.1.cmml" xref="S2.p4.15.m15.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.15.m15.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p4.15.m15.1d">italic_T</annotation></semantics></math> with parent <math alttext="y" class="ltx_Math" display="inline" id="S2.p4.16.m16.1"><semantics id="S2.p4.16.m16.1a"><mi id="S2.p4.16.m16.1.1" xref="S2.p4.16.m16.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.p4.16.m16.1b"><ci id="S2.p4.16.m16.1.1.cmml" xref="S2.p4.16.m16.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.16.m16.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.p4.16.m16.1d">italic_y</annotation></semantics></math>, the <em class="ltx_emph ltx_font_italic" id="S2.p4.29.4" style="color:#C22147;">boundary</em> of <math alttext="x" class="ltx_Math" display="inline" id="S2.p4.17.m17.1"><semantics id="S2.p4.17.m17.1a"><mi id="S2.p4.17.m17.1.1" xref="S2.p4.17.m17.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p4.17.m17.1b"><ci id="S2.p4.17.m17.1.1.cmml" xref="S2.p4.17.m17.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.17.m17.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p4.17.m17.1d">italic_x</annotation></semantics></math> is <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{\partial}_{\mathcal{T}}(x):=B_{x}\cap B_{y}" class="ltx_Math" display="inline" id="S2.p4.18.m18.2"><semantics id="S2.p4.18.m18.2a"><mrow id="S2.p4.18.m18.2.2" xref="S2.p4.18.m18.2.2.cmml"><mrow id="S2.p4.18.m18.2.2.1.1" xref="S2.p4.18.m18.2.2.1.2.cmml"><msub id="S2.p4.18.m18.2.2.1.1.1" xref="S2.p4.18.m18.2.2.1.1.1.cmml"><mi id="S2.p4.18.m18.2.2.1.1.1.2" mathcolor="#C22147" mathvariant="normal" xref="S2.p4.18.m18.2.2.1.1.1.2.cmml">∂</mi><mi class="ltx_font_mathcaligraphic" id="S2.p4.18.m18.2.2.1.1.1.3" mathcolor="#C22147" xref="S2.p4.18.m18.2.2.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.p4.18.m18.2.2.1.1a" xref="S2.p4.18.m18.2.2.1.2.cmml">⁡</mo><mrow id="S2.p4.18.m18.2.2.1.1.2" xref="S2.p4.18.m18.2.2.1.2.cmml"><mo id="S2.p4.18.m18.2.2.1.1.2.1" mathcolor="#C22147" stretchy="false" xref="S2.p4.18.m18.2.2.1.2.cmml">(</mo><mi id="S2.p4.18.m18.1.1" mathcolor="#C22147" xref="S2.p4.18.m18.1.1.cmml">x</mi><mo id="S2.p4.18.m18.2.2.1.1.2.2" mathcolor="#C22147" rspace="0.278em" stretchy="false" xref="S2.p4.18.m18.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p4.18.m18.2.2.2" rspace="0.278em" xref="S2.p4.18.m18.2.2.2.cmml">:=</mo><mrow id="S2.p4.18.m18.2.2.3" xref="S2.p4.18.m18.2.2.3.cmml"><msub id="S2.p4.18.m18.2.2.3.2" xref="S2.p4.18.m18.2.2.3.2.cmml"><mi id="S2.p4.18.m18.2.2.3.2.2" xref="S2.p4.18.m18.2.2.3.2.2.cmml">B</mi><mi id="S2.p4.18.m18.2.2.3.2.3" xref="S2.p4.18.m18.2.2.3.2.3.cmml">x</mi></msub><mo id="S2.p4.18.m18.2.2.3.1" xref="S2.p4.18.m18.2.2.3.1.cmml">∩</mo><msub id="S2.p4.18.m18.2.2.3.3" xref="S2.p4.18.m18.2.2.3.3.cmml"><mi id="S2.p4.18.m18.2.2.3.3.2" xref="S2.p4.18.m18.2.2.3.3.2.cmml">B</mi><mi id="S2.p4.18.m18.2.2.3.3.3" xref="S2.p4.18.m18.2.2.3.3.3.cmml">y</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.18.m18.2b"><apply id="S2.p4.18.m18.2.2.cmml" xref="S2.p4.18.m18.2.2"><csymbol cd="latexml" id="S2.p4.18.m18.2.2.2.cmml" xref="S2.p4.18.m18.2.2.2">assign</csymbol><apply id="S2.p4.18.m18.2.2.1.2.cmml" xref="S2.p4.18.m18.2.2.1.1"><apply id="S2.p4.18.m18.2.2.1.1.1.cmml" xref="S2.p4.18.m18.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.p4.18.m18.2.2.1.1.1.1.cmml" xref="S2.p4.18.m18.2.2.1.1.1">subscript</csymbol><partialdiff id="S2.p4.18.m18.2.2.1.1.1.2.cmml" xref="S2.p4.18.m18.2.2.1.1.1.2"></partialdiff><ci id="S2.p4.18.m18.2.2.1.1.1.3.cmml" xref="S2.p4.18.m18.2.2.1.1.1.3">𝒯</ci></apply><ci id="S2.p4.18.m18.1.1.cmml" xref="S2.p4.18.m18.1.1">𝑥</ci></apply><apply id="S2.p4.18.m18.2.2.3.cmml" xref="S2.p4.18.m18.2.2.3"><intersect id="S2.p4.18.m18.2.2.3.1.cmml" xref="S2.p4.18.m18.2.2.3.1"></intersect><apply id="S2.p4.18.m18.2.2.3.2.cmml" xref="S2.p4.18.m18.2.2.3.2"><csymbol cd="ambiguous" id="S2.p4.18.m18.2.2.3.2.1.cmml" xref="S2.p4.18.m18.2.2.3.2">subscript</csymbol><ci id="S2.p4.18.m18.2.2.3.2.2.cmml" xref="S2.p4.18.m18.2.2.3.2.2">𝐵</ci><ci id="S2.p4.18.m18.2.2.3.2.3.cmml" xref="S2.p4.18.m18.2.2.3.2.3">𝑥</ci></apply><apply id="S2.p4.18.m18.2.2.3.3.cmml" xref="S2.p4.18.m18.2.2.3.3"><csymbol cd="ambiguous" id="S2.p4.18.m18.2.2.3.3.1.cmml" xref="S2.p4.18.m18.2.2.3.3">subscript</csymbol><ci id="S2.p4.18.m18.2.2.3.3.2.cmml" xref="S2.p4.18.m18.2.2.3.3.2">𝐵</ci><ci id="S2.p4.18.m18.2.2.3.3.3.cmml" xref="S2.p4.18.m18.2.2.3.3.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.18.m18.2c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{\partial}_{\mathcal{T}}(x):=B_{x}\cap B_{y}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.18.m18.2d">∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) := italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.p4.29.5" style="color:#C22147;">interior</em> of a node <math alttext="x" class="ltx_Math" display="inline" id="S2.p4.19.m19.1"><semantics id="S2.p4.19.m19.1a"><mi id="S2.p4.19.m19.1.1" xref="S2.p4.19.m19.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p4.19.m19.1b"><ci id="S2.p4.19.m19.1.1.cmml" xref="S2.p4.19.m19.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.19.m19.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p4.19.m19.1d">italic_x</annotation></semantics></math> in <math alttext="T" class="ltx_Math" display="inline" id="S2.p4.20.m20.1"><semantics id="S2.p4.20.m20.1a"><mi id="S2.p4.20.m20.1.1" xref="S2.p4.20.m20.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p4.20.m20.1b"><ci id="S2.p4.20.m20.1.1.cmml" xref="S2.p4.20.m20.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.20.m20.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p4.20.m20.1d">italic_T</annotation></semantics></math> is <math alttext="\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{int}_{\mathcal{T}}(x):=(\bigcup_{x^{\prime}\in V% (T_{x})}B_{x^{\prime}})\setminus\operatorname{\partial}(x)" class="ltx_Math" display="inline" id="S2.p4.21.m21.6"><semantics id="S2.p4.21.m21.6a"><mrow id="S2.p4.21.m21.6.6" xref="S2.p4.21.m21.6.6.cmml"><mrow id="S2.p4.21.m21.5.5.1.1" xref="S2.p4.21.m21.5.5.1.2.cmml"><msub id="S2.p4.21.m21.5.5.1.1.1" xref="S2.p4.21.m21.5.5.1.1.1.cmml"><mi id="S2.p4.21.m21.5.5.1.1.1.2" mathcolor="#C22147" xref="S2.p4.21.m21.5.5.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.p4.21.m21.5.5.1.1.1.3" mathcolor="#C22147" xref="S2.p4.21.m21.5.5.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.p4.21.m21.5.5.1.1a" xref="S2.p4.21.m21.5.5.1.2.cmml">⁡</mo><mrow id="S2.p4.21.m21.5.5.1.1.2" xref="S2.p4.21.m21.5.5.1.2.cmml"><mo 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id="S2.p4.21.m21.1.1.1.1.2.cmml" xref="S2.p4.21.m21.1.1.1.1.2"></times><ci id="S2.p4.21.m21.1.1.1.1.3.cmml" xref="S2.p4.21.m21.1.1.1.1.3">𝑉</ci><apply id="S2.p4.21.m21.1.1.1.1.1.1.1.cmml" xref="S2.p4.21.m21.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p4.21.m21.1.1.1.1.1.1.1.1.cmml" xref="S2.p4.21.m21.1.1.1.1.1.1">subscript</csymbol><ci id="S2.p4.21.m21.1.1.1.1.1.1.1.2.cmml" xref="S2.p4.21.m21.1.1.1.1.1.1.1.2">𝑇</ci><ci id="S2.p4.21.m21.1.1.1.1.1.1.1.3.cmml" xref="S2.p4.21.m21.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply><apply id="S2.p4.21.m21.6.6.2.1.1.1.2.cmml" xref="S2.p4.21.m21.6.6.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.p4.21.m21.6.6.2.1.1.1.2.1.cmml" xref="S2.p4.21.m21.6.6.2.1.1.1.2">subscript</csymbol><ci id="S2.p4.21.m21.6.6.2.1.1.1.2.2.cmml" xref="S2.p4.21.m21.6.6.2.1.1.1.2.2">𝐵</ci><apply id="S2.p4.21.m21.6.6.2.1.1.1.2.3.cmml" xref="S2.p4.21.m21.6.6.2.1.1.1.2.3"><csymbol cd="ambiguous" id="S2.p4.21.m21.6.6.2.1.1.1.2.3.1.cmml" xref="S2.p4.21.m21.6.6.2.1.1.1.2.3">superscript</csymbol><ci id="S2.p4.21.m21.6.6.2.1.1.1.2.3.2.cmml" xref="S2.p4.21.m21.6.6.2.1.1.1.2.3.2">𝑥</ci><ci id="S2.p4.21.m21.6.6.2.1.1.1.2.3.3.cmml" xref="S2.p4.21.m21.6.6.2.1.1.1.2.3.3">′</ci></apply></apply></apply><apply id="S2.p4.21.m21.6.6.2.3.1.cmml" xref="S2.p4.21.m21.6.6.2.3.2"><partialdiff id="S2.p4.21.m21.3.3.cmml" xref="S2.p4.21.m21.3.3"></partialdiff><ci id="S2.p4.21.m21.4.4.cmml" xref="S2.p4.21.m21.4.4">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.21.m21.6c">\leavevmode\color[rgb]{0.76,0.13,0.28}\definecolor[named]{pgfstrokecolor}{rgb}% {0.76,0.13,0.28}\operatorname{int}_{\mathcal{T}}(x):=(\bigcup_{x^{\prime}\in V% (T_{x})}B_{x^{\prime}})\setminus\operatorname{\partial}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.p4.21.m21.6d">roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) := ( ⋃ start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ∖ ∂ ( italic_x )</annotation></semantics></math>. Note that, for the root <math alttext="x_{0}" class="ltx_Math" display="inline" id="S2.p4.22.m22.1"><semantics id="S2.p4.22.m22.1a"><msub id="S2.p4.22.m22.1.1" xref="S2.p4.22.m22.1.1.cmml"><mi id="S2.p4.22.m22.1.1.2" xref="S2.p4.22.m22.1.1.2.cmml">x</mi><mn id="S2.p4.22.m22.1.1.3" xref="S2.p4.22.m22.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p4.22.m22.1b"><apply id="S2.p4.22.m22.1.1.cmml" xref="S2.p4.22.m22.1.1"><csymbol cd="ambiguous" id="S2.p4.22.m22.1.1.1.cmml" xref="S2.p4.22.m22.1.1">subscript</csymbol><ci id="S2.p4.22.m22.1.1.2.cmml" xref="S2.p4.22.m22.1.1.2">𝑥</ci><cn id="S2.p4.22.m22.1.1.3.cmml" type="integer" xref="S2.p4.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.22.m22.1c">x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.22.m22.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="T" class="ltx_Math" display="inline" id="S2.p4.23.m23.1"><semantics id="S2.p4.23.m23.1a"><mi id="S2.p4.23.m23.1.1" xref="S2.p4.23.m23.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p4.23.m23.1b"><ci id="S2.p4.23.m23.1.1.cmml" xref="S2.p4.23.m23.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.23.m23.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p4.23.m23.1d">italic_T</annotation></semantics></math>, <math alttext="\operatorname{int}_{\mathcal{T}}(x_{0})=V(G)" class="ltx_Math" display="inline" id="S2.p4.24.m24.3"><semantics id="S2.p4.24.m24.3a"><mrow id="S2.p4.24.m24.3.3" xref="S2.p4.24.m24.3.3.cmml"><mrow id="S2.p4.24.m24.3.3.2.2" xref="S2.p4.24.m24.3.3.2.3.cmml"><msub id="S2.p4.24.m24.2.2.1.1.1" xref="S2.p4.24.m24.2.2.1.1.1.cmml"><mi id="S2.p4.24.m24.2.2.1.1.1.2" xref="S2.p4.24.m24.2.2.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.p4.24.m24.2.2.1.1.1.3" xref="S2.p4.24.m24.2.2.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.p4.24.m24.3.3.2.2a" xref="S2.p4.24.m24.3.3.2.3.cmml">⁡</mo><mrow id="S2.p4.24.m24.3.3.2.2.2" xref="S2.p4.24.m24.3.3.2.3.cmml"><mo id="S2.p4.24.m24.3.3.2.2.2.2" stretchy="false" xref="S2.p4.24.m24.3.3.2.3.cmml">(</mo><msub id="S2.p4.24.m24.3.3.2.2.2.1" xref="S2.p4.24.m24.3.3.2.2.2.1.cmml"><mi id="S2.p4.24.m24.3.3.2.2.2.1.2" xref="S2.p4.24.m24.3.3.2.2.2.1.2.cmml">x</mi><mn id="S2.p4.24.m24.3.3.2.2.2.1.3" xref="S2.p4.24.m24.3.3.2.2.2.1.3.cmml">0</mn></msub><mo id="S2.p4.24.m24.3.3.2.2.2.3" stretchy="false" xref="S2.p4.24.m24.3.3.2.3.cmml">)</mo></mrow></mrow><mo id="S2.p4.24.m24.3.3.3" xref="S2.p4.24.m24.3.3.3.cmml">=</mo><mrow id="S2.p4.24.m24.3.3.4" xref="S2.p4.24.m24.3.3.4.cmml"><mi id="S2.p4.24.m24.3.3.4.2" xref="S2.p4.24.m24.3.3.4.2.cmml">V</mi><mo id="S2.p4.24.m24.3.3.4.1" xref="S2.p4.24.m24.3.3.4.1.cmml">⁢</mo><mrow id="S2.p4.24.m24.3.3.4.3.2" xref="S2.p4.24.m24.3.3.4.cmml"><mo id="S2.p4.24.m24.3.3.4.3.2.1" stretchy="false" xref="S2.p4.24.m24.3.3.4.cmml">(</mo><mi id="S2.p4.24.m24.1.1" xref="S2.p4.24.m24.1.1.cmml">G</mi><mo id="S2.p4.24.m24.3.3.4.3.2.2" stretchy="false" xref="S2.p4.24.m24.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.24.m24.3b"><apply id="S2.p4.24.m24.3.3.cmml" xref="S2.p4.24.m24.3.3"><eq id="S2.p4.24.m24.3.3.3.cmml" xref="S2.p4.24.m24.3.3.3"></eq><apply id="S2.p4.24.m24.3.3.2.3.cmml" xref="S2.p4.24.m24.3.3.2.2"><apply id="S2.p4.24.m24.2.2.1.1.1.cmml" xref="S2.p4.24.m24.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.p4.24.m24.2.2.1.1.1.1.cmml" xref="S2.p4.24.m24.2.2.1.1.1">subscript</csymbol><ci id="S2.p4.24.m24.2.2.1.1.1.2.cmml" xref="S2.p4.24.m24.2.2.1.1.1.2">int</ci><ci id="S2.p4.24.m24.2.2.1.1.1.3.cmml" xref="S2.p4.24.m24.2.2.1.1.1.3">𝒯</ci></apply><apply id="S2.p4.24.m24.3.3.2.2.2.1.cmml" xref="S2.p4.24.m24.3.3.2.2.2.1"><csymbol cd="ambiguous" id="S2.p4.24.m24.3.3.2.2.2.1.1.cmml" xref="S2.p4.24.m24.3.3.2.2.2.1">subscript</csymbol><ci id="S2.p4.24.m24.3.3.2.2.2.1.2.cmml" xref="S2.p4.24.m24.3.3.2.2.2.1.2">𝑥</ci><cn id="S2.p4.24.m24.3.3.2.2.2.1.3.cmml" type="integer" xref="S2.p4.24.m24.3.3.2.2.2.1.3">0</cn></apply></apply><apply id="S2.p4.24.m24.3.3.4.cmml" xref="S2.p4.24.m24.3.3.4"><times id="S2.p4.24.m24.3.3.4.1.cmml" xref="S2.p4.24.m24.3.3.4.1"></times><ci id="S2.p4.24.m24.3.3.4.2.cmml" xref="S2.p4.24.m24.3.3.4.2">𝑉</ci><ci id="S2.p4.24.m24.1.1.cmml" xref="S2.p4.24.m24.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.24.m24.3c">\operatorname{int}_{\mathcal{T}}(x_{0})=V(G)</annotation><annotation encoding="application/x-llamapun" id="S2.p4.24.m24.3d">roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_V ( italic_G )</annotation></semantics></math>. From these definitions it follows that, if <math alttext="T_{y}\supseteq T_{x}" class="ltx_Math" display="inline" id="S2.p4.25.m25.1"><semantics id="S2.p4.25.m25.1a"><mrow id="S2.p4.25.m25.1.1" xref="S2.p4.25.m25.1.1.cmml"><msub id="S2.p4.25.m25.1.1.2" xref="S2.p4.25.m25.1.1.2.cmml"><mi id="S2.p4.25.m25.1.1.2.2" xref="S2.p4.25.m25.1.1.2.2.cmml">T</mi><mi id="S2.p4.25.m25.1.1.2.3" xref="S2.p4.25.m25.1.1.2.3.cmml">y</mi></msub><mo id="S2.p4.25.m25.1.1.1" xref="S2.p4.25.m25.1.1.cmml">⊇</mo><msub id="S2.p4.25.m25.1.1.3" xref="S2.p4.25.m25.1.1.3.cmml"><mi id="S2.p4.25.m25.1.1.3.2" xref="S2.p4.25.m25.1.1.3.2.cmml">T</mi><mi id="S2.p4.25.m25.1.1.3.3" xref="S2.p4.25.m25.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.25.m25.1b"><apply id="S2.p4.25.m25.1.1.cmml" xref="S2.p4.25.m25.1.1"><subset id="S2.p4.25.m25.1.1a.cmml" xref="S2.p4.25.m25.1.1"></subset><apply id="S2.p4.25.m25.1.1.3.cmml" xref="S2.p4.25.m25.1.1.3"><csymbol cd="ambiguous" id="S2.p4.25.m25.1.1.3.1.cmml" xref="S2.p4.25.m25.1.1.3">subscript</csymbol><ci id="S2.p4.25.m25.1.1.3.2.cmml" xref="S2.p4.25.m25.1.1.3.2">𝑇</ci><ci id="S2.p4.25.m25.1.1.3.3.cmml" xref="S2.p4.25.m25.1.1.3.3">𝑥</ci></apply><apply id="S2.p4.25.m25.1.1.2.cmml" xref="S2.p4.25.m25.1.1.2"><csymbol cd="ambiguous" id="S2.p4.25.m25.1.1.2.1.cmml" xref="S2.p4.25.m25.1.1.2">subscript</csymbol><ci id="S2.p4.25.m25.1.1.2.2.cmml" xref="S2.p4.25.m25.1.1.2.2">𝑇</ci><ci id="S2.p4.25.m25.1.1.2.3.cmml" xref="S2.p4.25.m25.1.1.2.3">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.25.m25.1c">T_{y}\supseteq T_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.25.m25.1d">italic_T start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⊇ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="\operatorname{int}_{\mathcal{T}}(y)\supseteq\operatorname{int}_{\mathcal{T}}(x)" class="ltx_Math" display="inline" id="S2.p4.26.m26.4"><semantics id="S2.p4.26.m26.4a"><mrow id="S2.p4.26.m26.4.4" xref="S2.p4.26.m26.4.4.cmml"><mrow id="S2.p4.26.m26.3.3.1.1" xref="S2.p4.26.m26.3.3.1.2.cmml"><msub id="S2.p4.26.m26.3.3.1.1.1" xref="S2.p4.26.m26.3.3.1.1.1.cmml"><mi id="S2.p4.26.m26.3.3.1.1.1.2" xref="S2.p4.26.m26.3.3.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.p4.26.m26.3.3.1.1.1.3" xref="S2.p4.26.m26.3.3.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.p4.26.m26.3.3.1.1a" xref="S2.p4.26.m26.3.3.1.2.cmml">⁡</mo><mrow id="S2.p4.26.m26.3.3.1.1.2" xref="S2.p4.26.m26.3.3.1.2.cmml"><mo id="S2.p4.26.m26.3.3.1.1.2.1" stretchy="false" xref="S2.p4.26.m26.3.3.1.2.cmml">(</mo><mi id="S2.p4.26.m26.1.1" xref="S2.p4.26.m26.1.1.cmml">y</mi><mo id="S2.p4.26.m26.3.3.1.1.2.2" stretchy="false" xref="S2.p4.26.m26.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p4.26.m26.4.4.3" xref="S2.p4.26.m26.4.4.cmml">⊇</mo><mrow id="S2.p4.26.m26.4.4.2.1" xref="S2.p4.26.m26.4.4.2.2.cmml"><msub id="S2.p4.26.m26.4.4.2.1.1" xref="S2.p4.26.m26.4.4.2.1.1.cmml"><mi id="S2.p4.26.m26.4.4.2.1.1.2" xref="S2.p4.26.m26.4.4.2.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.p4.26.m26.4.4.2.1.1.3" xref="S2.p4.26.m26.4.4.2.1.1.3.cmml">𝒯</mi></msub><mo id="S2.p4.26.m26.4.4.2.1a" xref="S2.p4.26.m26.4.4.2.2.cmml">⁡</mo><mrow id="S2.p4.26.m26.4.4.2.1.2" xref="S2.p4.26.m26.4.4.2.2.cmml"><mo id="S2.p4.26.m26.4.4.2.1.2.1" stretchy="false" xref="S2.p4.26.m26.4.4.2.2.cmml">(</mo><mi id="S2.p4.26.m26.2.2" xref="S2.p4.26.m26.2.2.cmml">x</mi><mo id="S2.p4.26.m26.4.4.2.1.2.2" stretchy="false" xref="S2.p4.26.m26.4.4.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.26.m26.4b"><apply id="S2.p4.26.m26.4.4.cmml" xref="S2.p4.26.m26.4.4"><subset id="S2.p4.26.m26.4.4a.cmml" xref="S2.p4.26.m26.4.4"></subset><apply id="S2.p4.26.m26.4.4.2.2.cmml" xref="S2.p4.26.m26.4.4.2.1"><apply id="S2.p4.26.m26.4.4.2.1.1.cmml" xref="S2.p4.26.m26.4.4.2.1.1"><csymbol cd="ambiguous" id="S2.p4.26.m26.4.4.2.1.1.1.cmml" xref="S2.p4.26.m26.4.4.2.1.1">subscript</csymbol><ci id="S2.p4.26.m26.4.4.2.1.1.2.cmml" xref="S2.p4.26.m26.4.4.2.1.1.2">int</ci><ci id="S2.p4.26.m26.4.4.2.1.1.3.cmml" xref="S2.p4.26.m26.4.4.2.1.1.3">𝒯</ci></apply><ci id="S2.p4.26.m26.2.2.cmml" xref="S2.p4.26.m26.2.2">𝑥</ci></apply><apply id="S2.p4.26.m26.3.3.1.2.cmml" xref="S2.p4.26.m26.3.3.1.1"><apply id="S2.p4.26.m26.3.3.1.1.1.cmml" xref="S2.p4.26.m26.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.p4.26.m26.3.3.1.1.1.1.cmml" xref="S2.p4.26.m26.3.3.1.1.1">subscript</csymbol><ci id="S2.p4.26.m26.3.3.1.1.1.2.cmml" xref="S2.p4.26.m26.3.3.1.1.1.2">int</ci><ci id="S2.p4.26.m26.3.3.1.1.1.3.cmml" xref="S2.p4.26.m26.3.3.1.1.1.3">𝒯</ci></apply><ci id="S2.p4.26.m26.1.1.cmml" xref="S2.p4.26.m26.1.1">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.26.m26.4c">\operatorname{int}_{\mathcal{T}}(y)\supseteq\operatorname{int}_{\mathcal{T}}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.p4.26.m26.4d">roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_y ) ⊇ roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math> and that <math alttext="\operatorname{int}_{\mathcal{T}}(y)\cup\operatorname{\partial}_{\mathcal{T}}(y% )\supseteq\operatorname{int}_{\mathcal{T}}(x)\cup\operatorname{\partial}_{% \mathcal{T}}(x)" class="ltx_Math" display="inline" id="S2.p4.27.m27.8"><semantics id="S2.p4.27.m27.8a"><mrow id="S2.p4.27.m27.8.8" xref="S2.p4.27.m27.8.8.cmml"><mrow id="S2.p4.27.m27.6.6.2" xref="S2.p4.27.m27.6.6.2.cmml"><mrow id="S2.p4.27.m27.5.5.1.1.1" xref="S2.p4.27.m27.5.5.1.1.2.cmml"><msub id="S2.p4.27.m27.5.5.1.1.1.1" xref="S2.p4.27.m27.5.5.1.1.1.1.cmml"><mi id="S2.p4.27.m27.5.5.1.1.1.1.2" xref="S2.p4.27.m27.5.5.1.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.p4.27.m27.5.5.1.1.1.1.3" xref="S2.p4.27.m27.5.5.1.1.1.1.3.cmml">𝒯</mi></msub><mo 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xref="S2.p4.27.m27.8.8.4.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.27.m27.8b"><apply id="S2.p4.27.m27.8.8.cmml" xref="S2.p4.27.m27.8.8"><subset id="S2.p4.27.m27.8.8a.cmml" xref="S2.p4.27.m27.8.8"></subset><apply id="S2.p4.27.m27.8.8.4.cmml" xref="S2.p4.27.m27.8.8.4"><union id="S2.p4.27.m27.8.8.4.3.cmml" xref="S2.p4.27.m27.8.8.4.3"></union><apply id="S2.p4.27.m27.7.7.3.1.2.cmml" xref="S2.p4.27.m27.7.7.3.1.1"><apply id="S2.p4.27.m27.7.7.3.1.1.1.cmml" xref="S2.p4.27.m27.7.7.3.1.1.1"><csymbol cd="ambiguous" id="S2.p4.27.m27.7.7.3.1.1.1.1.cmml" xref="S2.p4.27.m27.7.7.3.1.1.1">subscript</csymbol><ci id="S2.p4.27.m27.7.7.3.1.1.1.2.cmml" xref="S2.p4.27.m27.7.7.3.1.1.1.2">int</ci><ci id="S2.p4.27.m27.7.7.3.1.1.1.3.cmml" xref="S2.p4.27.m27.7.7.3.1.1.1.3">𝒯</ci></apply><ci id="S2.p4.27.m27.3.3.cmml" xref="S2.p4.27.m27.3.3">𝑥</ci></apply><apply id="S2.p4.27.m27.8.8.4.2.2.cmml" xref="S2.p4.27.m27.8.8.4.2.1"><apply 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xref="S2.p4.27.m27.5.5.1.1.1.1.3">𝒯</ci></apply><ci id="S2.p4.27.m27.1.1.cmml" xref="S2.p4.27.m27.1.1">𝑦</ci></apply><apply id="S2.p4.27.m27.6.6.2.2.2.cmml" xref="S2.p4.27.m27.6.6.2.2.1"><apply id="S2.p4.27.m27.6.6.2.2.1.1.cmml" xref="S2.p4.27.m27.6.6.2.2.1.1"><csymbol cd="ambiguous" id="S2.p4.27.m27.6.6.2.2.1.1.1.cmml" xref="S2.p4.27.m27.6.6.2.2.1.1">subscript</csymbol><partialdiff id="S2.p4.27.m27.6.6.2.2.1.1.2.cmml" xref="S2.p4.27.m27.6.6.2.2.1.1.2"></partialdiff><ci id="S2.p4.27.m27.6.6.2.2.1.1.3.cmml" xref="S2.p4.27.m27.6.6.2.2.1.1.3">𝒯</ci></apply><ci id="S2.p4.27.m27.2.2.cmml" xref="S2.p4.27.m27.2.2">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.27.m27.8c">\operatorname{int}_{\mathcal{T}}(y)\cup\operatorname{\partial}_{\mathcal{T}}(y% )\supseteq\operatorname{int}_{\mathcal{T}}(x)\cup\operatorname{\partial}_{% \mathcal{T}}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.p4.27.m27.8d">roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_y ) ∪ ∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_y ) ⊇ roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) ∪ ∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math>. In particular, these inclusion relations hold for every ancestor <math alttext="y" class="ltx_Math" display="inline" id="S2.p4.28.m28.1"><semantics id="S2.p4.28.m28.1a"><mi id="S2.p4.28.m28.1.1" xref="S2.p4.28.m28.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.p4.28.m28.1b"><ci id="S2.p4.28.m28.1.1.cmml" xref="S2.p4.28.m28.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.28.m28.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.p4.28.m28.1d">italic_y</annotation></semantics></math> of <math alttext="x" class="ltx_Math" display="inline" id="S2.p4.29.m29.1"><semantics id="S2.p4.29.m29.1a"><mi id="S2.p4.29.m29.1.1" xref="S2.p4.29.m29.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p4.29.m29.1b"><ci id="S2.p4.29.m29.1.1.cmml" xref="S2.p4.29.m29.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.29.m29.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p4.29.m29.1d">italic_x</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.4">The following lemma allows us to restrict a rooted tree decomposition of a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><mi id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.1b"><ci id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">italic_G</annotation></semantics></math> to the subgraph <math alttext="G[Y]" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><mrow id="S2.p5.2.m2.1.2" xref="S2.p5.2.m2.1.2.cmml"><mi id="S2.p5.2.m2.1.2.2" xref="S2.p5.2.m2.1.2.2.cmml">G</mi><mo id="S2.p5.2.m2.1.2.1" xref="S2.p5.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.p5.2.m2.1.2.3.2" xref="S2.p5.2.m2.1.2.3.1.cmml"><mo id="S2.p5.2.m2.1.2.3.2.1" stretchy="false" xref="S2.p5.2.m2.1.2.3.1.1.cmml">[</mo><mi id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml">Y</mi><mo id="S2.p5.2.m2.1.2.3.2.2" stretchy="false" xref="S2.p5.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><apply id="S2.p5.2.m2.1.2.cmml" xref="S2.p5.2.m2.1.2"><times id="S2.p5.2.m2.1.2.1.cmml" xref="S2.p5.2.m2.1.2.1"></times><ci id="S2.p5.2.m2.1.2.2.cmml" xref="S2.p5.2.m2.1.2.2">𝐺</ci><apply id="S2.p5.2.m2.1.2.3.1.cmml" xref="S2.p5.2.m2.1.2.3.2"><csymbol cd="latexml" id="S2.p5.2.m2.1.2.3.1.1.cmml" xref="S2.p5.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.2.m2.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">italic_G [ italic_Y ]</annotation></semantics></math> induced by one part of a separation <math alttext="(X,Y)" class="ltx_Math" display="inline" id="S2.p5.3.m3.2"><semantics id="S2.p5.3.m3.2a"><mrow id="S2.p5.3.m3.2.3.2" xref="S2.p5.3.m3.2.3.1.cmml"><mo id="S2.p5.3.m3.2.3.2.1" stretchy="false" xref="S2.p5.3.m3.2.3.1.cmml">(</mo><mi id="S2.p5.3.m3.1.1" xref="S2.p5.3.m3.1.1.cmml">X</mi><mo id="S2.p5.3.m3.2.3.2.2" xref="S2.p5.3.m3.2.3.1.cmml">,</mo><mi id="S2.p5.3.m3.2.2" xref="S2.p5.3.m3.2.2.cmml">Y</mi><mo id="S2.p5.3.m3.2.3.2.3" stretchy="false" xref="S2.p5.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.3.m3.2b"><interval closure="open" id="S2.p5.3.m3.2.3.1.cmml" xref="S2.p5.3.m3.2.3.2"><ci id="S2.p5.3.m3.1.1.cmml" xref="S2.p5.3.m3.1.1">𝑋</ci><ci id="S2.p5.3.m3.2.2.cmml" xref="S2.p5.3.m3.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.3.m3.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S2.p5.3.m3.2d">( italic_X , italic_Y )</annotation></semantics></math> in such a way that <math alttext="X\cap Y" class="ltx_Math" display="inline" id="S2.p5.4.m4.1"><semantics id="S2.p5.4.m4.1a"><mrow id="S2.p5.4.m4.1.1" xref="S2.p5.4.m4.1.1.cmml"><mi id="S2.p5.4.m4.1.1.2" xref="S2.p5.4.m4.1.1.2.cmml">X</mi><mo id="S2.p5.4.m4.1.1.1" xref="S2.p5.4.m4.1.1.1.cmml">∩</mo><mi id="S2.p5.4.m4.1.1.3" xref="S2.p5.4.m4.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.4.m4.1b"><apply id="S2.p5.4.m4.1.1.cmml" xref="S2.p5.4.m4.1.1"><intersect id="S2.p5.4.m4.1.1.1.cmml" xref="S2.p5.4.m4.1.1.1"></intersect><ci id="S2.p5.4.m4.1.1.2.cmml" xref="S2.p5.4.m4.1.1.2">𝑋</ci><ci id="S2.p5.4.m4.1.1.3.cmml" xref="S2.p5.4.m4.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.4.m4.1c">X\cap Y</annotation><annotation encoding="application/x-llamapun" id="S2.p5.4.m4.1d">italic_X ∩ italic_Y</annotation></semantics></math> is contained in a single bag (the root bag) of the resulting decomposition.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="Thmthm4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm4.1.1.1">Lemma 4</span></span><span class="ltx_text ltx_font_bold" id="Thmthm4.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm4.p1"> <p class="ltx_p" id="Thmthm4.p1.11"><span class="ltx_text ltx_font_italic" id="Thmthm4.p1.11.11">Let <math alttext="\mathcal{T}^{\prime}:=(B^{\prime}_{x}:x\in V(T^{\prime}))" class="ltx_math_unparsed" display="inline" id="Thmthm4.p1.1.1.m1.1"><semantics id="Thmthm4.p1.1.1.m1.1a"><mrow id="Thmthm4.p1.1.1.m1.1b"><msup id="Thmthm4.p1.1.1.m1.1.1"><mi class="ltx_font_mathcaligraphic" id="Thmthm4.p1.1.1.m1.1.1.2">𝒯</mi><mo id="Thmthm4.p1.1.1.m1.1.1.3">′</mo></msup><mo id="Thmthm4.p1.1.1.m1.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="Thmthm4.p1.1.1.m1.1.3"><mo id="Thmthm4.p1.1.1.m1.1.3.1" stretchy="false">(</mo><msubsup id="Thmthm4.p1.1.1.m1.1.3.2"><mi id="Thmthm4.p1.1.1.m1.1.3.2.2.2">B</mi><mi id="Thmthm4.p1.1.1.m1.1.3.2.3">x</mi><mo id="Thmthm4.p1.1.1.m1.1.3.2.2.3">′</mo></msubsup><mo id="Thmthm4.p1.1.1.m1.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="Thmthm4.p1.1.1.m1.1.3.4">x</mi><mo id="Thmthm4.p1.1.1.m1.1.3.5">∈</mo><mi id="Thmthm4.p1.1.1.m1.1.3.6">V</mi><mrow id="Thmthm4.p1.1.1.m1.1.3.7"><mo id="Thmthm4.p1.1.1.m1.1.3.7.1" stretchy="false">(</mo><msup id="Thmthm4.p1.1.1.m1.1.3.7.2"><mi id="Thmthm4.p1.1.1.m1.1.3.7.2.2">T</mi><mo id="Thmthm4.p1.1.1.m1.1.3.7.2.3">′</mo></msup><mo id="Thmthm4.p1.1.1.m1.1.3.7.3" stretchy="false">)</mo></mrow><mo id="Thmthm4.p1.1.1.m1.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmthm4.p1.1.1.m1.1c">\mathcal{T}^{\prime}:=(B^{\prime}_{x}:x\in V(T^{\prime}))</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.1.1.m1.1d">caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math> be a rooted tree decomposition of a graph <math alttext="G" class="ltx_Math" display="inline" id="Thmthm4.p1.2.2.m2.1"><semantics id="Thmthm4.p1.2.2.m2.1a"><mi id="Thmthm4.p1.2.2.m2.1.1" xref="Thmthm4.p1.2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.2.2.m2.1b"><ci id="Thmthm4.p1.2.2.m2.1.1.cmml" xref="Thmthm4.p1.2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.2.2.m2.1d">italic_G</annotation></semantics></math>, let <math alttext="(X,Y)" class="ltx_Math" display="inline" id="Thmthm4.p1.3.3.m3.2"><semantics id="Thmthm4.p1.3.3.m3.2a"><mrow id="Thmthm4.p1.3.3.m3.2.3.2" xref="Thmthm4.p1.3.3.m3.2.3.1.cmml"><mo id="Thmthm4.p1.3.3.m3.2.3.2.1" stretchy="false" xref="Thmthm4.p1.3.3.m3.2.3.1.cmml">(</mo><mi id="Thmthm4.p1.3.3.m3.1.1" xref="Thmthm4.p1.3.3.m3.1.1.cmml">X</mi><mo id="Thmthm4.p1.3.3.m3.2.3.2.2" xref="Thmthm4.p1.3.3.m3.2.3.1.cmml">,</mo><mi id="Thmthm4.p1.3.3.m3.2.2" xref="Thmthm4.p1.3.3.m3.2.2.cmml">Y</mi><mo id="Thmthm4.p1.3.3.m3.2.3.2.3" stretchy="false" xref="Thmthm4.p1.3.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.3.3.m3.2b"><interval closure="open" id="Thmthm4.p1.3.3.m3.2.3.1.cmml" xref="Thmthm4.p1.3.3.m3.2.3.2"><ci id="Thmthm4.p1.3.3.m3.1.1.cmml" xref="Thmthm4.p1.3.3.m3.1.1">𝑋</ci><ci id="Thmthm4.p1.3.3.m3.2.2.cmml" xref="Thmthm4.p1.3.3.m3.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.3.3.m3.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.3.3.m3.2d">( italic_X , italic_Y )</annotation></semantics></math> be a separation of <math alttext="G" class="ltx_Math" display="inline" id="Thmthm4.p1.4.4.m4.1"><semantics id="Thmthm4.p1.4.4.m4.1a"><mi id="Thmthm4.p1.4.4.m4.1.1" xref="Thmthm4.p1.4.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.4.4.m4.1b"><ci id="Thmthm4.p1.4.4.m4.1.1.cmml" xref="Thmthm4.p1.4.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.4.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.4.4.m4.1d">italic_G</annotation></semantics></math>, and let <math alttext="B_{x}:=(B^{\prime}_{x}\cap Y)\cup(\operatorname{int}_{\mathcal{T}^{\prime}}(x)% \cap X\cap Y)" class="ltx_Math" display="inline" id="Thmthm4.p1.5.5.m5.3"><semantics id="Thmthm4.p1.5.5.m5.3a"><mrow id="Thmthm4.p1.5.5.m5.3.3" xref="Thmthm4.p1.5.5.m5.3.3.cmml"><msub id="Thmthm4.p1.5.5.m5.3.3.4" xref="Thmthm4.p1.5.5.m5.3.3.4.cmml"><mi id="Thmthm4.p1.5.5.m5.3.3.4.2" xref="Thmthm4.p1.5.5.m5.3.3.4.2.cmml">B</mi><mi id="Thmthm4.p1.5.5.m5.3.3.4.3" xref="Thmthm4.p1.5.5.m5.3.3.4.3.cmml">x</mi></msub><mo id="Thmthm4.p1.5.5.m5.3.3.3" lspace="0.278em" rspace="0.278em" xref="Thmthm4.p1.5.5.m5.3.3.3.cmml">:=</mo><mrow id="Thmthm4.p1.5.5.m5.3.3.2" xref="Thmthm4.p1.5.5.m5.3.3.2.cmml"><mrow id="Thmthm4.p1.5.5.m5.2.2.1.1.1" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.cmml"><mo id="Thmthm4.p1.5.5.m5.2.2.1.1.1.2" stretchy="false" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.cmml">(</mo><mrow id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.cmml"><msubsup id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.cmml"><mi id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.2" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.2.cmml">B</mi><mi id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.3" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.3.cmml">x</mi><mo id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.3" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.1" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.1.cmml">∩</mo><mi id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.3" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.3.cmml">Y</mi></mrow><mo id="Thmthm4.p1.5.5.m5.2.2.1.1.1.3" stretchy="false" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="Thmthm4.p1.5.5.m5.3.3.2.3" xref="Thmthm4.p1.5.5.m5.3.3.2.3.cmml">∪</mo><mrow id="Thmthm4.p1.5.5.m5.3.3.2.2.1" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.cmml"><mo id="Thmthm4.p1.5.5.m5.3.3.2.2.1.2" stretchy="false" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.cmml">(</mo><mrow id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.cmml"><mrow id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.2.cmml"><msub id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.cmml"><mi id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.2" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.2.cmml">int</mi><msup id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.2" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.2.cmml">𝒯</mi><mo id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.3" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1a" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.2.cmml">⁡</mo><mrow id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.2" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.2.cmml"><mo id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.2.1" stretchy="false" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.2.cmml">(</mo><mi id="Thmthm4.p1.5.5.m5.1.1" xref="Thmthm4.p1.5.5.m5.1.1.cmml">x</mi><mo id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.2.2" stretchy="false" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.2" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.2.cmml">∩</mo><mi id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.3" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.3.cmml">X</mi><mo id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.2a" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.2.cmml">∩</mo><mi id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.4" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.4.cmml">Y</mi></mrow><mo id="Thmthm4.p1.5.5.m5.3.3.2.2.1.3" stretchy="false" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.5.5.m5.3b"><apply id="Thmthm4.p1.5.5.m5.3.3.cmml" xref="Thmthm4.p1.5.5.m5.3.3"><csymbol cd="latexml" id="Thmthm4.p1.5.5.m5.3.3.3.cmml" xref="Thmthm4.p1.5.5.m5.3.3.3">assign</csymbol><apply id="Thmthm4.p1.5.5.m5.3.3.4.cmml" xref="Thmthm4.p1.5.5.m5.3.3.4"><csymbol cd="ambiguous" id="Thmthm4.p1.5.5.m5.3.3.4.1.cmml" xref="Thmthm4.p1.5.5.m5.3.3.4">subscript</csymbol><ci id="Thmthm4.p1.5.5.m5.3.3.4.2.cmml" xref="Thmthm4.p1.5.5.m5.3.3.4.2">𝐵</ci><ci id="Thmthm4.p1.5.5.m5.3.3.4.3.cmml" xref="Thmthm4.p1.5.5.m5.3.3.4.3">𝑥</ci></apply><apply id="Thmthm4.p1.5.5.m5.3.3.2.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2"><union id="Thmthm4.p1.5.5.m5.3.3.2.3.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.3"></union><apply id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1"><intersect id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.1.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.1"></intersect><apply id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.1.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2">subscript</csymbol><apply id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.1.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2">superscript</csymbol><ci id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.2.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.2">𝐵</ci><ci id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.3.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.2.3">′</ci></apply><ci id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.3.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.2.3">𝑥</ci></apply><ci id="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.3.cmml" xref="Thmthm4.p1.5.5.m5.2.2.1.1.1.1.3">𝑌</ci></apply><apply id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1"><intersect id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.2.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.2"></intersect><apply id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.2.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1"><apply id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.1.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1">subscript</csymbol><ci id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.2.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.2">int</ci><apply id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.1.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3">superscript</csymbol><ci id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.2.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.2">𝒯</ci><ci id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.3.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.1.1.1.3.3">′</ci></apply></apply><ci id="Thmthm4.p1.5.5.m5.1.1.cmml" xref="Thmthm4.p1.5.5.m5.1.1">𝑥</ci></apply><ci id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.3.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.3">𝑋</ci><ci id="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.4.cmml" xref="Thmthm4.p1.5.5.m5.3.3.2.2.1.1.4">𝑌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.5.5.m5.3c">B_{x}:=(B^{\prime}_{x}\cap Y)\cup(\operatorname{int}_{\mathcal{T}^{\prime}}(x)% \cap X\cap Y)</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.5.5.m5.3d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT := ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_Y ) ∪ ( roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) ∩ italic_X ∩ italic_Y )</annotation></semantics></math> for each <math alttext="x\in V(T^{\prime})" class="ltx_Math" display="inline" id="Thmthm4.p1.6.6.m6.1"><semantics id="Thmthm4.p1.6.6.m6.1a"><mrow id="Thmthm4.p1.6.6.m6.1.1" xref="Thmthm4.p1.6.6.m6.1.1.cmml"><mi id="Thmthm4.p1.6.6.m6.1.1.3" xref="Thmthm4.p1.6.6.m6.1.1.3.cmml">x</mi><mo id="Thmthm4.p1.6.6.m6.1.1.2" xref="Thmthm4.p1.6.6.m6.1.1.2.cmml">∈</mo><mrow id="Thmthm4.p1.6.6.m6.1.1.1" xref="Thmthm4.p1.6.6.m6.1.1.1.cmml"><mi id="Thmthm4.p1.6.6.m6.1.1.1.3" xref="Thmthm4.p1.6.6.m6.1.1.1.3.cmml">V</mi><mo id="Thmthm4.p1.6.6.m6.1.1.1.2" xref="Thmthm4.p1.6.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="Thmthm4.p1.6.6.m6.1.1.1.1.1" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.cmml"><mo id="Thmthm4.p1.6.6.m6.1.1.1.1.1.2" stretchy="false" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="Thmthm4.p1.6.6.m6.1.1.1.1.1.1" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.cmml"><mi id="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.2" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.2.cmml">T</mi><mo id="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.3" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="Thmthm4.p1.6.6.m6.1.1.1.1.1.3" stretchy="false" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.6.6.m6.1b"><apply id="Thmthm4.p1.6.6.m6.1.1.cmml" xref="Thmthm4.p1.6.6.m6.1.1"><in id="Thmthm4.p1.6.6.m6.1.1.2.cmml" xref="Thmthm4.p1.6.6.m6.1.1.2"></in><ci id="Thmthm4.p1.6.6.m6.1.1.3.cmml" xref="Thmthm4.p1.6.6.m6.1.1.3">𝑥</ci><apply id="Thmthm4.p1.6.6.m6.1.1.1.cmml" xref="Thmthm4.p1.6.6.m6.1.1.1"><times id="Thmthm4.p1.6.6.m6.1.1.1.2.cmml" xref="Thmthm4.p1.6.6.m6.1.1.1.2"></times><ci id="Thmthm4.p1.6.6.m6.1.1.1.3.cmml" xref="Thmthm4.p1.6.6.m6.1.1.1.3">𝑉</ci><apply id="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.1.cmml" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1">superscript</csymbol><ci id="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.2.cmml" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.2">𝑇</ci><ci id="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.3.cmml" xref="Thmthm4.p1.6.6.m6.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.6.6.m6.1c">x\in V(T^{\prime})</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.6.6.m6.1d">italic_x ∈ italic_V ( italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then <math alttext="\mathcal{T}:=(B_{x}:x\in V(T^{\prime}))" class="ltx_math_unparsed" display="inline" id="Thmthm4.p1.7.7.m7.1"><semantics id="Thmthm4.p1.7.7.m7.1a"><mrow id="Thmthm4.p1.7.7.m7.1b"><mi class="ltx_font_mathcaligraphic" id="Thmthm4.p1.7.7.m7.1.1">𝒯</mi><mo id="Thmthm4.p1.7.7.m7.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="Thmthm4.p1.7.7.m7.1.3"><mo id="Thmthm4.p1.7.7.m7.1.3.1" stretchy="false">(</mo><msub id="Thmthm4.p1.7.7.m7.1.3.2"><mi id="Thmthm4.p1.7.7.m7.1.3.2.2">B</mi><mi id="Thmthm4.p1.7.7.m7.1.3.2.3">x</mi></msub><mo id="Thmthm4.p1.7.7.m7.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="Thmthm4.p1.7.7.m7.1.3.4">x</mi><mo id="Thmthm4.p1.7.7.m7.1.3.5">∈</mo><mi id="Thmthm4.p1.7.7.m7.1.3.6">V</mi><mrow id="Thmthm4.p1.7.7.m7.1.3.7"><mo id="Thmthm4.p1.7.7.m7.1.3.7.1" stretchy="false">(</mo><msup id="Thmthm4.p1.7.7.m7.1.3.7.2"><mi id="Thmthm4.p1.7.7.m7.1.3.7.2.2">T</mi><mo id="Thmthm4.p1.7.7.m7.1.3.7.2.3">′</mo></msup><mo id="Thmthm4.p1.7.7.m7.1.3.7.3" stretchy="false">)</mo></mrow><mo id="Thmthm4.p1.7.7.m7.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmthm4.p1.7.7.m7.1c">\mathcal{T}:=(B_{x}:x\in V(T^{\prime}))</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.7.7.m7.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math> is a tree decomposition of <math alttext="G[Y]" class="ltx_Math" display="inline" id="Thmthm4.p1.8.8.m8.1"><semantics id="Thmthm4.p1.8.8.m8.1a"><mrow id="Thmthm4.p1.8.8.m8.1.2" xref="Thmthm4.p1.8.8.m8.1.2.cmml"><mi id="Thmthm4.p1.8.8.m8.1.2.2" xref="Thmthm4.p1.8.8.m8.1.2.2.cmml">G</mi><mo id="Thmthm4.p1.8.8.m8.1.2.1" xref="Thmthm4.p1.8.8.m8.1.2.1.cmml">⁢</mo><mrow id="Thmthm4.p1.8.8.m8.1.2.3.2" xref="Thmthm4.p1.8.8.m8.1.2.3.1.cmml"><mo id="Thmthm4.p1.8.8.m8.1.2.3.2.1" stretchy="false" xref="Thmthm4.p1.8.8.m8.1.2.3.1.1.cmml">[</mo><mi id="Thmthm4.p1.8.8.m8.1.1" xref="Thmthm4.p1.8.8.m8.1.1.cmml">Y</mi><mo id="Thmthm4.p1.8.8.m8.1.2.3.2.2" stretchy="false" xref="Thmthm4.p1.8.8.m8.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.8.8.m8.1b"><apply id="Thmthm4.p1.8.8.m8.1.2.cmml" xref="Thmthm4.p1.8.8.m8.1.2"><times id="Thmthm4.p1.8.8.m8.1.2.1.cmml" xref="Thmthm4.p1.8.8.m8.1.2.1"></times><ci id="Thmthm4.p1.8.8.m8.1.2.2.cmml" xref="Thmthm4.p1.8.8.m8.1.2.2">𝐺</ci><apply id="Thmthm4.p1.8.8.m8.1.2.3.1.cmml" xref="Thmthm4.p1.8.8.m8.1.2.3.2"><csymbol cd="latexml" id="Thmthm4.p1.8.8.m8.1.2.3.1.1.cmml" xref="Thmthm4.p1.8.8.m8.1.2.3.2.1">delimited-[]</csymbol><ci id="Thmthm4.p1.8.8.m8.1.1.cmml" xref="Thmthm4.p1.8.8.m8.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.8.8.m8.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.8.8.m8.1d">italic_G [ italic_Y ]</annotation></semantics></math>. Furthermore, the root bag <math alttext="B_{x_{0}}" class="ltx_Math" display="inline" id="Thmthm4.p1.9.9.m9.1"><semantics id="Thmthm4.p1.9.9.m9.1a"><msub id="Thmthm4.p1.9.9.m9.1.1" xref="Thmthm4.p1.9.9.m9.1.1.cmml"><mi id="Thmthm4.p1.9.9.m9.1.1.2" xref="Thmthm4.p1.9.9.m9.1.1.2.cmml">B</mi><msub id="Thmthm4.p1.9.9.m9.1.1.3" xref="Thmthm4.p1.9.9.m9.1.1.3.cmml"><mi id="Thmthm4.p1.9.9.m9.1.1.3.2" xref="Thmthm4.p1.9.9.m9.1.1.3.2.cmml">x</mi><mn id="Thmthm4.p1.9.9.m9.1.1.3.3" xref="Thmthm4.p1.9.9.m9.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.9.9.m9.1b"><apply id="Thmthm4.p1.9.9.m9.1.1.cmml" xref="Thmthm4.p1.9.9.m9.1.1"><csymbol cd="ambiguous" id="Thmthm4.p1.9.9.m9.1.1.1.cmml" xref="Thmthm4.p1.9.9.m9.1.1">subscript</csymbol><ci id="Thmthm4.p1.9.9.m9.1.1.2.cmml" xref="Thmthm4.p1.9.9.m9.1.1.2">𝐵</ci><apply id="Thmthm4.p1.9.9.m9.1.1.3.cmml" xref="Thmthm4.p1.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="Thmthm4.p1.9.9.m9.1.1.3.1.cmml" xref="Thmthm4.p1.9.9.m9.1.1.3">subscript</csymbol><ci id="Thmthm4.p1.9.9.m9.1.1.3.2.cmml" xref="Thmthm4.p1.9.9.m9.1.1.3.2">𝑥</ci><cn id="Thmthm4.p1.9.9.m9.1.1.3.3.cmml" type="integer" xref="Thmthm4.p1.9.9.m9.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.9.9.m9.1c">B_{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.9.9.m9.1d">italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="Thmthm4.p1.10.10.m10.1"><semantics id="Thmthm4.p1.10.10.m10.1a"><mi class="ltx_font_mathcaligraphic" id="Thmthm4.p1.10.10.m10.1.1" xref="Thmthm4.p1.10.10.m10.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.10.10.m10.1b"><ci id="Thmthm4.p1.10.10.m10.1.1.cmml" xref="Thmthm4.p1.10.10.m10.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.10.10.m10.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.10.10.m10.1d">caligraphic_T</annotation></semantics></math> contains <math alttext="X\cap Y" class="ltx_Math" display="inline" id="Thmthm4.p1.11.11.m11.1"><semantics id="Thmthm4.p1.11.11.m11.1a"><mrow id="Thmthm4.p1.11.11.m11.1.1" xref="Thmthm4.p1.11.11.m11.1.1.cmml"><mi id="Thmthm4.p1.11.11.m11.1.1.2" xref="Thmthm4.p1.11.11.m11.1.1.2.cmml">X</mi><mo id="Thmthm4.p1.11.11.m11.1.1.1" xref="Thmthm4.p1.11.11.m11.1.1.1.cmml">∩</mo><mi id="Thmthm4.p1.11.11.m11.1.1.3" xref="Thmthm4.p1.11.11.m11.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmthm4.p1.11.11.m11.1b"><apply id="Thmthm4.p1.11.11.m11.1.1.cmml" xref="Thmthm4.p1.11.11.m11.1.1"><intersect id="Thmthm4.p1.11.11.m11.1.1.1.cmml" xref="Thmthm4.p1.11.11.m11.1.1.1"></intersect><ci id="Thmthm4.p1.11.11.m11.1.1.2.cmml" xref="Thmthm4.p1.11.11.m11.1.1.2">𝑋</ci><ci id="Thmthm4.p1.11.11.m11.1.1.3.cmml" xref="Thmthm4.p1.11.11.m11.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm4.p1.11.11.m11.1c">X\cap Y</annotation><annotation encoding="application/x-llamapun" id="Thmthm4.p1.11.11.m11.1d">italic_X ∩ italic_Y</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.1.p1"> <p class="ltx_p" id="S2.1.p1.15">The “furthermore” clause of the statement is immediate, since <math alttext="\operatorname{int}_{\mathcal{T}}(x_{0})=V(G)" class="ltx_Math" display="inline" id="S2.1.p1.1.m1.3"><semantics id="S2.1.p1.1.m1.3a"><mrow id="S2.1.p1.1.m1.3.3" xref="S2.1.p1.1.m1.3.3.cmml"><mrow id="S2.1.p1.1.m1.3.3.2.2" xref="S2.1.p1.1.m1.3.3.2.3.cmml"><msub id="S2.1.p1.1.m1.2.2.1.1.1" xref="S2.1.p1.1.m1.2.2.1.1.1.cmml"><mi id="S2.1.p1.1.m1.2.2.1.1.1.2" xref="S2.1.p1.1.m1.2.2.1.1.1.2.cmml">int</mi><mi 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id="S2.1.p1.1.m1.3.3.2.2.2.1.1.cmml" xref="S2.1.p1.1.m1.3.3.2.2.2.1">subscript</csymbol><ci id="S2.1.p1.1.m1.3.3.2.2.2.1.2.cmml" xref="S2.1.p1.1.m1.3.3.2.2.2.1.2">𝑥</ci><cn id="S2.1.p1.1.m1.3.3.2.2.2.1.3.cmml" type="integer" xref="S2.1.p1.1.m1.3.3.2.2.2.1.3">0</cn></apply></apply><apply id="S2.1.p1.1.m1.3.3.4.cmml" xref="S2.1.p1.1.m1.3.3.4"><times id="S2.1.p1.1.m1.3.3.4.1.cmml" xref="S2.1.p1.1.m1.3.3.4.1"></times><ci id="S2.1.p1.1.m1.3.3.4.2.cmml" xref="S2.1.p1.1.m1.3.3.4.2">𝑉</ci><ci id="S2.1.p1.1.m1.1.1.cmml" xref="S2.1.p1.1.m1.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.1.m1.3c">\operatorname{int}_{\mathcal{T}}(x_{0})=V(G)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.1.m1.3d">roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_V ( italic_G )</annotation></semantics></math>, so <math alttext="B_{x_{0}}\subseteq V(G)\cap X\cap Y=X\cap Y" class="ltx_Math" display="inline" id="S2.1.p1.2.m2.1"><semantics id="S2.1.p1.2.m2.1a"><mrow id="S2.1.p1.2.m2.1.2" xref="S2.1.p1.2.m2.1.2.cmml"><msub id="S2.1.p1.2.m2.1.2.2" xref="S2.1.p1.2.m2.1.2.2.cmml"><mi id="S2.1.p1.2.m2.1.2.2.2" xref="S2.1.p1.2.m2.1.2.2.2.cmml">B</mi><msub id="S2.1.p1.2.m2.1.2.2.3" xref="S2.1.p1.2.m2.1.2.2.3.cmml"><mi id="S2.1.p1.2.m2.1.2.2.3.2" xref="S2.1.p1.2.m2.1.2.2.3.2.cmml">x</mi><mn id="S2.1.p1.2.m2.1.2.2.3.3" xref="S2.1.p1.2.m2.1.2.2.3.3.cmml">0</mn></msub></msub><mo id="S2.1.p1.2.m2.1.2.3" xref="S2.1.p1.2.m2.1.2.3.cmml">⊆</mo><mrow id="S2.1.p1.2.m2.1.2.4" xref="S2.1.p1.2.m2.1.2.4.cmml"><mrow id="S2.1.p1.2.m2.1.2.4.2" xref="S2.1.p1.2.m2.1.2.4.2.cmml"><mi id="S2.1.p1.2.m2.1.2.4.2.2" xref="S2.1.p1.2.m2.1.2.4.2.2.cmml">V</mi><mo id="S2.1.p1.2.m2.1.2.4.2.1" xref="S2.1.p1.2.m2.1.2.4.2.1.cmml">⁢</mo><mrow id="S2.1.p1.2.m2.1.2.4.2.3.2" xref="S2.1.p1.2.m2.1.2.4.2.cmml"><mo id="S2.1.p1.2.m2.1.2.4.2.3.2.1" stretchy="false" xref="S2.1.p1.2.m2.1.2.4.2.cmml">(</mo><mi id="S2.1.p1.2.m2.1.1" xref="S2.1.p1.2.m2.1.1.cmml">G</mi><mo id="S2.1.p1.2.m2.1.2.4.2.3.2.2" stretchy="false" xref="S2.1.p1.2.m2.1.2.4.2.cmml">)</mo></mrow></mrow><mo id="S2.1.p1.2.m2.1.2.4.1" xref="S2.1.p1.2.m2.1.2.4.1.cmml">∩</mo><mi id="S2.1.p1.2.m2.1.2.4.3" xref="S2.1.p1.2.m2.1.2.4.3.cmml">X</mi><mo id="S2.1.p1.2.m2.1.2.4.1a" xref="S2.1.p1.2.m2.1.2.4.1.cmml">∩</mo><mi id="S2.1.p1.2.m2.1.2.4.4" xref="S2.1.p1.2.m2.1.2.4.4.cmml">Y</mi></mrow><mo id="S2.1.p1.2.m2.1.2.5" xref="S2.1.p1.2.m2.1.2.5.cmml">=</mo><mrow id="S2.1.p1.2.m2.1.2.6" xref="S2.1.p1.2.m2.1.2.6.cmml"><mi id="S2.1.p1.2.m2.1.2.6.2" xref="S2.1.p1.2.m2.1.2.6.2.cmml">X</mi><mo id="S2.1.p1.2.m2.1.2.6.1" xref="S2.1.p1.2.m2.1.2.6.1.cmml">∩</mo><mi id="S2.1.p1.2.m2.1.2.6.3" xref="S2.1.p1.2.m2.1.2.6.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.2.m2.1b"><apply id="S2.1.p1.2.m2.1.2.cmml" xref="S2.1.p1.2.m2.1.2"><and id="S2.1.p1.2.m2.1.2a.cmml" xref="S2.1.p1.2.m2.1.2"></and><apply id="S2.1.p1.2.m2.1.2b.cmml" xref="S2.1.p1.2.m2.1.2"><subset id="S2.1.p1.2.m2.1.2.3.cmml" xref="S2.1.p1.2.m2.1.2.3"></subset><apply id="S2.1.p1.2.m2.1.2.2.cmml" xref="S2.1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.1.p1.2.m2.1.2.2.1.cmml" xref="S2.1.p1.2.m2.1.2.2">subscript</csymbol><ci id="S2.1.p1.2.m2.1.2.2.2.cmml" xref="S2.1.p1.2.m2.1.2.2.2">𝐵</ci><apply id="S2.1.p1.2.m2.1.2.2.3.cmml" xref="S2.1.p1.2.m2.1.2.2.3"><csymbol cd="ambiguous" id="S2.1.p1.2.m2.1.2.2.3.1.cmml" xref="S2.1.p1.2.m2.1.2.2.3">subscript</csymbol><ci id="S2.1.p1.2.m2.1.2.2.3.2.cmml" xref="S2.1.p1.2.m2.1.2.2.3.2">𝑥</ci><cn id="S2.1.p1.2.m2.1.2.2.3.3.cmml" type="integer" xref="S2.1.p1.2.m2.1.2.2.3.3">0</cn></apply></apply><apply id="S2.1.p1.2.m2.1.2.4.cmml" xref="S2.1.p1.2.m2.1.2.4"><intersect id="S2.1.p1.2.m2.1.2.4.1.cmml" xref="S2.1.p1.2.m2.1.2.4.1"></intersect><apply id="S2.1.p1.2.m2.1.2.4.2.cmml" xref="S2.1.p1.2.m2.1.2.4.2"><times id="S2.1.p1.2.m2.1.2.4.2.1.cmml" xref="S2.1.p1.2.m2.1.2.4.2.1"></times><ci id="S2.1.p1.2.m2.1.2.4.2.2.cmml" xref="S2.1.p1.2.m2.1.2.4.2.2">𝑉</ci><ci id="S2.1.p1.2.m2.1.1.cmml" xref="S2.1.p1.2.m2.1.1">𝐺</ci></apply><ci id="S2.1.p1.2.m2.1.2.4.3.cmml" xref="S2.1.p1.2.m2.1.2.4.3">𝑋</ci><ci id="S2.1.p1.2.m2.1.2.4.4.cmml" xref="S2.1.p1.2.m2.1.2.4.4">𝑌</ci></apply></apply><apply id="S2.1.p1.2.m2.1.2c.cmml" xref="S2.1.p1.2.m2.1.2"><eq id="S2.1.p1.2.m2.1.2.5.cmml" xref="S2.1.p1.2.m2.1.2.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.1.p1.2.m2.1.2.4.cmml" id="S2.1.p1.2.m2.1.2d.cmml" xref="S2.1.p1.2.m2.1.2"></share><apply id="S2.1.p1.2.m2.1.2.6.cmml" xref="S2.1.p1.2.m2.1.2.6"><intersect id="S2.1.p1.2.m2.1.2.6.1.cmml" xref="S2.1.p1.2.m2.1.2.6.1"></intersect><ci id="S2.1.p1.2.m2.1.2.6.2.cmml" xref="S2.1.p1.2.m2.1.2.6.2">𝑋</ci><ci id="S2.1.p1.2.m2.1.2.6.3.cmml" xref="S2.1.p1.2.m2.1.2.6.3">𝑌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.2.m2.1c">B_{x_{0}}\subseteq V(G)\cap X\cap Y=X\cap Y</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.2.m2.1d">italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊆ italic_V ( italic_G ) ∩ italic_X ∩ italic_Y = italic_X ∩ italic_Y</annotation></semantics></math>. To show that <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S2.1.p1.3.m3.1"><semantics id="S2.1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.1.p1.3.m3.1.1" xref="S2.1.p1.3.m3.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.3.m3.1b"><ci id="S2.1.p1.3.m3.1.1.cmml" xref="S2.1.p1.3.m3.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.3.m3.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.3.m3.1d">caligraphic_T</annotation></semantics></math> is a tree decomposition of <math alttext="G" class="ltx_Math" display="inline" id="S2.1.p1.4.m4.1"><semantics id="S2.1.p1.4.m4.1a"><mi id="S2.1.p1.4.m4.1.1" xref="S2.1.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.4.m4.1b"><ci id="S2.1.p1.4.m4.1.1.cmml" xref="S2.1.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.4.m4.1d">italic_G</annotation></semantics></math> we must show that <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S2.1.p1.5.m5.1"><semantics id="S2.1.p1.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S2.1.p1.5.m5.1.1" xref="S2.1.p1.5.m5.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.5.m5.1b"><ci id="S2.1.p1.5.m5.1.1.cmml" xref="S2.1.p1.5.m5.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.5.m5.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.5.m5.1d">caligraphic_T</annotation></semantics></math> satisfies Properties <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S1.I1.i1" title="Item (i) ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(i)</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S1.I1.i2" title="Item (ii) ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(ii)</span></a> of tree decompositions. Let <math alttext="vw" class="ltx_Math" display="inline" id="S2.1.p1.6.m6.1"><semantics id="S2.1.p1.6.m6.1a"><mrow id="S2.1.p1.6.m6.1.1" xref="S2.1.p1.6.m6.1.1.cmml"><mi id="S2.1.p1.6.m6.1.1.2" xref="S2.1.p1.6.m6.1.1.2.cmml">v</mi><mo id="S2.1.p1.6.m6.1.1.1" xref="S2.1.p1.6.m6.1.1.1.cmml">⁢</mo><mi id="S2.1.p1.6.m6.1.1.3" xref="S2.1.p1.6.m6.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.6.m6.1b"><apply id="S2.1.p1.6.m6.1.1.cmml" xref="S2.1.p1.6.m6.1.1"><times id="S2.1.p1.6.m6.1.1.1.cmml" xref="S2.1.p1.6.m6.1.1.1"></times><ci id="S2.1.p1.6.m6.1.1.2.cmml" xref="S2.1.p1.6.m6.1.1.2">𝑣</ci><ci id="S2.1.p1.6.m6.1.1.3.cmml" xref="S2.1.p1.6.m6.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.6.m6.1c">vw</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.6.m6.1d">italic_v italic_w</annotation></semantics></math> be an edge of <math alttext="G[Y]" class="ltx_Math" display="inline" id="S2.1.p1.7.m7.1"><semantics id="S2.1.p1.7.m7.1a"><mrow id="S2.1.p1.7.m7.1.2" xref="S2.1.p1.7.m7.1.2.cmml"><mi id="S2.1.p1.7.m7.1.2.2" xref="S2.1.p1.7.m7.1.2.2.cmml">G</mi><mo id="S2.1.p1.7.m7.1.2.1" xref="S2.1.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S2.1.p1.7.m7.1.2.3.2" xref="S2.1.p1.7.m7.1.2.3.1.cmml"><mo id="S2.1.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S2.1.p1.7.m7.1.2.3.1.1.cmml">[</mo><mi id="S2.1.p1.7.m7.1.1" xref="S2.1.p1.7.m7.1.1.cmml">Y</mi><mo id="S2.1.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S2.1.p1.7.m7.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.7.m7.1b"><apply id="S2.1.p1.7.m7.1.2.cmml" xref="S2.1.p1.7.m7.1.2"><times id="S2.1.p1.7.m7.1.2.1.cmml" xref="S2.1.p1.7.m7.1.2.1"></times><ci id="S2.1.p1.7.m7.1.2.2.cmml" xref="S2.1.p1.7.m7.1.2.2">𝐺</ci><apply id="S2.1.p1.7.m7.1.2.3.1.cmml" xref="S2.1.p1.7.m7.1.2.3.2"><csymbol cd="latexml" id="S2.1.p1.7.m7.1.2.3.1.1.cmml" xref="S2.1.p1.7.m7.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.1.p1.7.m7.1.1.cmml" xref="S2.1.p1.7.m7.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.7.m7.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.7.m7.1d">italic_G [ italic_Y ]</annotation></semantics></math>. Since <math alttext="vw" class="ltx_Math" display="inline" id="S2.1.p1.8.m8.1"><semantics id="S2.1.p1.8.m8.1a"><mrow id="S2.1.p1.8.m8.1.1" xref="S2.1.p1.8.m8.1.1.cmml"><mi id="S2.1.p1.8.m8.1.1.2" xref="S2.1.p1.8.m8.1.1.2.cmml">v</mi><mo id="S2.1.p1.8.m8.1.1.1" xref="S2.1.p1.8.m8.1.1.1.cmml">⁢</mo><mi id="S2.1.p1.8.m8.1.1.3" xref="S2.1.p1.8.m8.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.8.m8.1b"><apply id="S2.1.p1.8.m8.1.1.cmml" xref="S2.1.p1.8.m8.1.1"><times id="S2.1.p1.8.m8.1.1.1.cmml" xref="S2.1.p1.8.m8.1.1.1"></times><ci id="S2.1.p1.8.m8.1.1.2.cmml" xref="S2.1.p1.8.m8.1.1.2">𝑣</ci><ci id="S2.1.p1.8.m8.1.1.3.cmml" xref="S2.1.p1.8.m8.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.8.m8.1c">vw</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.8.m8.1d">italic_v italic_w</annotation></semantics></math> is also an edge of <math alttext="G" class="ltx_Math" display="inline" id="S2.1.p1.9.m9.1"><semantics id="S2.1.p1.9.m9.1a"><mi id="S2.1.p1.9.m9.1.1" xref="S2.1.p1.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.9.m9.1b"><ci id="S2.1.p1.9.m9.1.1.cmml" xref="S2.1.p1.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.9.m9.1d">italic_G</annotation></semantics></math> and <math alttext="\mathcal{T^{\prime}}" class="ltx_Math" display="inline" id="S2.1.p1.10.m10.1"><semantics id="S2.1.p1.10.m10.1a"><msup id="S2.1.p1.10.m10.1.1" xref="S2.1.p1.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.1.p1.10.m10.1.1.2" xref="S2.1.p1.10.m10.1.1.2.cmml">𝒯</mi><mo id="S2.1.p1.10.m10.1.1.3" xref="S2.1.p1.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.1.p1.10.m10.1b"><apply id="S2.1.p1.10.m10.1.1.cmml" xref="S2.1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S2.1.p1.10.m10.1.1.1.cmml" xref="S2.1.p1.10.m10.1.1">superscript</csymbol><ci id="S2.1.p1.10.m10.1.1.2.cmml" xref="S2.1.p1.10.m10.1.1.2">𝒯</ci><ci id="S2.1.p1.10.m10.1.1.3.cmml" xref="S2.1.p1.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.10.m10.1c">\mathcal{T^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.10.m10.1d">caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a tree decomposition of <math alttext="G" class="ltx_Math" display="inline" id="S2.1.p1.11.m11.1"><semantics id="S2.1.p1.11.m11.1a"><mi id="S2.1.p1.11.m11.1.1" xref="S2.1.p1.11.m11.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.11.m11.1b"><ci id="S2.1.p1.11.m11.1.1.cmml" xref="S2.1.p1.11.m11.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.11.m11.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.11.m11.1d">italic_G</annotation></semantics></math>, <math alttext="\{v,w\}\subseteq B^{\prime}_{x}" class="ltx_Math" display="inline" id="S2.1.p1.12.m12.2"><semantics id="S2.1.p1.12.m12.2a"><mrow id="S2.1.p1.12.m12.2.3" xref="S2.1.p1.12.m12.2.3.cmml"><mrow id="S2.1.p1.12.m12.2.3.2.2" xref="S2.1.p1.12.m12.2.3.2.1.cmml"><mo id="S2.1.p1.12.m12.2.3.2.2.1" stretchy="false" xref="S2.1.p1.12.m12.2.3.2.1.cmml">{</mo><mi id="S2.1.p1.12.m12.1.1" xref="S2.1.p1.12.m12.1.1.cmml">v</mi><mo id="S2.1.p1.12.m12.2.3.2.2.2" xref="S2.1.p1.12.m12.2.3.2.1.cmml">,</mo><mi id="S2.1.p1.12.m12.2.2" xref="S2.1.p1.12.m12.2.2.cmml">w</mi><mo id="S2.1.p1.12.m12.2.3.2.2.3" stretchy="false" xref="S2.1.p1.12.m12.2.3.2.1.cmml">}</mo></mrow><mo id="S2.1.p1.12.m12.2.3.1" xref="S2.1.p1.12.m12.2.3.1.cmml">⊆</mo><msubsup id="S2.1.p1.12.m12.2.3.3" xref="S2.1.p1.12.m12.2.3.3.cmml"><mi id="S2.1.p1.12.m12.2.3.3.2.2" xref="S2.1.p1.12.m12.2.3.3.2.2.cmml">B</mi><mi id="S2.1.p1.12.m12.2.3.3.3" xref="S2.1.p1.12.m12.2.3.3.3.cmml">x</mi><mo id="S2.1.p1.12.m12.2.3.3.2.3" xref="S2.1.p1.12.m12.2.3.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.12.m12.2b"><apply id="S2.1.p1.12.m12.2.3.cmml" xref="S2.1.p1.12.m12.2.3"><subset id="S2.1.p1.12.m12.2.3.1.cmml" xref="S2.1.p1.12.m12.2.3.1"></subset><set id="S2.1.p1.12.m12.2.3.2.1.cmml" xref="S2.1.p1.12.m12.2.3.2.2"><ci id="S2.1.p1.12.m12.1.1.cmml" xref="S2.1.p1.12.m12.1.1">𝑣</ci><ci id="S2.1.p1.12.m12.2.2.cmml" xref="S2.1.p1.12.m12.2.2">𝑤</ci></set><apply id="S2.1.p1.12.m12.2.3.3.cmml" xref="S2.1.p1.12.m12.2.3.3"><csymbol cd="ambiguous" id="S2.1.p1.12.m12.2.3.3.1.cmml" xref="S2.1.p1.12.m12.2.3.3">subscript</csymbol><apply id="S2.1.p1.12.m12.2.3.3.2.cmml" xref="S2.1.p1.12.m12.2.3.3"><csymbol cd="ambiguous" id="S2.1.p1.12.m12.2.3.3.2.1.cmml" xref="S2.1.p1.12.m12.2.3.3">superscript</csymbol><ci id="S2.1.p1.12.m12.2.3.3.2.2.cmml" xref="S2.1.p1.12.m12.2.3.3.2.2">𝐵</ci><ci id="S2.1.p1.12.m12.2.3.3.2.3.cmml" xref="S2.1.p1.12.m12.2.3.3.2.3">′</ci></apply><ci id="S2.1.p1.12.m12.2.3.3.3.cmml" xref="S2.1.p1.12.m12.2.3.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.12.m12.2c">\{v,w\}\subseteq B^{\prime}_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.12.m12.2d">{ italic_v , italic_w } ⊆ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="x\in V(T^{\prime})" class="ltx_Math" display="inline" id="S2.1.p1.13.m13.1"><semantics id="S2.1.p1.13.m13.1a"><mrow id="S2.1.p1.13.m13.1.1" xref="S2.1.p1.13.m13.1.1.cmml"><mi id="S2.1.p1.13.m13.1.1.3" xref="S2.1.p1.13.m13.1.1.3.cmml">x</mi><mo id="S2.1.p1.13.m13.1.1.2" xref="S2.1.p1.13.m13.1.1.2.cmml">∈</mo><mrow id="S2.1.p1.13.m13.1.1.1" xref="S2.1.p1.13.m13.1.1.1.cmml"><mi id="S2.1.p1.13.m13.1.1.1.3" xref="S2.1.p1.13.m13.1.1.1.3.cmml">V</mi><mo id="S2.1.p1.13.m13.1.1.1.2" xref="S2.1.p1.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="S2.1.p1.13.m13.1.1.1.1.1" xref="S2.1.p1.13.m13.1.1.1.1.1.1.cmml"><mo id="S2.1.p1.13.m13.1.1.1.1.1.2" stretchy="false" xref="S2.1.p1.13.m13.1.1.1.1.1.1.cmml">(</mo><msup id="S2.1.p1.13.m13.1.1.1.1.1.1" xref="S2.1.p1.13.m13.1.1.1.1.1.1.cmml"><mi id="S2.1.p1.13.m13.1.1.1.1.1.1.2" xref="S2.1.p1.13.m13.1.1.1.1.1.1.2.cmml">T</mi><mo id="S2.1.p1.13.m13.1.1.1.1.1.1.3" xref="S2.1.p1.13.m13.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.1.p1.13.m13.1.1.1.1.1.3" stretchy="false" xref="S2.1.p1.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.13.m13.1b"><apply id="S2.1.p1.13.m13.1.1.cmml" xref="S2.1.p1.13.m13.1.1"><in id="S2.1.p1.13.m13.1.1.2.cmml" xref="S2.1.p1.13.m13.1.1.2"></in><ci id="S2.1.p1.13.m13.1.1.3.cmml" xref="S2.1.p1.13.m13.1.1.3">𝑥</ci><apply id="S2.1.p1.13.m13.1.1.1.cmml" xref="S2.1.p1.13.m13.1.1.1"><times id="S2.1.p1.13.m13.1.1.1.2.cmml" xref="S2.1.p1.13.m13.1.1.1.2"></times><ci id="S2.1.p1.13.m13.1.1.1.3.cmml" xref="S2.1.p1.13.m13.1.1.1.3">𝑉</ci><apply id="S2.1.p1.13.m13.1.1.1.1.1.1.cmml" xref="S2.1.p1.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.13.m13.1.1.1.1.1.1.1.cmml" xref="S2.1.p1.13.m13.1.1.1.1.1">superscript</csymbol><ci id="S2.1.p1.13.m13.1.1.1.1.1.1.2.cmml" xref="S2.1.p1.13.m13.1.1.1.1.1.1.2">𝑇</ci><ci id="S2.1.p1.13.m13.1.1.1.1.1.1.3.cmml" xref="S2.1.p1.13.m13.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.13.m13.1c">x\in V(T^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.13.m13.1d">italic_x ∈ italic_V ( italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, so <math alttext="\{v,w\}\subseteq B^{\prime}_{x}\cap Y\subseteq B_{x}" class="ltx_Math" display="inline" id="S2.1.p1.14.m14.2"><semantics id="S2.1.p1.14.m14.2a"><mrow id="S2.1.p1.14.m14.2.3" xref="S2.1.p1.14.m14.2.3.cmml"><mrow id="S2.1.p1.14.m14.2.3.2.2" xref="S2.1.p1.14.m14.2.3.2.1.cmml"><mo id="S2.1.p1.14.m14.2.3.2.2.1" stretchy="false" xref="S2.1.p1.14.m14.2.3.2.1.cmml">{</mo><mi id="S2.1.p1.14.m14.1.1" xref="S2.1.p1.14.m14.1.1.cmml">v</mi><mo id="S2.1.p1.14.m14.2.3.2.2.2" xref="S2.1.p1.14.m14.2.3.2.1.cmml">,</mo><mi id="S2.1.p1.14.m14.2.2" xref="S2.1.p1.14.m14.2.2.cmml">w</mi><mo id="S2.1.p1.14.m14.2.3.2.2.3" stretchy="false" xref="S2.1.p1.14.m14.2.3.2.1.cmml">}</mo></mrow><mo id="S2.1.p1.14.m14.2.3.3" xref="S2.1.p1.14.m14.2.3.3.cmml">⊆</mo><mrow id="S2.1.p1.14.m14.2.3.4" xref="S2.1.p1.14.m14.2.3.4.cmml"><msubsup id="S2.1.p1.14.m14.2.3.4.2" xref="S2.1.p1.14.m14.2.3.4.2.cmml"><mi id="S2.1.p1.14.m14.2.3.4.2.2.2" xref="S2.1.p1.14.m14.2.3.4.2.2.2.cmml">B</mi><mi id="S2.1.p1.14.m14.2.3.4.2.3" xref="S2.1.p1.14.m14.2.3.4.2.3.cmml">x</mi><mo id="S2.1.p1.14.m14.2.3.4.2.2.3" xref="S2.1.p1.14.m14.2.3.4.2.2.3.cmml">′</mo></msubsup><mo id="S2.1.p1.14.m14.2.3.4.1" xref="S2.1.p1.14.m14.2.3.4.1.cmml">∩</mo><mi id="S2.1.p1.14.m14.2.3.4.3" xref="S2.1.p1.14.m14.2.3.4.3.cmml">Y</mi></mrow><mo id="S2.1.p1.14.m14.2.3.5" xref="S2.1.p1.14.m14.2.3.5.cmml">⊆</mo><msub id="S2.1.p1.14.m14.2.3.6" xref="S2.1.p1.14.m14.2.3.6.cmml"><mi id="S2.1.p1.14.m14.2.3.6.2" xref="S2.1.p1.14.m14.2.3.6.2.cmml">B</mi><mi id="S2.1.p1.14.m14.2.3.6.3" xref="S2.1.p1.14.m14.2.3.6.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.14.m14.2b"><apply id="S2.1.p1.14.m14.2.3.cmml" xref="S2.1.p1.14.m14.2.3"><and id="S2.1.p1.14.m14.2.3a.cmml" xref="S2.1.p1.14.m14.2.3"></and><apply id="S2.1.p1.14.m14.2.3b.cmml" xref="S2.1.p1.14.m14.2.3"><subset id="S2.1.p1.14.m14.2.3.3.cmml" xref="S2.1.p1.14.m14.2.3.3"></subset><set id="S2.1.p1.14.m14.2.3.2.1.cmml" xref="S2.1.p1.14.m14.2.3.2.2"><ci id="S2.1.p1.14.m14.1.1.cmml" xref="S2.1.p1.14.m14.1.1">𝑣</ci><ci id="S2.1.p1.14.m14.2.2.cmml" xref="S2.1.p1.14.m14.2.2">𝑤</ci></set><apply id="S2.1.p1.14.m14.2.3.4.cmml" xref="S2.1.p1.14.m14.2.3.4"><intersect id="S2.1.p1.14.m14.2.3.4.1.cmml" xref="S2.1.p1.14.m14.2.3.4.1"></intersect><apply id="S2.1.p1.14.m14.2.3.4.2.cmml" xref="S2.1.p1.14.m14.2.3.4.2"><csymbol cd="ambiguous" id="S2.1.p1.14.m14.2.3.4.2.1.cmml" xref="S2.1.p1.14.m14.2.3.4.2">subscript</csymbol><apply id="S2.1.p1.14.m14.2.3.4.2.2.cmml" xref="S2.1.p1.14.m14.2.3.4.2"><csymbol cd="ambiguous" id="S2.1.p1.14.m14.2.3.4.2.2.1.cmml" xref="S2.1.p1.14.m14.2.3.4.2">superscript</csymbol><ci id="S2.1.p1.14.m14.2.3.4.2.2.2.cmml" xref="S2.1.p1.14.m14.2.3.4.2.2.2">𝐵</ci><ci id="S2.1.p1.14.m14.2.3.4.2.2.3.cmml" xref="S2.1.p1.14.m14.2.3.4.2.2.3">′</ci></apply><ci id="S2.1.p1.14.m14.2.3.4.2.3.cmml" xref="S2.1.p1.14.m14.2.3.4.2.3">𝑥</ci></apply><ci id="S2.1.p1.14.m14.2.3.4.3.cmml" xref="S2.1.p1.14.m14.2.3.4.3">𝑌</ci></apply></apply><apply id="S2.1.p1.14.m14.2.3c.cmml" xref="S2.1.p1.14.m14.2.3"><subset id="S2.1.p1.14.m14.2.3.5.cmml" xref="S2.1.p1.14.m14.2.3.5"></subset><share href="https://arxiv.org/html/2503.17112v1#S2.1.p1.14.m14.2.3.4.cmml" id="S2.1.p1.14.m14.2.3d.cmml" xref="S2.1.p1.14.m14.2.3"></share><apply id="S2.1.p1.14.m14.2.3.6.cmml" xref="S2.1.p1.14.m14.2.3.6"><csymbol cd="ambiguous" id="S2.1.p1.14.m14.2.3.6.1.cmml" xref="S2.1.p1.14.m14.2.3.6">subscript</csymbol><ci id="S2.1.p1.14.m14.2.3.6.2.cmml" xref="S2.1.p1.14.m14.2.3.6.2">𝐵</ci><ci id="S2.1.p1.14.m14.2.3.6.3.cmml" xref="S2.1.p1.14.m14.2.3.6.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.14.m14.2c">\{v,w\}\subseteq B^{\prime}_{x}\cap Y\subseteq B_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.14.m14.2d">{ italic_v , italic_w } ⊆ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_Y ⊆ italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S2.1.p1.15.m15.1"><semantics id="S2.1.p1.15.m15.1a"><mi class="ltx_font_mathcaligraphic" id="S2.1.p1.15.m15.1.1" xref="S2.1.p1.15.m15.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.15.m15.1b"><ci id="S2.1.p1.15.m15.1.1.cmml" xref="S2.1.p1.15.m15.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.15.m15.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.15.m15.1d">caligraphic_T</annotation></semantics></math> has Property <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S1.I1.i1" title="Item (i) ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(i)</span></a> of tree decompositions.</p> </div> <div class="ltx_para" id="S2.2.p2"> <p class="ltx_p" id="S2.2.p2.23">Suppose, for the sake of contradiction, that <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S2.2.p2.1.m1.1"><semantics id="S2.2.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.2.p2.1.m1.1.1" xref="S2.2.p2.1.m1.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S2.2.p2.1.m1.1b"><ci id="S2.2.p2.1.m1.1.1.cmml" xref="S2.2.p2.1.m1.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.1.m1.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.1.m1.1d">caligraphic_T</annotation></semantics></math> violates Property <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S1.I1.i2" title="Item (ii) ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(ii)</span></a>. Then there exists some <math alttext="v\in Y" class="ltx_Math" display="inline" id="S2.2.p2.2.m2.1"><semantics id="S2.2.p2.2.m2.1a"><mrow id="S2.2.p2.2.m2.1.1" xref="S2.2.p2.2.m2.1.1.cmml"><mi id="S2.2.p2.2.m2.1.1.2" xref="S2.2.p2.2.m2.1.1.2.cmml">v</mi><mo id="S2.2.p2.2.m2.1.1.1" xref="S2.2.p2.2.m2.1.1.1.cmml">∈</mo><mi id="S2.2.p2.2.m2.1.1.3" xref="S2.2.p2.2.m2.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.2.m2.1b"><apply id="S2.2.p2.2.m2.1.1.cmml" xref="S2.2.p2.2.m2.1.1"><in id="S2.2.p2.2.m2.1.1.1.cmml" xref="S2.2.p2.2.m2.1.1.1"></in><ci id="S2.2.p2.2.m2.1.1.2.cmml" xref="S2.2.p2.2.m2.1.1.2">𝑣</ci><ci id="S2.2.p2.2.m2.1.1.3.cmml" xref="S2.2.p2.2.m2.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.2.m2.1c">v\in Y</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.2.m2.1d">italic_v ∈ italic_Y</annotation></semantics></math>, some <math alttext="r\geq 2" class="ltx_Math" display="inline" id="S2.2.p2.3.m3.1"><semantics id="S2.2.p2.3.m3.1a"><mrow id="S2.2.p2.3.m3.1.1" xref="S2.2.p2.3.m3.1.1.cmml"><mi id="S2.2.p2.3.m3.1.1.2" xref="S2.2.p2.3.m3.1.1.2.cmml">r</mi><mo id="S2.2.p2.3.m3.1.1.1" xref="S2.2.p2.3.m3.1.1.1.cmml">≥</mo><mn id="S2.2.p2.3.m3.1.1.3" xref="S2.2.p2.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.3.m3.1b"><apply id="S2.2.p2.3.m3.1.1.cmml" xref="S2.2.p2.3.m3.1.1"><geq id="S2.2.p2.3.m3.1.1.1.cmml" xref="S2.2.p2.3.m3.1.1.1"></geq><ci id="S2.2.p2.3.m3.1.1.2.cmml" xref="S2.2.p2.3.m3.1.1.2">𝑟</ci><cn id="S2.2.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.2.p2.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.3.m3.1c">r\geq 2</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.3.m3.1d">italic_r ≥ 2</annotation></semantics></math>, and some path <math alttext="P=x_{0},\ldots,x_{r}" class="ltx_Math" display="inline" id="S2.2.p2.4.m4.3"><semantics id="S2.2.p2.4.m4.3a"><mrow id="S2.2.p2.4.m4.3.3" xref="S2.2.p2.4.m4.3.3.cmml"><mi id="S2.2.p2.4.m4.3.3.4" xref="S2.2.p2.4.m4.3.3.4.cmml">P</mi><mo id="S2.2.p2.4.m4.3.3.3" xref="S2.2.p2.4.m4.3.3.3.cmml">=</mo><mrow id="S2.2.p2.4.m4.3.3.2.2" xref="S2.2.p2.4.m4.3.3.2.3.cmml"><msub id="S2.2.p2.4.m4.2.2.1.1.1" xref="S2.2.p2.4.m4.2.2.1.1.1.cmml"><mi id="S2.2.p2.4.m4.2.2.1.1.1.2" xref="S2.2.p2.4.m4.2.2.1.1.1.2.cmml">x</mi><mn id="S2.2.p2.4.m4.2.2.1.1.1.3" xref="S2.2.p2.4.m4.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S2.2.p2.4.m4.3.3.2.2.3" xref="S2.2.p2.4.m4.3.3.2.3.cmml">,</mo><mi id="S2.2.p2.4.m4.1.1" mathvariant="normal" xref="S2.2.p2.4.m4.1.1.cmml">…</mi><mo id="S2.2.p2.4.m4.3.3.2.2.4" xref="S2.2.p2.4.m4.3.3.2.3.cmml">,</mo><msub id="S2.2.p2.4.m4.3.3.2.2.2" xref="S2.2.p2.4.m4.3.3.2.2.2.cmml"><mi id="S2.2.p2.4.m4.3.3.2.2.2.2" xref="S2.2.p2.4.m4.3.3.2.2.2.2.cmml">x</mi><mi id="S2.2.p2.4.m4.3.3.2.2.2.3" xref="S2.2.p2.4.m4.3.3.2.2.2.3.cmml">r</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.4.m4.3b"><apply id="S2.2.p2.4.m4.3.3.cmml" xref="S2.2.p2.4.m4.3.3"><eq id="S2.2.p2.4.m4.3.3.3.cmml" xref="S2.2.p2.4.m4.3.3.3"></eq><ci id="S2.2.p2.4.m4.3.3.4.cmml" xref="S2.2.p2.4.m4.3.3.4">𝑃</ci><list id="S2.2.p2.4.m4.3.3.2.3.cmml" xref="S2.2.p2.4.m4.3.3.2.2"><apply id="S2.2.p2.4.m4.2.2.1.1.1.cmml" xref="S2.2.p2.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.2.p2.4.m4.2.2.1.1.1.1.cmml" xref="S2.2.p2.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S2.2.p2.4.m4.2.2.1.1.1.2.cmml" xref="S2.2.p2.4.m4.2.2.1.1.1.2">𝑥</ci><cn id="S2.2.p2.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S2.2.p2.4.m4.2.2.1.1.1.3">0</cn></apply><ci id="S2.2.p2.4.m4.1.1.cmml" xref="S2.2.p2.4.m4.1.1">…</ci><apply id="S2.2.p2.4.m4.3.3.2.2.2.cmml" xref="S2.2.p2.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.2.p2.4.m4.3.3.2.2.2.1.cmml" xref="S2.2.p2.4.m4.3.3.2.2.2">subscript</csymbol><ci id="S2.2.p2.4.m4.3.3.2.2.2.2.cmml" xref="S2.2.p2.4.m4.3.3.2.2.2.2">𝑥</ci><ci id="S2.2.p2.4.m4.3.3.2.2.2.3.cmml" xref="S2.2.p2.4.m4.3.3.2.2.2.3">𝑟</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.4.m4.3c">P=x_{0},\ldots,x_{r}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.4.m4.3d">italic_P = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S2.2.p2.5.m5.1"><semantics id="S2.2.p2.5.m5.1a"><msup id="S2.2.p2.5.m5.1.1" xref="S2.2.p2.5.m5.1.1.cmml"><mi id="S2.2.p2.5.m5.1.1.2" xref="S2.2.p2.5.m5.1.1.2.cmml">T</mi><mo id="S2.2.p2.5.m5.1.1.3" xref="S2.2.p2.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.2.p2.5.m5.1b"><apply id="S2.2.p2.5.m5.1.1.cmml" xref="S2.2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.2.p2.5.m5.1.1.1.cmml" xref="S2.2.p2.5.m5.1.1">superscript</csymbol><ci id="S2.2.p2.5.m5.1.1.2.cmml" xref="S2.2.p2.5.m5.1.1.2">𝑇</ci><ci id="S2.2.p2.5.m5.1.1.3.cmml" xref="S2.2.p2.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.5.m5.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.5.m5.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="v\in B_{x_{0}}" class="ltx_Math" display="inline" id="S2.2.p2.6.m6.1"><semantics id="S2.2.p2.6.m6.1a"><mrow id="S2.2.p2.6.m6.1.1" xref="S2.2.p2.6.m6.1.1.cmml"><mi id="S2.2.p2.6.m6.1.1.2" xref="S2.2.p2.6.m6.1.1.2.cmml">v</mi><mo id="S2.2.p2.6.m6.1.1.1" xref="S2.2.p2.6.m6.1.1.1.cmml">∈</mo><msub id="S2.2.p2.6.m6.1.1.3" xref="S2.2.p2.6.m6.1.1.3.cmml"><mi id="S2.2.p2.6.m6.1.1.3.2" xref="S2.2.p2.6.m6.1.1.3.2.cmml">B</mi><msub id="S2.2.p2.6.m6.1.1.3.3" xref="S2.2.p2.6.m6.1.1.3.3.cmml"><mi id="S2.2.p2.6.m6.1.1.3.3.2" xref="S2.2.p2.6.m6.1.1.3.3.2.cmml">x</mi><mn id="S2.2.p2.6.m6.1.1.3.3.3" xref="S2.2.p2.6.m6.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.6.m6.1b"><apply id="S2.2.p2.6.m6.1.1.cmml" xref="S2.2.p2.6.m6.1.1"><in id="S2.2.p2.6.m6.1.1.1.cmml" xref="S2.2.p2.6.m6.1.1.1"></in><ci id="S2.2.p2.6.m6.1.1.2.cmml" xref="S2.2.p2.6.m6.1.1.2">𝑣</ci><apply id="S2.2.p2.6.m6.1.1.3.cmml" xref="S2.2.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.6.m6.1.1.3.1.cmml" xref="S2.2.p2.6.m6.1.1.3">subscript</csymbol><ci id="S2.2.p2.6.m6.1.1.3.2.cmml" xref="S2.2.p2.6.m6.1.1.3.2">𝐵</ci><apply id="S2.2.p2.6.m6.1.1.3.3.cmml" xref="S2.2.p2.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p2.6.m6.1.1.3.3.1.cmml" xref="S2.2.p2.6.m6.1.1.3.3">subscript</csymbol><ci id="S2.2.p2.6.m6.1.1.3.3.2.cmml" xref="S2.2.p2.6.m6.1.1.3.3.2">𝑥</ci><cn id="S2.2.p2.6.m6.1.1.3.3.3.cmml" type="integer" xref="S2.2.p2.6.m6.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.6.m6.1c">v\in B_{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.6.m6.1d">italic_v ∈ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="v\in B_{x_{r}}" class="ltx_Math" display="inline" id="S2.2.p2.7.m7.1"><semantics id="S2.2.p2.7.m7.1a"><mrow id="S2.2.p2.7.m7.1.1" xref="S2.2.p2.7.m7.1.1.cmml"><mi id="S2.2.p2.7.m7.1.1.2" xref="S2.2.p2.7.m7.1.1.2.cmml">v</mi><mo id="S2.2.p2.7.m7.1.1.1" xref="S2.2.p2.7.m7.1.1.1.cmml">∈</mo><msub id="S2.2.p2.7.m7.1.1.3" xref="S2.2.p2.7.m7.1.1.3.cmml"><mi id="S2.2.p2.7.m7.1.1.3.2" xref="S2.2.p2.7.m7.1.1.3.2.cmml">B</mi><msub id="S2.2.p2.7.m7.1.1.3.3" xref="S2.2.p2.7.m7.1.1.3.3.cmml"><mi id="S2.2.p2.7.m7.1.1.3.3.2" xref="S2.2.p2.7.m7.1.1.3.3.2.cmml">x</mi><mi id="S2.2.p2.7.m7.1.1.3.3.3" xref="S2.2.p2.7.m7.1.1.3.3.3.cmml">r</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.7.m7.1b"><apply id="S2.2.p2.7.m7.1.1.cmml" xref="S2.2.p2.7.m7.1.1"><in id="S2.2.p2.7.m7.1.1.1.cmml" xref="S2.2.p2.7.m7.1.1.1"></in><ci id="S2.2.p2.7.m7.1.1.2.cmml" xref="S2.2.p2.7.m7.1.1.2">𝑣</ci><apply id="S2.2.p2.7.m7.1.1.3.cmml" xref="S2.2.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.7.m7.1.1.3.1.cmml" xref="S2.2.p2.7.m7.1.1.3">subscript</csymbol><ci id="S2.2.p2.7.m7.1.1.3.2.cmml" xref="S2.2.p2.7.m7.1.1.3.2">𝐵</ci><apply id="S2.2.p2.7.m7.1.1.3.3.cmml" xref="S2.2.p2.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p2.7.m7.1.1.3.3.1.cmml" xref="S2.2.p2.7.m7.1.1.3.3">subscript</csymbol><ci id="S2.2.p2.7.m7.1.1.3.3.2.cmml" xref="S2.2.p2.7.m7.1.1.3.3.2">𝑥</ci><ci id="S2.2.p2.7.m7.1.1.3.3.3.cmml" xref="S2.2.p2.7.m7.1.1.3.3.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.7.m7.1c">v\in B_{x_{r}}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.7.m7.1d">italic_v ∈ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="v\not\in B_{x_{i}}" class="ltx_Math" display="inline" id="S2.2.p2.8.m8.1"><semantics id="S2.2.p2.8.m8.1a"><mrow id="S2.2.p2.8.m8.1.1" xref="S2.2.p2.8.m8.1.1.cmml"><mi id="S2.2.p2.8.m8.1.1.2" xref="S2.2.p2.8.m8.1.1.2.cmml">v</mi><mo id="S2.2.p2.8.m8.1.1.1" xref="S2.2.p2.8.m8.1.1.1.cmml">∉</mo><msub id="S2.2.p2.8.m8.1.1.3" xref="S2.2.p2.8.m8.1.1.3.cmml"><mi id="S2.2.p2.8.m8.1.1.3.2" xref="S2.2.p2.8.m8.1.1.3.2.cmml">B</mi><msub id="S2.2.p2.8.m8.1.1.3.3" xref="S2.2.p2.8.m8.1.1.3.3.cmml"><mi id="S2.2.p2.8.m8.1.1.3.3.2" xref="S2.2.p2.8.m8.1.1.3.3.2.cmml">x</mi><mi id="S2.2.p2.8.m8.1.1.3.3.3" xref="S2.2.p2.8.m8.1.1.3.3.3.cmml">i</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.8.m8.1b"><apply id="S2.2.p2.8.m8.1.1.cmml" xref="S2.2.p2.8.m8.1.1"><notin id="S2.2.p2.8.m8.1.1.1.cmml" xref="S2.2.p2.8.m8.1.1.1"></notin><ci id="S2.2.p2.8.m8.1.1.2.cmml" xref="S2.2.p2.8.m8.1.1.2">𝑣</ci><apply id="S2.2.p2.8.m8.1.1.3.cmml" xref="S2.2.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.8.m8.1.1.3.1.cmml" xref="S2.2.p2.8.m8.1.1.3">subscript</csymbol><ci id="S2.2.p2.8.m8.1.1.3.2.cmml" xref="S2.2.p2.8.m8.1.1.3.2">𝐵</ci><apply id="S2.2.p2.8.m8.1.1.3.3.cmml" xref="S2.2.p2.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p2.8.m8.1.1.3.3.1.cmml" xref="S2.2.p2.8.m8.1.1.3.3">subscript</csymbol><ci id="S2.2.p2.8.m8.1.1.3.3.2.cmml" xref="S2.2.p2.8.m8.1.1.3.3.2">𝑥</ci><ci id="S2.2.p2.8.m8.1.1.3.3.3.cmml" xref="S2.2.p2.8.m8.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.8.m8.1c">v\not\in B_{x_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.8.m8.1d">italic_v ∉ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="i\in\{1,\ldots,r-1\}" class="ltx_Math" display="inline" id="S2.2.p2.9.m9.3"><semantics id="S2.2.p2.9.m9.3a"><mrow id="S2.2.p2.9.m9.3.3" xref="S2.2.p2.9.m9.3.3.cmml"><mi id="S2.2.p2.9.m9.3.3.3" xref="S2.2.p2.9.m9.3.3.3.cmml">i</mi><mo id="S2.2.p2.9.m9.3.3.2" xref="S2.2.p2.9.m9.3.3.2.cmml">∈</mo><mrow id="S2.2.p2.9.m9.3.3.1.1" xref="S2.2.p2.9.m9.3.3.1.2.cmml"><mo id="S2.2.p2.9.m9.3.3.1.1.2" stretchy="false" xref="S2.2.p2.9.m9.3.3.1.2.cmml">{</mo><mn id="S2.2.p2.9.m9.1.1" xref="S2.2.p2.9.m9.1.1.cmml">1</mn><mo id="S2.2.p2.9.m9.3.3.1.1.3" xref="S2.2.p2.9.m9.3.3.1.2.cmml">,</mo><mi id="S2.2.p2.9.m9.2.2" mathvariant="normal" xref="S2.2.p2.9.m9.2.2.cmml">…</mi><mo id="S2.2.p2.9.m9.3.3.1.1.4" xref="S2.2.p2.9.m9.3.3.1.2.cmml">,</mo><mrow id="S2.2.p2.9.m9.3.3.1.1.1" xref="S2.2.p2.9.m9.3.3.1.1.1.cmml"><mi id="S2.2.p2.9.m9.3.3.1.1.1.2" xref="S2.2.p2.9.m9.3.3.1.1.1.2.cmml">r</mi><mo id="S2.2.p2.9.m9.3.3.1.1.1.1" xref="S2.2.p2.9.m9.3.3.1.1.1.1.cmml">−</mo><mn id="S2.2.p2.9.m9.3.3.1.1.1.3" xref="S2.2.p2.9.m9.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S2.2.p2.9.m9.3.3.1.1.5" stretchy="false" xref="S2.2.p2.9.m9.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.9.m9.3b"><apply id="S2.2.p2.9.m9.3.3.cmml" xref="S2.2.p2.9.m9.3.3"><in id="S2.2.p2.9.m9.3.3.2.cmml" xref="S2.2.p2.9.m9.3.3.2"></in><ci id="S2.2.p2.9.m9.3.3.3.cmml" xref="S2.2.p2.9.m9.3.3.3">𝑖</ci><set id="S2.2.p2.9.m9.3.3.1.2.cmml" xref="S2.2.p2.9.m9.3.3.1.1"><cn id="S2.2.p2.9.m9.1.1.cmml" type="integer" xref="S2.2.p2.9.m9.1.1">1</cn><ci id="S2.2.p2.9.m9.2.2.cmml" xref="S2.2.p2.9.m9.2.2">…</ci><apply id="S2.2.p2.9.m9.3.3.1.1.1.cmml" xref="S2.2.p2.9.m9.3.3.1.1.1"><minus id="S2.2.p2.9.m9.3.3.1.1.1.1.cmml" xref="S2.2.p2.9.m9.3.3.1.1.1.1"></minus><ci id="S2.2.p2.9.m9.3.3.1.1.1.2.cmml" xref="S2.2.p2.9.m9.3.3.1.1.1.2">𝑟</ci><cn id="S2.2.p2.9.m9.3.3.1.1.1.3.cmml" type="integer" xref="S2.2.p2.9.m9.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.9.m9.3c">i\in\{1,\ldots,r-1\}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.9.m9.3d">italic_i ∈ { 1 , … , italic_r - 1 }</annotation></semantics></math>. Let <math alttext="P" class="ltx_Math" display="inline" id="S2.2.p2.10.m10.1"><semantics id="S2.2.p2.10.m10.1a"><mi id="S2.2.p2.10.m10.1.1" xref="S2.2.p2.10.m10.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.2.p2.10.m10.1b"><ci id="S2.2.p2.10.m10.1.1.cmml" xref="S2.2.p2.10.m10.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.10.m10.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.10.m10.1d">italic_P</annotation></semantics></math> be chosen so that its length, <math alttext="r" class="ltx_Math" display="inline" id="S2.2.p2.11.m11.1"><semantics id="S2.2.p2.11.m11.1a"><mi id="S2.2.p2.11.m11.1.1" xref="S2.2.p2.11.m11.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.2.p2.11.m11.1b"><ci id="S2.2.p2.11.m11.1.1.cmml" xref="S2.2.p2.11.m11.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.11.m11.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.11.m11.1d">italic_r</annotation></semantics></math>, is minimum. Then <math alttext="v\not\in B_{x_{i}}" class="ltx_Math" display="inline" id="S2.2.p2.12.m12.1"><semantics id="S2.2.p2.12.m12.1a"><mrow id="S2.2.p2.12.m12.1.1" xref="S2.2.p2.12.m12.1.1.cmml"><mi id="S2.2.p2.12.m12.1.1.2" xref="S2.2.p2.12.m12.1.1.2.cmml">v</mi><mo id="S2.2.p2.12.m12.1.1.1" xref="S2.2.p2.12.m12.1.1.1.cmml">∉</mo><msub id="S2.2.p2.12.m12.1.1.3" xref="S2.2.p2.12.m12.1.1.3.cmml"><mi id="S2.2.p2.12.m12.1.1.3.2" xref="S2.2.p2.12.m12.1.1.3.2.cmml">B</mi><msub id="S2.2.p2.12.m12.1.1.3.3" xref="S2.2.p2.12.m12.1.1.3.3.cmml"><mi id="S2.2.p2.12.m12.1.1.3.3.2" xref="S2.2.p2.12.m12.1.1.3.3.2.cmml">x</mi><mi id="S2.2.p2.12.m12.1.1.3.3.3" xref="S2.2.p2.12.m12.1.1.3.3.3.cmml">i</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.12.m12.1b"><apply id="S2.2.p2.12.m12.1.1.cmml" xref="S2.2.p2.12.m12.1.1"><notin id="S2.2.p2.12.m12.1.1.1.cmml" xref="S2.2.p2.12.m12.1.1.1"></notin><ci id="S2.2.p2.12.m12.1.1.2.cmml" xref="S2.2.p2.12.m12.1.1.2">𝑣</ci><apply id="S2.2.p2.12.m12.1.1.3.cmml" xref="S2.2.p2.12.m12.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.12.m12.1.1.3.1.cmml" xref="S2.2.p2.12.m12.1.1.3">subscript</csymbol><ci id="S2.2.p2.12.m12.1.1.3.2.cmml" xref="S2.2.p2.12.m12.1.1.3.2">𝐵</ci><apply id="S2.2.p2.12.m12.1.1.3.3.cmml" xref="S2.2.p2.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p2.12.m12.1.1.3.3.1.cmml" xref="S2.2.p2.12.m12.1.1.3.3">subscript</csymbol><ci id="S2.2.p2.12.m12.1.1.3.3.2.cmml" xref="S2.2.p2.12.m12.1.1.3.3.2">𝑥</ci><ci id="S2.2.p2.12.m12.1.1.3.3.3.cmml" xref="S2.2.p2.12.m12.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.12.m12.1c">v\not\in B_{x_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.12.m12.1d">italic_v ∉ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for each <math alttext="i\in\{1,\ldots,r-1\}" class="ltx_Math" display="inline" id="S2.2.p2.13.m13.3"><semantics id="S2.2.p2.13.m13.3a"><mrow id="S2.2.p2.13.m13.3.3" xref="S2.2.p2.13.m13.3.3.cmml"><mi id="S2.2.p2.13.m13.3.3.3" xref="S2.2.p2.13.m13.3.3.3.cmml">i</mi><mo id="S2.2.p2.13.m13.3.3.2" xref="S2.2.p2.13.m13.3.3.2.cmml">∈</mo><mrow id="S2.2.p2.13.m13.3.3.1.1" xref="S2.2.p2.13.m13.3.3.1.2.cmml"><mo id="S2.2.p2.13.m13.3.3.1.1.2" stretchy="false" xref="S2.2.p2.13.m13.3.3.1.2.cmml">{</mo><mn id="S2.2.p2.13.m13.1.1" xref="S2.2.p2.13.m13.1.1.cmml">1</mn><mo id="S2.2.p2.13.m13.3.3.1.1.3" xref="S2.2.p2.13.m13.3.3.1.2.cmml">,</mo><mi id="S2.2.p2.13.m13.2.2" mathvariant="normal" xref="S2.2.p2.13.m13.2.2.cmml">…</mi><mo id="S2.2.p2.13.m13.3.3.1.1.4" xref="S2.2.p2.13.m13.3.3.1.2.cmml">,</mo><mrow id="S2.2.p2.13.m13.3.3.1.1.1" xref="S2.2.p2.13.m13.3.3.1.1.1.cmml"><mi id="S2.2.p2.13.m13.3.3.1.1.1.2" xref="S2.2.p2.13.m13.3.3.1.1.1.2.cmml">r</mi><mo id="S2.2.p2.13.m13.3.3.1.1.1.1" xref="S2.2.p2.13.m13.3.3.1.1.1.1.cmml">−</mo><mn id="S2.2.p2.13.m13.3.3.1.1.1.3" xref="S2.2.p2.13.m13.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S2.2.p2.13.m13.3.3.1.1.5" stretchy="false" xref="S2.2.p2.13.m13.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.13.m13.3b"><apply id="S2.2.p2.13.m13.3.3.cmml" xref="S2.2.p2.13.m13.3.3"><in id="S2.2.p2.13.m13.3.3.2.cmml" xref="S2.2.p2.13.m13.3.3.2"></in><ci id="S2.2.p2.13.m13.3.3.3.cmml" xref="S2.2.p2.13.m13.3.3.3">𝑖</ci><set id="S2.2.p2.13.m13.3.3.1.2.cmml" xref="S2.2.p2.13.m13.3.3.1.1"><cn id="S2.2.p2.13.m13.1.1.cmml" type="integer" xref="S2.2.p2.13.m13.1.1">1</cn><ci id="S2.2.p2.13.m13.2.2.cmml" xref="S2.2.p2.13.m13.2.2">…</ci><apply id="S2.2.p2.13.m13.3.3.1.1.1.cmml" xref="S2.2.p2.13.m13.3.3.1.1.1"><minus id="S2.2.p2.13.m13.3.3.1.1.1.1.cmml" xref="S2.2.p2.13.m13.3.3.1.1.1.1"></minus><ci id="S2.2.p2.13.m13.3.3.1.1.1.2.cmml" xref="S2.2.p2.13.m13.3.3.1.1.1.2">𝑟</ci><cn id="S2.2.p2.13.m13.3.3.1.1.1.3.cmml" type="integer" xref="S2.2.p2.13.m13.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.13.m13.3c">i\in\{1,\ldots,r-1\}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.13.m13.3d">italic_i ∈ { 1 , … , italic_r - 1 }</annotation></semantics></math>. Since <math alttext="\mathcal{T}^{\prime}" class="ltx_Math" display="inline" id="S2.2.p2.14.m14.1"><semantics id="S2.2.p2.14.m14.1a"><msup id="S2.2.p2.14.m14.1.1" xref="S2.2.p2.14.m14.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.2.p2.14.m14.1.1.2" xref="S2.2.p2.14.m14.1.1.2.cmml">𝒯</mi><mo id="S2.2.p2.14.m14.1.1.3" xref="S2.2.p2.14.m14.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.2.p2.14.m14.1b"><apply id="S2.2.p2.14.m14.1.1.cmml" xref="S2.2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S2.2.p2.14.m14.1.1.1.cmml" xref="S2.2.p2.14.m14.1.1">superscript</csymbol><ci id="S2.2.p2.14.m14.1.1.2.cmml" xref="S2.2.p2.14.m14.1.1.2">𝒯</ci><ci id="S2.2.p2.14.m14.1.1.3.cmml" xref="S2.2.p2.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.14.m14.1c">\mathcal{T}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.14.m14.1d">caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a tree decomposition of <math alttext="G" class="ltx_Math" display="inline" id="S2.2.p2.15.m15.1"><semantics id="S2.2.p2.15.m15.1a"><mi id="S2.2.p2.15.m15.1.1" xref="S2.2.p2.15.m15.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.2.p2.15.m15.1b"><ci id="S2.2.p2.15.m15.1.1.cmml" xref="S2.2.p2.15.m15.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.15.m15.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.15.m15.1d">italic_G</annotation></semantics></math>, <math alttext="v\not\in B^{\prime}_{x_{0}}" class="ltx_Math" display="inline" id="S2.2.p2.16.m16.1"><semantics id="S2.2.p2.16.m16.1a"><mrow id="S2.2.p2.16.m16.1.1" xref="S2.2.p2.16.m16.1.1.cmml"><mi id="S2.2.p2.16.m16.1.1.2" xref="S2.2.p2.16.m16.1.1.2.cmml">v</mi><mo id="S2.2.p2.16.m16.1.1.1" xref="S2.2.p2.16.m16.1.1.1.cmml">∉</mo><msubsup id="S2.2.p2.16.m16.1.1.3" xref="S2.2.p2.16.m16.1.1.3.cmml"><mi id="S2.2.p2.16.m16.1.1.3.2.2" xref="S2.2.p2.16.m16.1.1.3.2.2.cmml">B</mi><msub id="S2.2.p2.16.m16.1.1.3.3" xref="S2.2.p2.16.m16.1.1.3.3.cmml"><mi id="S2.2.p2.16.m16.1.1.3.3.2" xref="S2.2.p2.16.m16.1.1.3.3.2.cmml">x</mi><mn id="S2.2.p2.16.m16.1.1.3.3.3" xref="S2.2.p2.16.m16.1.1.3.3.3.cmml">0</mn></msub><mo id="S2.2.p2.16.m16.1.1.3.2.3" xref="S2.2.p2.16.m16.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.16.m16.1b"><apply id="S2.2.p2.16.m16.1.1.cmml" xref="S2.2.p2.16.m16.1.1"><notin id="S2.2.p2.16.m16.1.1.1.cmml" xref="S2.2.p2.16.m16.1.1.1"></notin><ci id="S2.2.p2.16.m16.1.1.2.cmml" xref="S2.2.p2.16.m16.1.1.2">𝑣</ci><apply id="S2.2.p2.16.m16.1.1.3.cmml" xref="S2.2.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.16.m16.1.1.3.1.cmml" xref="S2.2.p2.16.m16.1.1.3">subscript</csymbol><apply id="S2.2.p2.16.m16.1.1.3.2.cmml" xref="S2.2.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.16.m16.1.1.3.2.1.cmml" xref="S2.2.p2.16.m16.1.1.3">superscript</csymbol><ci id="S2.2.p2.16.m16.1.1.3.2.2.cmml" xref="S2.2.p2.16.m16.1.1.3.2.2">𝐵</ci><ci id="S2.2.p2.16.m16.1.1.3.2.3.cmml" xref="S2.2.p2.16.m16.1.1.3.2.3">′</ci></apply><apply id="S2.2.p2.16.m16.1.1.3.3.cmml" xref="S2.2.p2.16.m16.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p2.16.m16.1.1.3.3.1.cmml" xref="S2.2.p2.16.m16.1.1.3.3">subscript</csymbol><ci id="S2.2.p2.16.m16.1.1.3.3.2.cmml" xref="S2.2.p2.16.m16.1.1.3.3.2">𝑥</ci><cn id="S2.2.p2.16.m16.1.1.3.3.3.cmml" type="integer" xref="S2.2.p2.16.m16.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.16.m16.1c">v\not\in B^{\prime}_{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.16.m16.1d">italic_v ∉ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="v\not\in B^{\prime}_{x_{r}}" class="ltx_Math" display="inline" id="S2.2.p2.17.m17.1"><semantics id="S2.2.p2.17.m17.1a"><mrow id="S2.2.p2.17.m17.1.1" xref="S2.2.p2.17.m17.1.1.cmml"><mi id="S2.2.p2.17.m17.1.1.2" xref="S2.2.p2.17.m17.1.1.2.cmml">v</mi><mo id="S2.2.p2.17.m17.1.1.1" xref="S2.2.p2.17.m17.1.1.1.cmml">∉</mo><msubsup id="S2.2.p2.17.m17.1.1.3" xref="S2.2.p2.17.m17.1.1.3.cmml"><mi id="S2.2.p2.17.m17.1.1.3.2.2" xref="S2.2.p2.17.m17.1.1.3.2.2.cmml">B</mi><msub id="S2.2.p2.17.m17.1.1.3.3" xref="S2.2.p2.17.m17.1.1.3.3.cmml"><mi id="S2.2.p2.17.m17.1.1.3.3.2" xref="S2.2.p2.17.m17.1.1.3.3.2.cmml">x</mi><mi id="S2.2.p2.17.m17.1.1.3.3.3" xref="S2.2.p2.17.m17.1.1.3.3.3.cmml">r</mi></msub><mo id="S2.2.p2.17.m17.1.1.3.2.3" xref="S2.2.p2.17.m17.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.17.m17.1b"><apply id="S2.2.p2.17.m17.1.1.cmml" xref="S2.2.p2.17.m17.1.1"><notin id="S2.2.p2.17.m17.1.1.1.cmml" xref="S2.2.p2.17.m17.1.1.1"></notin><ci id="S2.2.p2.17.m17.1.1.2.cmml" xref="S2.2.p2.17.m17.1.1.2">𝑣</ci><apply id="S2.2.p2.17.m17.1.1.3.cmml" xref="S2.2.p2.17.m17.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.17.m17.1.1.3.1.cmml" xref="S2.2.p2.17.m17.1.1.3">subscript</csymbol><apply id="S2.2.p2.17.m17.1.1.3.2.cmml" xref="S2.2.p2.17.m17.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.17.m17.1.1.3.2.1.cmml" xref="S2.2.p2.17.m17.1.1.3">superscript</csymbol><ci id="S2.2.p2.17.m17.1.1.3.2.2.cmml" xref="S2.2.p2.17.m17.1.1.3.2.2">𝐵</ci><ci id="S2.2.p2.17.m17.1.1.3.2.3.cmml" xref="S2.2.p2.17.m17.1.1.3.2.3">′</ci></apply><apply id="S2.2.p2.17.m17.1.1.3.3.cmml" xref="S2.2.p2.17.m17.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p2.17.m17.1.1.3.3.1.cmml" xref="S2.2.p2.17.m17.1.1.3.3">subscript</csymbol><ci id="S2.2.p2.17.m17.1.1.3.3.2.cmml" xref="S2.2.p2.17.m17.1.1.3.3.2">𝑥</ci><ci id="S2.2.p2.17.m17.1.1.3.3.3.cmml" xref="S2.2.p2.17.m17.1.1.3.3.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.17.m17.1c">v\not\in B^{\prime}_{x_{r}}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.17.m17.1d">italic_v ∉ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> since, otherwise <math alttext="v\in B_{x_{i}}\subseteq B^{\prime}_{x_{i}}\cap Y" class="ltx_Math" display="inline" id="S2.2.p2.18.m18.1"><semantics id="S2.2.p2.18.m18.1a"><mrow id="S2.2.p2.18.m18.1.1" xref="S2.2.p2.18.m18.1.1.cmml"><mi id="S2.2.p2.18.m18.1.1.2" xref="S2.2.p2.18.m18.1.1.2.cmml">v</mi><mo id="S2.2.p2.18.m18.1.1.3" xref="S2.2.p2.18.m18.1.1.3.cmml">∈</mo><msub id="S2.2.p2.18.m18.1.1.4" xref="S2.2.p2.18.m18.1.1.4.cmml"><mi id="S2.2.p2.18.m18.1.1.4.2" xref="S2.2.p2.18.m18.1.1.4.2.cmml">B</mi><msub id="S2.2.p2.18.m18.1.1.4.3" xref="S2.2.p2.18.m18.1.1.4.3.cmml"><mi id="S2.2.p2.18.m18.1.1.4.3.2" xref="S2.2.p2.18.m18.1.1.4.3.2.cmml">x</mi><mi id="S2.2.p2.18.m18.1.1.4.3.3" xref="S2.2.p2.18.m18.1.1.4.3.3.cmml">i</mi></msub></msub><mo id="S2.2.p2.18.m18.1.1.5" xref="S2.2.p2.18.m18.1.1.5.cmml">⊆</mo><mrow id="S2.2.p2.18.m18.1.1.6" xref="S2.2.p2.18.m18.1.1.6.cmml"><msubsup id="S2.2.p2.18.m18.1.1.6.2" xref="S2.2.p2.18.m18.1.1.6.2.cmml"><mi id="S2.2.p2.18.m18.1.1.6.2.2.2" xref="S2.2.p2.18.m18.1.1.6.2.2.2.cmml">B</mi><msub id="S2.2.p2.18.m18.1.1.6.2.3" xref="S2.2.p2.18.m18.1.1.6.2.3.cmml"><mi id="S2.2.p2.18.m18.1.1.6.2.3.2" xref="S2.2.p2.18.m18.1.1.6.2.3.2.cmml">x</mi><mi id="S2.2.p2.18.m18.1.1.6.2.3.3" xref="S2.2.p2.18.m18.1.1.6.2.3.3.cmml">i</mi></msub><mo id="S2.2.p2.18.m18.1.1.6.2.2.3" xref="S2.2.p2.18.m18.1.1.6.2.2.3.cmml">′</mo></msubsup><mo id="S2.2.p2.18.m18.1.1.6.1" xref="S2.2.p2.18.m18.1.1.6.1.cmml">∩</mo><mi id="S2.2.p2.18.m18.1.1.6.3" xref="S2.2.p2.18.m18.1.1.6.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.18.m18.1b"><apply id="S2.2.p2.18.m18.1.1.cmml" xref="S2.2.p2.18.m18.1.1"><and id="S2.2.p2.18.m18.1.1a.cmml" xref="S2.2.p2.18.m18.1.1"></and><apply id="S2.2.p2.18.m18.1.1b.cmml" xref="S2.2.p2.18.m18.1.1"><in id="S2.2.p2.18.m18.1.1.3.cmml" xref="S2.2.p2.18.m18.1.1.3"></in><ci id="S2.2.p2.18.m18.1.1.2.cmml" xref="S2.2.p2.18.m18.1.1.2">𝑣</ci><apply id="S2.2.p2.18.m18.1.1.4.cmml" xref="S2.2.p2.18.m18.1.1.4"><csymbol cd="ambiguous" id="S2.2.p2.18.m18.1.1.4.1.cmml" xref="S2.2.p2.18.m18.1.1.4">subscript</csymbol><ci id="S2.2.p2.18.m18.1.1.4.2.cmml" xref="S2.2.p2.18.m18.1.1.4.2">𝐵</ci><apply id="S2.2.p2.18.m18.1.1.4.3.cmml" xref="S2.2.p2.18.m18.1.1.4.3"><csymbol cd="ambiguous" id="S2.2.p2.18.m18.1.1.4.3.1.cmml" xref="S2.2.p2.18.m18.1.1.4.3">subscript</csymbol><ci id="S2.2.p2.18.m18.1.1.4.3.2.cmml" xref="S2.2.p2.18.m18.1.1.4.3.2">𝑥</ci><ci id="S2.2.p2.18.m18.1.1.4.3.3.cmml" xref="S2.2.p2.18.m18.1.1.4.3.3">𝑖</ci></apply></apply></apply><apply id="S2.2.p2.18.m18.1.1c.cmml" xref="S2.2.p2.18.m18.1.1"><subset id="S2.2.p2.18.m18.1.1.5.cmml" xref="S2.2.p2.18.m18.1.1.5"></subset><share href="https://arxiv.org/html/2503.17112v1#S2.2.p2.18.m18.1.1.4.cmml" id="S2.2.p2.18.m18.1.1d.cmml" xref="S2.2.p2.18.m18.1.1"></share><apply id="S2.2.p2.18.m18.1.1.6.cmml" xref="S2.2.p2.18.m18.1.1.6"><intersect id="S2.2.p2.18.m18.1.1.6.1.cmml" xref="S2.2.p2.18.m18.1.1.6.1"></intersect><apply id="S2.2.p2.18.m18.1.1.6.2.cmml" xref="S2.2.p2.18.m18.1.1.6.2"><csymbol cd="ambiguous" id="S2.2.p2.18.m18.1.1.6.2.1.cmml" xref="S2.2.p2.18.m18.1.1.6.2">subscript</csymbol><apply id="S2.2.p2.18.m18.1.1.6.2.2.cmml" xref="S2.2.p2.18.m18.1.1.6.2"><csymbol cd="ambiguous" id="S2.2.p2.18.m18.1.1.6.2.2.1.cmml" xref="S2.2.p2.18.m18.1.1.6.2">superscript</csymbol><ci id="S2.2.p2.18.m18.1.1.6.2.2.2.cmml" xref="S2.2.p2.18.m18.1.1.6.2.2.2">𝐵</ci><ci id="S2.2.p2.18.m18.1.1.6.2.2.3.cmml" xref="S2.2.p2.18.m18.1.1.6.2.2.3">′</ci></apply><apply id="S2.2.p2.18.m18.1.1.6.2.3.cmml" xref="S2.2.p2.18.m18.1.1.6.2.3"><csymbol cd="ambiguous" id="S2.2.p2.18.m18.1.1.6.2.3.1.cmml" xref="S2.2.p2.18.m18.1.1.6.2.3">subscript</csymbol><ci id="S2.2.p2.18.m18.1.1.6.2.3.2.cmml" xref="S2.2.p2.18.m18.1.1.6.2.3.2">𝑥</ci><ci id="S2.2.p2.18.m18.1.1.6.2.3.3.cmml" xref="S2.2.p2.18.m18.1.1.6.2.3.3">𝑖</ci></apply></apply><ci id="S2.2.p2.18.m18.1.1.6.3.cmml" xref="S2.2.p2.18.m18.1.1.6.3">𝑌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.18.m18.1c">v\in B_{x_{i}}\subseteq B^{\prime}_{x_{i}}\cap Y</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.18.m18.1d">italic_v ∈ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊆ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ italic_Y</annotation></semantics></math> for each <math alttext="i\in\{0,\ldots,r\}" class="ltx_Math" display="inline" id="S2.2.p2.19.m19.3"><semantics id="S2.2.p2.19.m19.3a"><mrow id="S2.2.p2.19.m19.3.4" xref="S2.2.p2.19.m19.3.4.cmml"><mi id="S2.2.p2.19.m19.3.4.2" xref="S2.2.p2.19.m19.3.4.2.cmml">i</mi><mo id="S2.2.p2.19.m19.3.4.1" xref="S2.2.p2.19.m19.3.4.1.cmml">∈</mo><mrow id="S2.2.p2.19.m19.3.4.3.2" xref="S2.2.p2.19.m19.3.4.3.1.cmml"><mo id="S2.2.p2.19.m19.3.4.3.2.1" stretchy="false" xref="S2.2.p2.19.m19.3.4.3.1.cmml">{</mo><mn id="S2.2.p2.19.m19.1.1" xref="S2.2.p2.19.m19.1.1.cmml">0</mn><mo id="S2.2.p2.19.m19.3.4.3.2.2" xref="S2.2.p2.19.m19.3.4.3.1.cmml">,</mo><mi id="S2.2.p2.19.m19.2.2" mathvariant="normal" xref="S2.2.p2.19.m19.2.2.cmml">…</mi><mo id="S2.2.p2.19.m19.3.4.3.2.3" xref="S2.2.p2.19.m19.3.4.3.1.cmml">,</mo><mi id="S2.2.p2.19.m19.3.3" xref="S2.2.p2.19.m19.3.3.cmml">r</mi><mo id="S2.2.p2.19.m19.3.4.3.2.4" stretchy="false" xref="S2.2.p2.19.m19.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.19.m19.3b"><apply id="S2.2.p2.19.m19.3.4.cmml" xref="S2.2.p2.19.m19.3.4"><in id="S2.2.p2.19.m19.3.4.1.cmml" xref="S2.2.p2.19.m19.3.4.1"></in><ci id="S2.2.p2.19.m19.3.4.2.cmml" xref="S2.2.p2.19.m19.3.4.2">𝑖</ci><set id="S2.2.p2.19.m19.3.4.3.1.cmml" xref="S2.2.p2.19.m19.3.4.3.2"><cn id="S2.2.p2.19.m19.1.1.cmml" type="integer" xref="S2.2.p2.19.m19.1.1">0</cn><ci id="S2.2.p2.19.m19.2.2.cmml" xref="S2.2.p2.19.m19.2.2">…</ci><ci id="S2.2.p2.19.m19.3.3.cmml" xref="S2.2.p2.19.m19.3.3">𝑟</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.19.m19.3c">i\in\{0,\ldots,r\}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.19.m19.3d">italic_i ∈ { 0 , … , italic_r }</annotation></semantics></math>. Assume, without loss of generality that <math alttext="v\not\in B^{\prime}_{x_{r}}" class="ltx_Math" display="inline" id="S2.2.p2.20.m20.1"><semantics id="S2.2.p2.20.m20.1a"><mrow id="S2.2.p2.20.m20.1.1" xref="S2.2.p2.20.m20.1.1.cmml"><mi id="S2.2.p2.20.m20.1.1.2" xref="S2.2.p2.20.m20.1.1.2.cmml">v</mi><mo id="S2.2.p2.20.m20.1.1.1" xref="S2.2.p2.20.m20.1.1.1.cmml">∉</mo><msubsup id="S2.2.p2.20.m20.1.1.3" xref="S2.2.p2.20.m20.1.1.3.cmml"><mi id="S2.2.p2.20.m20.1.1.3.2.2" xref="S2.2.p2.20.m20.1.1.3.2.2.cmml">B</mi><msub id="S2.2.p2.20.m20.1.1.3.3" xref="S2.2.p2.20.m20.1.1.3.3.cmml"><mi id="S2.2.p2.20.m20.1.1.3.3.2" xref="S2.2.p2.20.m20.1.1.3.3.2.cmml">x</mi><mi id="S2.2.p2.20.m20.1.1.3.3.3" xref="S2.2.p2.20.m20.1.1.3.3.3.cmml">r</mi></msub><mo id="S2.2.p2.20.m20.1.1.3.2.3" xref="S2.2.p2.20.m20.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.20.m20.1b"><apply id="S2.2.p2.20.m20.1.1.cmml" xref="S2.2.p2.20.m20.1.1"><notin id="S2.2.p2.20.m20.1.1.1.cmml" xref="S2.2.p2.20.m20.1.1.1"></notin><ci id="S2.2.p2.20.m20.1.1.2.cmml" xref="S2.2.p2.20.m20.1.1.2">𝑣</ci><apply id="S2.2.p2.20.m20.1.1.3.cmml" xref="S2.2.p2.20.m20.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.20.m20.1.1.3.1.cmml" xref="S2.2.p2.20.m20.1.1.3">subscript</csymbol><apply id="S2.2.p2.20.m20.1.1.3.2.cmml" xref="S2.2.p2.20.m20.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.20.m20.1.1.3.2.1.cmml" xref="S2.2.p2.20.m20.1.1.3">superscript</csymbol><ci id="S2.2.p2.20.m20.1.1.3.2.2.cmml" xref="S2.2.p2.20.m20.1.1.3.2.2">𝐵</ci><ci id="S2.2.p2.20.m20.1.1.3.2.3.cmml" xref="S2.2.p2.20.m20.1.1.3.2.3">′</ci></apply><apply id="S2.2.p2.20.m20.1.1.3.3.cmml" xref="S2.2.p2.20.m20.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p2.20.m20.1.1.3.3.1.cmml" xref="S2.2.p2.20.m20.1.1.3.3">subscript</csymbol><ci id="S2.2.p2.20.m20.1.1.3.3.2.cmml" xref="S2.2.p2.20.m20.1.1.3.3.2">𝑥</ci><ci id="S2.2.p2.20.m20.1.1.3.3.3.cmml" xref="S2.2.p2.20.m20.1.1.3.3.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.20.m20.1c">v\not\in B^{\prime}_{x_{r}}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.20.m20.1d">italic_v ∉ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{r})\cap X\cap Y" class="ltx_Math" display="inline" id="S2.2.p2.21.m21.2"><semantics id="S2.2.p2.21.m21.2a"><mrow id="S2.2.p2.21.m21.2.2" xref="S2.2.p2.21.m21.2.2.cmml"><mi id="S2.2.p2.21.m21.2.2.4" xref="S2.2.p2.21.m21.2.2.4.cmml">v</mi><mo id="S2.2.p2.21.m21.2.2.3" xref="S2.2.p2.21.m21.2.2.3.cmml">∈</mo><mrow id="S2.2.p2.21.m21.2.2.2" xref="S2.2.p2.21.m21.2.2.2.cmml"><mrow id="S2.2.p2.21.m21.2.2.2.2.2" xref="S2.2.p2.21.m21.2.2.2.2.3.cmml"><msub id="S2.2.p2.21.m21.1.1.1.1.1.1" xref="S2.2.p2.21.m21.1.1.1.1.1.1.cmml"><mi id="S2.2.p2.21.m21.1.1.1.1.1.1.2" xref="S2.2.p2.21.m21.1.1.1.1.1.1.2.cmml">int</mi><msup id="S2.2.p2.21.m21.1.1.1.1.1.1.3" xref="S2.2.p2.21.m21.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.2.p2.21.m21.1.1.1.1.1.1.3.2" xref="S2.2.p2.21.m21.1.1.1.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.2.p2.21.m21.1.1.1.1.1.1.3.3" xref="S2.2.p2.21.m21.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.2.p2.21.m21.2.2.2.2.2a" xref="S2.2.p2.21.m21.2.2.2.2.3.cmml">⁡</mo><mrow id="S2.2.p2.21.m21.2.2.2.2.2.2" xref="S2.2.p2.21.m21.2.2.2.2.3.cmml"><mo id="S2.2.p2.21.m21.2.2.2.2.2.2.2" stretchy="false" xref="S2.2.p2.21.m21.2.2.2.2.3.cmml">(</mo><msub id="S2.2.p2.21.m21.2.2.2.2.2.2.1" xref="S2.2.p2.21.m21.2.2.2.2.2.2.1.cmml"><mi id="S2.2.p2.21.m21.2.2.2.2.2.2.1.2" xref="S2.2.p2.21.m21.2.2.2.2.2.2.1.2.cmml">x</mi><mi id="S2.2.p2.21.m21.2.2.2.2.2.2.1.3" xref="S2.2.p2.21.m21.2.2.2.2.2.2.1.3.cmml">r</mi></msub><mo id="S2.2.p2.21.m21.2.2.2.2.2.2.3" stretchy="false" xref="S2.2.p2.21.m21.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.2.p2.21.m21.2.2.2.3" xref="S2.2.p2.21.m21.2.2.2.3.cmml">∩</mo><mi id="S2.2.p2.21.m21.2.2.2.4" xref="S2.2.p2.21.m21.2.2.2.4.cmml">X</mi><mo id="S2.2.p2.21.m21.2.2.2.3a" xref="S2.2.p2.21.m21.2.2.2.3.cmml">∩</mo><mi id="S2.2.p2.21.m21.2.2.2.5" xref="S2.2.p2.21.m21.2.2.2.5.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.21.m21.2b"><apply id="S2.2.p2.21.m21.2.2.cmml" xref="S2.2.p2.21.m21.2.2"><in id="S2.2.p2.21.m21.2.2.3.cmml" xref="S2.2.p2.21.m21.2.2.3"></in><ci id="S2.2.p2.21.m21.2.2.4.cmml" xref="S2.2.p2.21.m21.2.2.4">𝑣</ci><apply id="S2.2.p2.21.m21.2.2.2.cmml" xref="S2.2.p2.21.m21.2.2.2"><intersect id="S2.2.p2.21.m21.2.2.2.3.cmml" xref="S2.2.p2.21.m21.2.2.2.3"></intersect><apply id="S2.2.p2.21.m21.2.2.2.2.3.cmml" xref="S2.2.p2.21.m21.2.2.2.2.2"><apply id="S2.2.p2.21.m21.1.1.1.1.1.1.cmml" xref="S2.2.p2.21.m21.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.2.p2.21.m21.1.1.1.1.1.1.1.cmml" xref="S2.2.p2.21.m21.1.1.1.1.1.1">subscript</csymbol><ci id="S2.2.p2.21.m21.1.1.1.1.1.1.2.cmml" xref="S2.2.p2.21.m21.1.1.1.1.1.1.2">int</ci><apply id="S2.2.p2.21.m21.1.1.1.1.1.1.3.cmml" xref="S2.2.p2.21.m21.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.21.m21.1.1.1.1.1.1.3.1.cmml" xref="S2.2.p2.21.m21.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.2.p2.21.m21.1.1.1.1.1.1.3.2.cmml" xref="S2.2.p2.21.m21.1.1.1.1.1.1.3.2">𝒯</ci><ci id="S2.2.p2.21.m21.1.1.1.1.1.1.3.3.cmml" xref="S2.2.p2.21.m21.1.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S2.2.p2.21.m21.2.2.2.2.2.2.1.cmml" xref="S2.2.p2.21.m21.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S2.2.p2.21.m21.2.2.2.2.2.2.1.1.cmml" xref="S2.2.p2.21.m21.2.2.2.2.2.2.1">subscript</csymbol><ci id="S2.2.p2.21.m21.2.2.2.2.2.2.1.2.cmml" xref="S2.2.p2.21.m21.2.2.2.2.2.2.1.2">𝑥</ci><ci id="S2.2.p2.21.m21.2.2.2.2.2.2.1.3.cmml" xref="S2.2.p2.21.m21.2.2.2.2.2.2.1.3">𝑟</ci></apply></apply><ci id="S2.2.p2.21.m21.2.2.2.4.cmml" xref="S2.2.p2.21.m21.2.2.2.4">𝑋</ci><ci id="S2.2.p2.21.m21.2.2.2.5.cmml" xref="S2.2.p2.21.m21.2.2.2.5">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.21.m21.2c">v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{r})\cap X\cap Y</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.21.m21.2d">italic_v ∈ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) ∩ italic_X ∩ italic_Y</annotation></semantics></math> since <math alttext="v\in B_{x_{r}}" class="ltx_Math" display="inline" id="S2.2.p2.22.m22.1"><semantics id="S2.2.p2.22.m22.1a"><mrow id="S2.2.p2.22.m22.1.1" xref="S2.2.p2.22.m22.1.1.cmml"><mi id="S2.2.p2.22.m22.1.1.2" xref="S2.2.p2.22.m22.1.1.2.cmml">v</mi><mo id="S2.2.p2.22.m22.1.1.1" xref="S2.2.p2.22.m22.1.1.1.cmml">∈</mo><msub id="S2.2.p2.22.m22.1.1.3" xref="S2.2.p2.22.m22.1.1.3.cmml"><mi id="S2.2.p2.22.m22.1.1.3.2" xref="S2.2.p2.22.m22.1.1.3.2.cmml">B</mi><msub id="S2.2.p2.22.m22.1.1.3.3" xref="S2.2.p2.22.m22.1.1.3.3.cmml"><mi id="S2.2.p2.22.m22.1.1.3.3.2" xref="S2.2.p2.22.m22.1.1.3.3.2.cmml">x</mi><mi id="S2.2.p2.22.m22.1.1.3.3.3" xref="S2.2.p2.22.m22.1.1.3.3.3.cmml">r</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.22.m22.1b"><apply id="S2.2.p2.22.m22.1.1.cmml" xref="S2.2.p2.22.m22.1.1"><in id="S2.2.p2.22.m22.1.1.1.cmml" xref="S2.2.p2.22.m22.1.1.1"></in><ci id="S2.2.p2.22.m22.1.1.2.cmml" xref="S2.2.p2.22.m22.1.1.2">𝑣</ci><apply id="S2.2.p2.22.m22.1.1.3.cmml" xref="S2.2.p2.22.m22.1.1.3"><csymbol cd="ambiguous" id="S2.2.p2.22.m22.1.1.3.1.cmml" xref="S2.2.p2.22.m22.1.1.3">subscript</csymbol><ci id="S2.2.p2.22.m22.1.1.3.2.cmml" xref="S2.2.p2.22.m22.1.1.3.2">𝐵</ci><apply id="S2.2.p2.22.m22.1.1.3.3.cmml" xref="S2.2.p2.22.m22.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p2.22.m22.1.1.3.3.1.cmml" xref="S2.2.p2.22.m22.1.1.3.3">subscript</csymbol><ci id="S2.2.p2.22.m22.1.1.3.3.2.cmml" xref="S2.2.p2.22.m22.1.1.3.3.2">𝑥</ci><ci id="S2.2.p2.22.m22.1.1.3.3.3.cmml" xref="S2.2.p2.22.m22.1.1.3.3.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.22.m22.1c">v\in B_{x_{r}}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.22.m22.1d">italic_v ∈ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. In particular, <math alttext="v\in X\cap Y" class="ltx_Math" display="inline" id="S2.2.p2.23.m23.1"><semantics id="S2.2.p2.23.m23.1a"><mrow id="S2.2.p2.23.m23.1.1" xref="S2.2.p2.23.m23.1.1.cmml"><mi id="S2.2.p2.23.m23.1.1.2" xref="S2.2.p2.23.m23.1.1.2.cmml">v</mi><mo id="S2.2.p2.23.m23.1.1.1" xref="S2.2.p2.23.m23.1.1.1.cmml">∈</mo><mrow id="S2.2.p2.23.m23.1.1.3" xref="S2.2.p2.23.m23.1.1.3.cmml"><mi id="S2.2.p2.23.m23.1.1.3.2" xref="S2.2.p2.23.m23.1.1.3.2.cmml">X</mi><mo id="S2.2.p2.23.m23.1.1.3.1" xref="S2.2.p2.23.m23.1.1.3.1.cmml">∩</mo><mi id="S2.2.p2.23.m23.1.1.3.3" xref="S2.2.p2.23.m23.1.1.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p2.23.m23.1b"><apply id="S2.2.p2.23.m23.1.1.cmml" xref="S2.2.p2.23.m23.1.1"><in id="S2.2.p2.23.m23.1.1.1.cmml" xref="S2.2.p2.23.m23.1.1.1"></in><ci id="S2.2.p2.23.m23.1.1.2.cmml" xref="S2.2.p2.23.m23.1.1.2">𝑣</ci><apply id="S2.2.p2.23.m23.1.1.3.cmml" xref="S2.2.p2.23.m23.1.1.3"><intersect id="S2.2.p2.23.m23.1.1.3.1.cmml" xref="S2.2.p2.23.m23.1.1.3.1"></intersect><ci id="S2.2.p2.23.m23.1.1.3.2.cmml" xref="S2.2.p2.23.m23.1.1.3.2">𝑋</ci><ci id="S2.2.p2.23.m23.1.1.3.3.cmml" xref="S2.2.p2.23.m23.1.1.3.3">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p2.23.m23.1c">v\in X\cap Y</annotation><annotation encoding="application/x-llamapun" id="S2.2.p2.23.m23.1d">italic_v ∈ italic_X ∩ italic_Y</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.3.p3"> <p class="ltx_p" id="S2.3.p3.26">Define <math alttext="i^{*}" class="ltx_Math" display="inline" id="S2.3.p3.1.m1.1"><semantics id="S2.3.p3.1.m1.1a"><msup id="S2.3.p3.1.m1.1.1" xref="S2.3.p3.1.m1.1.1.cmml"><mi id="S2.3.p3.1.m1.1.1.2" xref="S2.3.p3.1.m1.1.1.2.cmml">i</mi><mo id="S2.3.p3.1.m1.1.1.3" xref="S2.3.p3.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.3.p3.1.m1.1b"><apply id="S2.3.p3.1.m1.1.1.cmml" xref="S2.3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.3.p3.1.m1.1.1.1.cmml" xref="S2.3.p3.1.m1.1.1">superscript</csymbol><ci id="S2.3.p3.1.m1.1.1.2.cmml" xref="S2.3.p3.1.m1.1.1.2">𝑖</ci><times id="S2.3.p3.1.m1.1.1.3.cmml" xref="S2.3.p3.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.1.m1.1c">i^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.1.m1.1d">italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> so that <math alttext="x_{i^{*}}" class="ltx_Math" display="inline" id="S2.3.p3.2.m2.1"><semantics id="S2.3.p3.2.m2.1a"><msub id="S2.3.p3.2.m2.1.1" xref="S2.3.p3.2.m2.1.1.cmml"><mi id="S2.3.p3.2.m2.1.1.2" xref="S2.3.p3.2.m2.1.1.2.cmml">x</mi><msup id="S2.3.p3.2.m2.1.1.3" xref="S2.3.p3.2.m2.1.1.3.cmml"><mi id="S2.3.p3.2.m2.1.1.3.2" xref="S2.3.p3.2.m2.1.1.3.2.cmml">i</mi><mo id="S2.3.p3.2.m2.1.1.3.3" xref="S2.3.p3.2.m2.1.1.3.3.cmml">∗</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.2.m2.1b"><apply id="S2.3.p3.2.m2.1.1.cmml" xref="S2.3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.3.p3.2.m2.1.1.1.cmml" xref="S2.3.p3.2.m2.1.1">subscript</csymbol><ci id="S2.3.p3.2.m2.1.1.2.cmml" xref="S2.3.p3.2.m2.1.1.2">𝑥</ci><apply id="S2.3.p3.2.m2.1.1.3.cmml" xref="S2.3.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.2.m2.1.1.3.1.cmml" xref="S2.3.p3.2.m2.1.1.3">superscript</csymbol><ci id="S2.3.p3.2.m2.1.1.3.2.cmml" xref="S2.3.p3.2.m2.1.1.3.2">𝑖</ci><times id="S2.3.p3.2.m2.1.1.3.3.cmml" xref="S2.3.p3.2.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.2.m2.1c">x_{i^{*}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is the unique vertex in <math alttext="P" class="ltx_Math" display="inline" id="S2.3.p3.3.m3.1"><semantics id="S2.3.p3.3.m3.1a"><mi id="S2.3.p3.3.m3.1.1" xref="S2.3.p3.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S2.3.p3.3.m3.1b"><ci id="S2.3.p3.3.m3.1.1.cmml" xref="S2.3.p3.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.3.m3.1d">italic_P</annotation></semantics></math> that is an ancestor of both <math alttext="x_{0}" class="ltx_Math" display="inline" id="S2.3.p3.4.m4.1"><semantics id="S2.3.p3.4.m4.1a"><msub id="S2.3.p3.4.m4.1.1" xref="S2.3.p3.4.m4.1.1.cmml"><mi id="S2.3.p3.4.m4.1.1.2" xref="S2.3.p3.4.m4.1.1.2.cmml">x</mi><mn id="S2.3.p3.4.m4.1.1.3" xref="S2.3.p3.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.4.m4.1b"><apply id="S2.3.p3.4.m4.1.1.cmml" xref="S2.3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S2.3.p3.4.m4.1.1.1.cmml" xref="S2.3.p3.4.m4.1.1">subscript</csymbol><ci id="S2.3.p3.4.m4.1.1.2.cmml" xref="S2.3.p3.4.m4.1.1.2">𝑥</ci><cn id="S2.3.p3.4.m4.1.1.3.cmml" type="integer" xref="S2.3.p3.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.4.m4.1c">x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.4.m4.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x_{r}" class="ltx_Math" display="inline" id="S2.3.p3.5.m5.1"><semantics id="S2.3.p3.5.m5.1a"><msub id="S2.3.p3.5.m5.1.1" xref="S2.3.p3.5.m5.1.1.cmml"><mi id="S2.3.p3.5.m5.1.1.2" xref="S2.3.p3.5.m5.1.1.2.cmml">x</mi><mi id="S2.3.p3.5.m5.1.1.3" xref="S2.3.p3.5.m5.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.5.m5.1b"><apply id="S2.3.p3.5.m5.1.1.cmml" xref="S2.3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S2.3.p3.5.m5.1.1.1.cmml" xref="S2.3.p3.5.m5.1.1">subscript</csymbol><ci id="S2.3.p3.5.m5.1.1.2.cmml" xref="S2.3.p3.5.m5.1.1.2">𝑥</ci><ci id="S2.3.p3.5.m5.1.1.3.cmml" xref="S2.3.p3.5.m5.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.5.m5.1c">x_{r}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.5.m5.1d">italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\operatorname{int}_{\mathcal{T}^{\prime}}(x_{i^{*}})\supseteq\operatorname{int% }_{\mathcal{T}^{\prime}}(x_{i})" class="ltx_Math" display="inline" id="S2.3.p3.6.m6.4"><semantics id="S2.3.p3.6.m6.4a"><mrow id="S2.3.p3.6.m6.4.4" xref="S2.3.p3.6.m6.4.4.cmml"><mrow id="S2.3.p3.6.m6.2.2.2.2" xref="S2.3.p3.6.m6.2.2.2.3.cmml"><msub id="S2.3.p3.6.m6.1.1.1.1.1" xref="S2.3.p3.6.m6.1.1.1.1.1.cmml"><mi id="S2.3.p3.6.m6.1.1.1.1.1.2" xref="S2.3.p3.6.m6.1.1.1.1.1.2.cmml">int</mi><msup id="S2.3.p3.6.m6.1.1.1.1.1.3" xref="S2.3.p3.6.m6.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.6.m6.1.1.1.1.1.3.2" xref="S2.3.p3.6.m6.1.1.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.6.m6.1.1.1.1.1.3.3" xref="S2.3.p3.6.m6.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.6.m6.2.2.2.2a" xref="S2.3.p3.6.m6.2.2.2.3.cmml">⁡</mo><mrow id="S2.3.p3.6.m6.2.2.2.2.2" xref="S2.3.p3.6.m6.2.2.2.3.cmml"><mo id="S2.3.p3.6.m6.2.2.2.2.2.2" stretchy="false" xref="S2.3.p3.6.m6.2.2.2.3.cmml">(</mo><msub id="S2.3.p3.6.m6.2.2.2.2.2.1" xref="S2.3.p3.6.m6.2.2.2.2.2.1.cmml"><mi id="S2.3.p3.6.m6.2.2.2.2.2.1.2" xref="S2.3.p3.6.m6.2.2.2.2.2.1.2.cmml">x</mi><msup id="S2.3.p3.6.m6.2.2.2.2.2.1.3" xref="S2.3.p3.6.m6.2.2.2.2.2.1.3.cmml"><mi id="S2.3.p3.6.m6.2.2.2.2.2.1.3.2" xref="S2.3.p3.6.m6.2.2.2.2.2.1.3.2.cmml">i</mi><mo id="S2.3.p3.6.m6.2.2.2.2.2.1.3.3" xref="S2.3.p3.6.m6.2.2.2.2.2.1.3.3.cmml">∗</mo></msup></msub><mo id="S2.3.p3.6.m6.2.2.2.2.2.3" stretchy="false" xref="S2.3.p3.6.m6.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.3.p3.6.m6.4.4.5" xref="S2.3.p3.6.m6.4.4.cmml">⊇</mo><mrow id="S2.3.p3.6.m6.4.4.4.2" xref="S2.3.p3.6.m6.4.4.4.3.cmml"><msub id="S2.3.p3.6.m6.3.3.3.1.1" xref="S2.3.p3.6.m6.3.3.3.1.1.cmml"><mi id="S2.3.p3.6.m6.3.3.3.1.1.2" xref="S2.3.p3.6.m6.3.3.3.1.1.2.cmml">int</mi><msup id="S2.3.p3.6.m6.3.3.3.1.1.3" xref="S2.3.p3.6.m6.3.3.3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.6.m6.3.3.3.1.1.3.2" xref="S2.3.p3.6.m6.3.3.3.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.6.m6.3.3.3.1.1.3.3" xref="S2.3.p3.6.m6.3.3.3.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.6.m6.4.4.4.2a" xref="S2.3.p3.6.m6.4.4.4.3.cmml">⁡</mo><mrow id="S2.3.p3.6.m6.4.4.4.2.2" xref="S2.3.p3.6.m6.4.4.4.3.cmml"><mo id="S2.3.p3.6.m6.4.4.4.2.2.2" stretchy="false" xref="S2.3.p3.6.m6.4.4.4.3.cmml">(</mo><msub id="S2.3.p3.6.m6.4.4.4.2.2.1" xref="S2.3.p3.6.m6.4.4.4.2.2.1.cmml"><mi id="S2.3.p3.6.m6.4.4.4.2.2.1.2" xref="S2.3.p3.6.m6.4.4.4.2.2.1.2.cmml">x</mi><mi id="S2.3.p3.6.m6.4.4.4.2.2.1.3" xref="S2.3.p3.6.m6.4.4.4.2.2.1.3.cmml">i</mi></msub><mo id="S2.3.p3.6.m6.4.4.4.2.2.3" stretchy="false" xref="S2.3.p3.6.m6.4.4.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.6.m6.4b"><apply id="S2.3.p3.6.m6.4.4.cmml" xref="S2.3.p3.6.m6.4.4"><subset id="S2.3.p3.6.m6.4.4a.cmml" xref="S2.3.p3.6.m6.4.4"></subset><apply id="S2.3.p3.6.m6.4.4.4.3.cmml" xref="S2.3.p3.6.m6.4.4.4.2"><apply id="S2.3.p3.6.m6.3.3.3.1.1.cmml" xref="S2.3.p3.6.m6.3.3.3.1.1"><csymbol cd="ambiguous" id="S2.3.p3.6.m6.3.3.3.1.1.1.cmml" xref="S2.3.p3.6.m6.3.3.3.1.1">subscript</csymbol><ci id="S2.3.p3.6.m6.3.3.3.1.1.2.cmml" xref="S2.3.p3.6.m6.3.3.3.1.1.2">int</ci><apply id="S2.3.p3.6.m6.3.3.3.1.1.3.cmml" xref="S2.3.p3.6.m6.3.3.3.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.6.m6.3.3.3.1.1.3.1.cmml" xref="S2.3.p3.6.m6.3.3.3.1.1.3">superscript</csymbol><ci id="S2.3.p3.6.m6.3.3.3.1.1.3.2.cmml" xref="S2.3.p3.6.m6.3.3.3.1.1.3.2">𝒯</ci><ci id="S2.3.p3.6.m6.3.3.3.1.1.3.3.cmml" xref="S2.3.p3.6.m6.3.3.3.1.1.3.3">′</ci></apply></apply><apply id="S2.3.p3.6.m6.4.4.4.2.2.1.cmml" xref="S2.3.p3.6.m6.4.4.4.2.2.1"><csymbol cd="ambiguous" id="S2.3.p3.6.m6.4.4.4.2.2.1.1.cmml" xref="S2.3.p3.6.m6.4.4.4.2.2.1">subscript</csymbol><ci id="S2.3.p3.6.m6.4.4.4.2.2.1.2.cmml" xref="S2.3.p3.6.m6.4.4.4.2.2.1.2">𝑥</ci><ci id="S2.3.p3.6.m6.4.4.4.2.2.1.3.cmml" xref="S2.3.p3.6.m6.4.4.4.2.2.1.3">𝑖</ci></apply></apply><apply id="S2.3.p3.6.m6.2.2.2.3.cmml" xref="S2.3.p3.6.m6.2.2.2.2"><apply id="S2.3.p3.6.m6.1.1.1.1.1.cmml" xref="S2.3.p3.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.3.p3.6.m6.1.1.1.1.1.1.cmml" xref="S2.3.p3.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S2.3.p3.6.m6.1.1.1.1.1.2.cmml" xref="S2.3.p3.6.m6.1.1.1.1.1.2">int</ci><apply id="S2.3.p3.6.m6.1.1.1.1.1.3.cmml" xref="S2.3.p3.6.m6.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.6.m6.1.1.1.1.1.3.1.cmml" xref="S2.3.p3.6.m6.1.1.1.1.1.3">superscript</csymbol><ci id="S2.3.p3.6.m6.1.1.1.1.1.3.2.cmml" xref="S2.3.p3.6.m6.1.1.1.1.1.3.2">𝒯</ci><ci id="S2.3.p3.6.m6.1.1.1.1.1.3.3.cmml" xref="S2.3.p3.6.m6.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S2.3.p3.6.m6.2.2.2.2.2.1.cmml" xref="S2.3.p3.6.m6.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S2.3.p3.6.m6.2.2.2.2.2.1.1.cmml" xref="S2.3.p3.6.m6.2.2.2.2.2.1">subscript</csymbol><ci id="S2.3.p3.6.m6.2.2.2.2.2.1.2.cmml" xref="S2.3.p3.6.m6.2.2.2.2.2.1.2">𝑥</ci><apply id="S2.3.p3.6.m6.2.2.2.2.2.1.3.cmml" xref="S2.3.p3.6.m6.2.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S2.3.p3.6.m6.2.2.2.2.2.1.3.1.cmml" xref="S2.3.p3.6.m6.2.2.2.2.2.1.3">superscript</csymbol><ci id="S2.3.p3.6.m6.2.2.2.2.2.1.3.2.cmml" xref="S2.3.p3.6.m6.2.2.2.2.2.1.3.2">𝑖</ci><times id="S2.3.p3.6.m6.2.2.2.2.2.1.3.3.cmml" xref="S2.3.p3.6.m6.2.2.2.2.2.1.3.3"></times></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.6.m6.4c">\operatorname{int}_{\mathcal{T}^{\prime}}(x_{i^{*}})\supseteq\operatorname{int% }_{\mathcal{T}^{\prime}}(x_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.6.m6.4d">roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ⊇ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> for each <math alttext="i\in\{0,\ldots,r\}" class="ltx_Math" display="inline" id="S2.3.p3.7.m7.3"><semantics id="S2.3.p3.7.m7.3a"><mrow id="S2.3.p3.7.m7.3.4" xref="S2.3.p3.7.m7.3.4.cmml"><mi id="S2.3.p3.7.m7.3.4.2" xref="S2.3.p3.7.m7.3.4.2.cmml">i</mi><mo id="S2.3.p3.7.m7.3.4.1" xref="S2.3.p3.7.m7.3.4.1.cmml">∈</mo><mrow id="S2.3.p3.7.m7.3.4.3.2" xref="S2.3.p3.7.m7.3.4.3.1.cmml"><mo id="S2.3.p3.7.m7.3.4.3.2.1" stretchy="false" xref="S2.3.p3.7.m7.3.4.3.1.cmml">{</mo><mn id="S2.3.p3.7.m7.1.1" xref="S2.3.p3.7.m7.1.1.cmml">0</mn><mo id="S2.3.p3.7.m7.3.4.3.2.2" xref="S2.3.p3.7.m7.3.4.3.1.cmml">,</mo><mi id="S2.3.p3.7.m7.2.2" mathvariant="normal" xref="S2.3.p3.7.m7.2.2.cmml">…</mi><mo id="S2.3.p3.7.m7.3.4.3.2.3" xref="S2.3.p3.7.m7.3.4.3.1.cmml">,</mo><mi id="S2.3.p3.7.m7.3.3" xref="S2.3.p3.7.m7.3.3.cmml">r</mi><mo id="S2.3.p3.7.m7.3.4.3.2.4" stretchy="false" xref="S2.3.p3.7.m7.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.7.m7.3b"><apply id="S2.3.p3.7.m7.3.4.cmml" xref="S2.3.p3.7.m7.3.4"><in id="S2.3.p3.7.m7.3.4.1.cmml" xref="S2.3.p3.7.m7.3.4.1"></in><ci id="S2.3.p3.7.m7.3.4.2.cmml" xref="S2.3.p3.7.m7.3.4.2">𝑖</ci><set id="S2.3.p3.7.m7.3.4.3.1.cmml" xref="S2.3.p3.7.m7.3.4.3.2"><cn id="S2.3.p3.7.m7.1.1.cmml" type="integer" xref="S2.3.p3.7.m7.1.1">0</cn><ci id="S2.3.p3.7.m7.2.2.cmml" xref="S2.3.p3.7.m7.2.2">…</ci><ci id="S2.3.p3.7.m7.3.3.cmml" xref="S2.3.p3.7.m7.3.3">𝑟</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.7.m7.3c">i\in\{0,\ldots,r\}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.7.m7.3d">italic_i ∈ { 0 , … , italic_r }</annotation></semantics></math>. In particular <math alttext="v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{r})\cap X\cap Y\subseteq% \operatorname{int}_{\mathcal{T}^{\prime}}(x_{i^{*}})\cap X\cap Y\subseteq B_{x% _{i}}" class="ltx_Math" display="inline" id="S2.3.p3.8.m8.4"><semantics id="S2.3.p3.8.m8.4a"><mrow id="S2.3.p3.8.m8.4.4" xref="S2.3.p3.8.m8.4.4.cmml"><mi id="S2.3.p3.8.m8.4.4.6" xref="S2.3.p3.8.m8.4.4.6.cmml">v</mi><mo id="S2.3.p3.8.m8.4.4.7" xref="S2.3.p3.8.m8.4.4.7.cmml">∈</mo><mrow id="S2.3.p3.8.m8.2.2.2" xref="S2.3.p3.8.m8.2.2.2.cmml"><mrow id="S2.3.p3.8.m8.2.2.2.2.2" xref="S2.3.p3.8.m8.2.2.2.2.3.cmml"><msub id="S2.3.p3.8.m8.1.1.1.1.1.1" xref="S2.3.p3.8.m8.1.1.1.1.1.1.cmml"><mi id="S2.3.p3.8.m8.1.1.1.1.1.1.2" xref="S2.3.p3.8.m8.1.1.1.1.1.1.2.cmml">int</mi><msup id="S2.3.p3.8.m8.1.1.1.1.1.1.3" xref="S2.3.p3.8.m8.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.8.m8.1.1.1.1.1.1.3.2" xref="S2.3.p3.8.m8.1.1.1.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.8.m8.1.1.1.1.1.1.3.3" xref="S2.3.p3.8.m8.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.8.m8.2.2.2.2.2a" xref="S2.3.p3.8.m8.2.2.2.2.3.cmml">⁡</mo><mrow id="S2.3.p3.8.m8.2.2.2.2.2.2" xref="S2.3.p3.8.m8.2.2.2.2.3.cmml"><mo id="S2.3.p3.8.m8.2.2.2.2.2.2.2" stretchy="false" xref="S2.3.p3.8.m8.2.2.2.2.3.cmml">(</mo><msub id="S2.3.p3.8.m8.2.2.2.2.2.2.1" xref="S2.3.p3.8.m8.2.2.2.2.2.2.1.cmml"><mi id="S2.3.p3.8.m8.2.2.2.2.2.2.1.2" xref="S2.3.p3.8.m8.2.2.2.2.2.2.1.2.cmml">x</mi><mi id="S2.3.p3.8.m8.2.2.2.2.2.2.1.3" xref="S2.3.p3.8.m8.2.2.2.2.2.2.1.3.cmml">r</mi></msub><mo id="S2.3.p3.8.m8.2.2.2.2.2.2.3" stretchy="false" xref="S2.3.p3.8.m8.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.3.p3.8.m8.2.2.2.3" xref="S2.3.p3.8.m8.2.2.2.3.cmml">∩</mo><mi id="S2.3.p3.8.m8.2.2.2.4" xref="S2.3.p3.8.m8.2.2.2.4.cmml">X</mi><mo id="S2.3.p3.8.m8.2.2.2.3a" xref="S2.3.p3.8.m8.2.2.2.3.cmml">∩</mo><mi id="S2.3.p3.8.m8.2.2.2.5" xref="S2.3.p3.8.m8.2.2.2.5.cmml">Y</mi></mrow><mo id="S2.3.p3.8.m8.4.4.8" xref="S2.3.p3.8.m8.4.4.8.cmml">⊆</mo><mrow id="S2.3.p3.8.m8.4.4.4" xref="S2.3.p3.8.m8.4.4.4.cmml"><mrow id="S2.3.p3.8.m8.4.4.4.2.2" xref="S2.3.p3.8.m8.4.4.4.2.3.cmml"><msub id="S2.3.p3.8.m8.3.3.3.1.1.1" xref="S2.3.p3.8.m8.3.3.3.1.1.1.cmml"><mi id="S2.3.p3.8.m8.3.3.3.1.1.1.2" xref="S2.3.p3.8.m8.3.3.3.1.1.1.2.cmml">int</mi><msup id="S2.3.p3.8.m8.3.3.3.1.1.1.3" xref="S2.3.p3.8.m8.3.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.8.m8.3.3.3.1.1.1.3.2" xref="S2.3.p3.8.m8.3.3.3.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.8.m8.3.3.3.1.1.1.3.3" xref="S2.3.p3.8.m8.3.3.3.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.8.m8.4.4.4.2.2a" xref="S2.3.p3.8.m8.4.4.4.2.3.cmml">⁡</mo><mrow id="S2.3.p3.8.m8.4.4.4.2.2.2" xref="S2.3.p3.8.m8.4.4.4.2.3.cmml"><mo id="S2.3.p3.8.m8.4.4.4.2.2.2.2" stretchy="false" xref="S2.3.p3.8.m8.4.4.4.2.3.cmml">(</mo><msub id="S2.3.p3.8.m8.4.4.4.2.2.2.1" xref="S2.3.p3.8.m8.4.4.4.2.2.2.1.cmml"><mi id="S2.3.p3.8.m8.4.4.4.2.2.2.1.2" 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xref="S2.3.p3.8.m8.4.4"><subset id="S2.3.p3.8.m8.4.4.9.cmml" xref="S2.3.p3.8.m8.4.4.9"></subset><share href="https://arxiv.org/html/2503.17112v1#S2.3.p3.8.m8.4.4.4.cmml" id="S2.3.p3.8.m8.4.4f.cmml" xref="S2.3.p3.8.m8.4.4"></share><apply id="S2.3.p3.8.m8.4.4.10.cmml" xref="S2.3.p3.8.m8.4.4.10"><csymbol cd="ambiguous" id="S2.3.p3.8.m8.4.4.10.1.cmml" xref="S2.3.p3.8.m8.4.4.10">subscript</csymbol><ci id="S2.3.p3.8.m8.4.4.10.2.cmml" xref="S2.3.p3.8.m8.4.4.10.2">𝐵</ci><apply id="S2.3.p3.8.m8.4.4.10.3.cmml" xref="S2.3.p3.8.m8.4.4.10.3"><csymbol cd="ambiguous" id="S2.3.p3.8.m8.4.4.10.3.1.cmml" xref="S2.3.p3.8.m8.4.4.10.3">subscript</csymbol><ci id="S2.3.p3.8.m8.4.4.10.3.2.cmml" xref="S2.3.p3.8.m8.4.4.10.3.2">𝑥</ci><ci id="S2.3.p3.8.m8.4.4.10.3.3.cmml" xref="S2.3.p3.8.m8.4.4.10.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.8.m8.4c">v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{r})\cap X\cap Y\subseteq% \operatorname{int}_{\mathcal{T}^{\prime}}(x_{i^{*}})\cap X\cap Y\subseteq B_{x% _{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.8.m8.4d">italic_v ∈ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) ∩ italic_X ∩ italic_Y ⊆ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ∩ italic_X ∩ italic_Y ⊆ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for each <math alttext="i\in\{i^{*},\ldots,r\}" class="ltx_Math" display="inline" id="S2.3.p3.9.m9.3"><semantics id="S2.3.p3.9.m9.3a"><mrow id="S2.3.p3.9.m9.3.3" xref="S2.3.p3.9.m9.3.3.cmml"><mi id="S2.3.p3.9.m9.3.3.3" xref="S2.3.p3.9.m9.3.3.3.cmml">i</mi><mo id="S2.3.p3.9.m9.3.3.2" xref="S2.3.p3.9.m9.3.3.2.cmml">∈</mo><mrow id="S2.3.p3.9.m9.3.3.1.1" xref="S2.3.p3.9.m9.3.3.1.2.cmml"><mo id="S2.3.p3.9.m9.3.3.1.1.2" stretchy="false" xref="S2.3.p3.9.m9.3.3.1.2.cmml">{</mo><msup id="S2.3.p3.9.m9.3.3.1.1.1" xref="S2.3.p3.9.m9.3.3.1.1.1.cmml"><mi id="S2.3.p3.9.m9.3.3.1.1.1.2" xref="S2.3.p3.9.m9.3.3.1.1.1.2.cmml">i</mi><mo id="S2.3.p3.9.m9.3.3.1.1.1.3" xref="S2.3.p3.9.m9.3.3.1.1.1.3.cmml">∗</mo></msup><mo id="S2.3.p3.9.m9.3.3.1.1.3" xref="S2.3.p3.9.m9.3.3.1.2.cmml">,</mo><mi id="S2.3.p3.9.m9.1.1" mathvariant="normal" xref="S2.3.p3.9.m9.1.1.cmml">…</mi><mo id="S2.3.p3.9.m9.3.3.1.1.4" xref="S2.3.p3.9.m9.3.3.1.2.cmml">,</mo><mi id="S2.3.p3.9.m9.2.2" xref="S2.3.p3.9.m9.2.2.cmml">r</mi><mo id="S2.3.p3.9.m9.3.3.1.1.5" stretchy="false" xref="S2.3.p3.9.m9.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.9.m9.3b"><apply id="S2.3.p3.9.m9.3.3.cmml" xref="S2.3.p3.9.m9.3.3"><in id="S2.3.p3.9.m9.3.3.2.cmml" xref="S2.3.p3.9.m9.3.3.2"></in><ci id="S2.3.p3.9.m9.3.3.3.cmml" xref="S2.3.p3.9.m9.3.3.3">𝑖</ci><set id="S2.3.p3.9.m9.3.3.1.2.cmml" xref="S2.3.p3.9.m9.3.3.1.1"><apply id="S2.3.p3.9.m9.3.3.1.1.1.cmml" xref="S2.3.p3.9.m9.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.3.p3.9.m9.3.3.1.1.1.1.cmml" xref="S2.3.p3.9.m9.3.3.1.1.1">superscript</csymbol><ci id="S2.3.p3.9.m9.3.3.1.1.1.2.cmml" xref="S2.3.p3.9.m9.3.3.1.1.1.2">𝑖</ci><times id="S2.3.p3.9.m9.3.3.1.1.1.3.cmml" xref="S2.3.p3.9.m9.3.3.1.1.1.3"></times></apply><ci id="S2.3.p3.9.m9.1.1.cmml" xref="S2.3.p3.9.m9.1.1">…</ci><ci id="S2.3.p3.9.m9.2.2.cmml" xref="S2.3.p3.9.m9.2.2">𝑟</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.9.m9.3c">i\in\{i^{*},\ldots,r\}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.9.m9.3d">italic_i ∈ { italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , … , italic_r }</annotation></semantics></math>. Therefore <math alttext="i^{*}=r" class="ltx_Math" display="inline" id="S2.3.p3.10.m10.1"><semantics id="S2.3.p3.10.m10.1a"><mrow id="S2.3.p3.10.m10.1.1" xref="S2.3.p3.10.m10.1.1.cmml"><msup id="S2.3.p3.10.m10.1.1.2" xref="S2.3.p3.10.m10.1.1.2.cmml"><mi id="S2.3.p3.10.m10.1.1.2.2" xref="S2.3.p3.10.m10.1.1.2.2.cmml">i</mi><mo id="S2.3.p3.10.m10.1.1.2.3" xref="S2.3.p3.10.m10.1.1.2.3.cmml">∗</mo></msup><mo id="S2.3.p3.10.m10.1.1.1" xref="S2.3.p3.10.m10.1.1.1.cmml">=</mo><mi id="S2.3.p3.10.m10.1.1.3" xref="S2.3.p3.10.m10.1.1.3.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.10.m10.1b"><apply id="S2.3.p3.10.m10.1.1.cmml" xref="S2.3.p3.10.m10.1.1"><eq id="S2.3.p3.10.m10.1.1.1.cmml" xref="S2.3.p3.10.m10.1.1.1"></eq><apply id="S2.3.p3.10.m10.1.1.2.cmml" xref="S2.3.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S2.3.p3.10.m10.1.1.2.1.cmml" xref="S2.3.p3.10.m10.1.1.2">superscript</csymbol><ci id="S2.3.p3.10.m10.1.1.2.2.cmml" xref="S2.3.p3.10.m10.1.1.2.2">𝑖</ci><times id="S2.3.p3.10.m10.1.1.2.3.cmml" xref="S2.3.p3.10.m10.1.1.2.3"></times></apply><ci id="S2.3.p3.10.m10.1.1.3.cmml" xref="S2.3.p3.10.m10.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.10.m10.1c">i^{*}=r</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.10.m10.1d">italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_r</annotation></semantics></math> and <math alttext="x_{r}" class="ltx_Math" display="inline" id="S2.3.p3.11.m11.1"><semantics id="S2.3.p3.11.m11.1a"><msub id="S2.3.p3.11.m11.1.1" xref="S2.3.p3.11.m11.1.1.cmml"><mi id="S2.3.p3.11.m11.1.1.2" xref="S2.3.p3.11.m11.1.1.2.cmml">x</mi><mi id="S2.3.p3.11.m11.1.1.3" xref="S2.3.p3.11.m11.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.11.m11.1b"><apply id="S2.3.p3.11.m11.1.1.cmml" xref="S2.3.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S2.3.p3.11.m11.1.1.1.cmml" xref="S2.3.p3.11.m11.1.1">subscript</csymbol><ci id="S2.3.p3.11.m11.1.1.2.cmml" xref="S2.3.p3.11.m11.1.1.2">𝑥</ci><ci id="S2.3.p3.11.m11.1.1.3.cmml" xref="S2.3.p3.11.m11.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.11.m11.1c">x_{r}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.11.m11.1d">italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> is an ancestor of <math alttext="x_{0}" class="ltx_Math" display="inline" id="S2.3.p3.12.m12.1"><semantics id="S2.3.p3.12.m12.1a"><msub id="S2.3.p3.12.m12.1.1" xref="S2.3.p3.12.m12.1.1.cmml"><mi id="S2.3.p3.12.m12.1.1.2" xref="S2.3.p3.12.m12.1.1.2.cmml">x</mi><mn id="S2.3.p3.12.m12.1.1.3" xref="S2.3.p3.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.12.m12.1b"><apply id="S2.3.p3.12.m12.1.1.cmml" xref="S2.3.p3.12.m12.1.1"><csymbol cd="ambiguous" id="S2.3.p3.12.m12.1.1.1.cmml" xref="S2.3.p3.12.m12.1.1">subscript</csymbol><ci id="S2.3.p3.12.m12.1.1.2.cmml" xref="S2.3.p3.12.m12.1.1.2">𝑥</ci><cn id="S2.3.p3.12.m12.1.1.3.cmml" type="integer" xref="S2.3.p3.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.12.m12.1c">x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.12.m12.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore <math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.3.p3.13.m13.1"><semantics id="S2.3.p3.13.m13.1a"><msub id="S2.3.p3.13.m13.1.1" xref="S2.3.p3.13.m13.1.1.cmml"><mi id="S2.3.p3.13.m13.1.1.2" xref="S2.3.p3.13.m13.1.1.2.cmml">x</mi><mn id="S2.3.p3.13.m13.1.1.3" xref="S2.3.p3.13.m13.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.13.m13.1b"><apply id="S2.3.p3.13.m13.1.1.cmml" xref="S2.3.p3.13.m13.1.1"><csymbol cd="ambiguous" id="S2.3.p3.13.m13.1.1.1.cmml" xref="S2.3.p3.13.m13.1.1">subscript</csymbol><ci id="S2.3.p3.13.m13.1.1.2.cmml" xref="S2.3.p3.13.m13.1.1.2">𝑥</ci><cn id="S2.3.p3.13.m13.1.1.3.cmml" type="integer" xref="S2.3.p3.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.13.m13.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.13.m13.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is the parent of <math alttext="x_{0}" class="ltx_Math" display="inline" id="S2.3.p3.14.m14.1"><semantics id="S2.3.p3.14.m14.1a"><msub id="S2.3.p3.14.m14.1.1" xref="S2.3.p3.14.m14.1.1.cmml"><mi id="S2.3.p3.14.m14.1.1.2" xref="S2.3.p3.14.m14.1.1.2.cmml">x</mi><mn id="S2.3.p3.14.m14.1.1.3" xref="S2.3.p3.14.m14.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.14.m14.1b"><apply id="S2.3.p3.14.m14.1.1.cmml" xref="S2.3.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S2.3.p3.14.m14.1.1.1.cmml" xref="S2.3.p3.14.m14.1.1">subscript</csymbol><ci id="S2.3.p3.14.m14.1.1.2.cmml" xref="S2.3.p3.14.m14.1.1.2">𝑥</ci><cn id="S2.3.p3.14.m14.1.1.3.cmml" type="integer" xref="S2.3.p3.14.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.14.m14.1c">x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.14.m14.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="v\in B_{x_{0}}" class="ltx_Math" display="inline" id="S2.3.p3.15.m15.1"><semantics id="S2.3.p3.15.m15.1a"><mrow id="S2.3.p3.15.m15.1.1" xref="S2.3.p3.15.m15.1.1.cmml"><mi id="S2.3.p3.15.m15.1.1.2" xref="S2.3.p3.15.m15.1.1.2.cmml">v</mi><mo id="S2.3.p3.15.m15.1.1.1" xref="S2.3.p3.15.m15.1.1.1.cmml">∈</mo><msub id="S2.3.p3.15.m15.1.1.3" xref="S2.3.p3.15.m15.1.1.3.cmml"><mi id="S2.3.p3.15.m15.1.1.3.2" xref="S2.3.p3.15.m15.1.1.3.2.cmml">B</mi><msub id="S2.3.p3.15.m15.1.1.3.3" xref="S2.3.p3.15.m15.1.1.3.3.cmml"><mi id="S2.3.p3.15.m15.1.1.3.3.2" xref="S2.3.p3.15.m15.1.1.3.3.2.cmml">x</mi><mn id="S2.3.p3.15.m15.1.1.3.3.3" xref="S2.3.p3.15.m15.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.15.m15.1b"><apply id="S2.3.p3.15.m15.1.1.cmml" xref="S2.3.p3.15.m15.1.1"><in id="S2.3.p3.15.m15.1.1.1.cmml" xref="S2.3.p3.15.m15.1.1.1"></in><ci id="S2.3.p3.15.m15.1.1.2.cmml" xref="S2.3.p3.15.m15.1.1.2">𝑣</ci><apply id="S2.3.p3.15.m15.1.1.3.cmml" xref="S2.3.p3.15.m15.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.15.m15.1.1.3.1.cmml" xref="S2.3.p3.15.m15.1.1.3">subscript</csymbol><ci id="S2.3.p3.15.m15.1.1.3.2.cmml" xref="S2.3.p3.15.m15.1.1.3.2">𝐵</ci><apply id="S2.3.p3.15.m15.1.1.3.3.cmml" xref="S2.3.p3.15.m15.1.1.3.3"><csymbol cd="ambiguous" id="S2.3.p3.15.m15.1.1.3.3.1.cmml" xref="S2.3.p3.15.m15.1.1.3.3">subscript</csymbol><ci id="S2.3.p3.15.m15.1.1.3.3.2.cmml" xref="S2.3.p3.15.m15.1.1.3.3.2">𝑥</ci><cn id="S2.3.p3.15.m15.1.1.3.3.3.cmml" type="integer" xref="S2.3.p3.15.m15.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.15.m15.1c">v\in B_{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.15.m15.1d">italic_v ∈ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="v\not\in B_{x_{1}}" class="ltx_Math" display="inline" id="S2.3.p3.16.m16.1"><semantics id="S2.3.p3.16.m16.1a"><mrow id="S2.3.p3.16.m16.1.1" xref="S2.3.p3.16.m16.1.1.cmml"><mi id="S2.3.p3.16.m16.1.1.2" xref="S2.3.p3.16.m16.1.1.2.cmml">v</mi><mo id="S2.3.p3.16.m16.1.1.1" xref="S2.3.p3.16.m16.1.1.1.cmml">∉</mo><msub id="S2.3.p3.16.m16.1.1.3" xref="S2.3.p3.16.m16.1.1.3.cmml"><mi id="S2.3.p3.16.m16.1.1.3.2" xref="S2.3.p3.16.m16.1.1.3.2.cmml">B</mi><msub id="S2.3.p3.16.m16.1.1.3.3" xref="S2.3.p3.16.m16.1.1.3.3.cmml"><mi id="S2.3.p3.16.m16.1.1.3.3.2" xref="S2.3.p3.16.m16.1.1.3.3.2.cmml">x</mi><mn id="S2.3.p3.16.m16.1.1.3.3.3" xref="S2.3.p3.16.m16.1.1.3.3.3.cmml">1</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.16.m16.1b"><apply id="S2.3.p3.16.m16.1.1.cmml" xref="S2.3.p3.16.m16.1.1"><notin id="S2.3.p3.16.m16.1.1.1.cmml" xref="S2.3.p3.16.m16.1.1.1"></notin><ci id="S2.3.p3.16.m16.1.1.2.cmml" xref="S2.3.p3.16.m16.1.1.2">𝑣</ci><apply id="S2.3.p3.16.m16.1.1.3.cmml" xref="S2.3.p3.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.16.m16.1.1.3.1.cmml" xref="S2.3.p3.16.m16.1.1.3">subscript</csymbol><ci id="S2.3.p3.16.m16.1.1.3.2.cmml" xref="S2.3.p3.16.m16.1.1.3.2">𝐵</ci><apply id="S2.3.p3.16.m16.1.1.3.3.cmml" xref="S2.3.p3.16.m16.1.1.3.3"><csymbol cd="ambiguous" id="S2.3.p3.16.m16.1.1.3.3.1.cmml" xref="S2.3.p3.16.m16.1.1.3.3">subscript</csymbol><ci id="S2.3.p3.16.m16.1.1.3.3.2.cmml" xref="S2.3.p3.16.m16.1.1.3.3.2">𝑥</ci><cn id="S2.3.p3.16.m16.1.1.3.3.3.cmml" type="integer" xref="S2.3.p3.16.m16.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.16.m16.1c">v\not\in B_{x_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.16.m16.1d">italic_v ∉ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. If <math alttext="v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{0})" class="ltx_Math" display="inline" id="S2.3.p3.17.m17.2"><semantics id="S2.3.p3.17.m17.2a"><mrow id="S2.3.p3.17.m17.2.2" xref="S2.3.p3.17.m17.2.2.cmml"><mi id="S2.3.p3.17.m17.2.2.4" xref="S2.3.p3.17.m17.2.2.4.cmml">v</mi><mo id="S2.3.p3.17.m17.2.2.3" xref="S2.3.p3.17.m17.2.2.3.cmml">∈</mo><mrow id="S2.3.p3.17.m17.2.2.2.2" xref="S2.3.p3.17.m17.2.2.2.3.cmml"><msub id="S2.3.p3.17.m17.1.1.1.1.1" xref="S2.3.p3.17.m17.1.1.1.1.1.cmml"><mi id="S2.3.p3.17.m17.1.1.1.1.1.2" xref="S2.3.p3.17.m17.1.1.1.1.1.2.cmml">int</mi><msup id="S2.3.p3.17.m17.1.1.1.1.1.3" xref="S2.3.p3.17.m17.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.17.m17.1.1.1.1.1.3.2" xref="S2.3.p3.17.m17.1.1.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.17.m17.1.1.1.1.1.3.3" xref="S2.3.p3.17.m17.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.17.m17.2.2.2.2a" xref="S2.3.p3.17.m17.2.2.2.3.cmml">⁡</mo><mrow id="S2.3.p3.17.m17.2.2.2.2.2" xref="S2.3.p3.17.m17.2.2.2.3.cmml"><mo id="S2.3.p3.17.m17.2.2.2.2.2.2" stretchy="false" xref="S2.3.p3.17.m17.2.2.2.3.cmml">(</mo><msub id="S2.3.p3.17.m17.2.2.2.2.2.1" xref="S2.3.p3.17.m17.2.2.2.2.2.1.cmml"><mi id="S2.3.p3.17.m17.2.2.2.2.2.1.2" xref="S2.3.p3.17.m17.2.2.2.2.2.1.2.cmml">x</mi><mn id="S2.3.p3.17.m17.2.2.2.2.2.1.3" xref="S2.3.p3.17.m17.2.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S2.3.p3.17.m17.2.2.2.2.2.3" stretchy="false" xref="S2.3.p3.17.m17.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.17.m17.2b"><apply id="S2.3.p3.17.m17.2.2.cmml" xref="S2.3.p3.17.m17.2.2"><in id="S2.3.p3.17.m17.2.2.3.cmml" xref="S2.3.p3.17.m17.2.2.3"></in><ci id="S2.3.p3.17.m17.2.2.4.cmml" xref="S2.3.p3.17.m17.2.2.4">𝑣</ci><apply id="S2.3.p3.17.m17.2.2.2.3.cmml" xref="S2.3.p3.17.m17.2.2.2.2"><apply id="S2.3.p3.17.m17.1.1.1.1.1.cmml" xref="S2.3.p3.17.m17.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.3.p3.17.m17.1.1.1.1.1.1.cmml" xref="S2.3.p3.17.m17.1.1.1.1.1">subscript</csymbol><ci id="S2.3.p3.17.m17.1.1.1.1.1.2.cmml" xref="S2.3.p3.17.m17.1.1.1.1.1.2">int</ci><apply id="S2.3.p3.17.m17.1.1.1.1.1.3.cmml" xref="S2.3.p3.17.m17.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.17.m17.1.1.1.1.1.3.1.cmml" xref="S2.3.p3.17.m17.1.1.1.1.1.3">superscript</csymbol><ci id="S2.3.p3.17.m17.1.1.1.1.1.3.2.cmml" xref="S2.3.p3.17.m17.1.1.1.1.1.3.2">𝒯</ci><ci id="S2.3.p3.17.m17.1.1.1.1.1.3.3.cmml" xref="S2.3.p3.17.m17.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S2.3.p3.17.m17.2.2.2.2.2.1.cmml" xref="S2.3.p3.17.m17.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S2.3.p3.17.m17.2.2.2.2.2.1.1.cmml" xref="S2.3.p3.17.m17.2.2.2.2.2.1">subscript</csymbol><ci id="S2.3.p3.17.m17.2.2.2.2.2.1.2.cmml" xref="S2.3.p3.17.m17.2.2.2.2.2.1.2">𝑥</ci><cn id="S2.3.p3.17.m17.2.2.2.2.2.1.3.cmml" type="integer" xref="S2.3.p3.17.m17.2.2.2.2.2.1.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.17.m17.2c">v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{0})</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.17.m17.2d">italic_v ∈ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math> then <math alttext="v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{0})\cap X\cap Y\subseteq% \operatorname{int}_{\mathcal{T}^{\prime}}(x_{1})\cap X\cap Y\subseteq B_{x_{1}}" class="ltx_Math" display="inline" id="S2.3.p3.18.m18.4"><semantics id="S2.3.p3.18.m18.4a"><mrow id="S2.3.p3.18.m18.4.4" xref="S2.3.p3.18.m18.4.4.cmml"><mi id="S2.3.p3.18.m18.4.4.6" xref="S2.3.p3.18.m18.4.4.6.cmml">v</mi><mo id="S2.3.p3.18.m18.4.4.7" xref="S2.3.p3.18.m18.4.4.7.cmml">∈</mo><mrow id="S2.3.p3.18.m18.2.2.2" xref="S2.3.p3.18.m18.2.2.2.cmml"><mrow id="S2.3.p3.18.m18.2.2.2.2.2" xref="S2.3.p3.18.m18.2.2.2.2.3.cmml"><msub id="S2.3.p3.18.m18.1.1.1.1.1.1" xref="S2.3.p3.18.m18.1.1.1.1.1.1.cmml"><mi id="S2.3.p3.18.m18.1.1.1.1.1.1.2" xref="S2.3.p3.18.m18.1.1.1.1.1.1.2.cmml">int</mi><msup id="S2.3.p3.18.m18.1.1.1.1.1.1.3" xref="S2.3.p3.18.m18.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.18.m18.1.1.1.1.1.1.3.2" xref="S2.3.p3.18.m18.1.1.1.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.18.m18.1.1.1.1.1.1.3.3" xref="S2.3.p3.18.m18.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.18.m18.2.2.2.2.2a" xref="S2.3.p3.18.m18.2.2.2.2.3.cmml">⁡</mo><mrow id="S2.3.p3.18.m18.2.2.2.2.2.2" xref="S2.3.p3.18.m18.2.2.2.2.3.cmml"><mo id="S2.3.p3.18.m18.2.2.2.2.2.2.2" stretchy="false" xref="S2.3.p3.18.m18.2.2.2.2.3.cmml">(</mo><msub id="S2.3.p3.18.m18.2.2.2.2.2.2.1" xref="S2.3.p3.18.m18.2.2.2.2.2.2.1.cmml"><mi id="S2.3.p3.18.m18.2.2.2.2.2.2.1.2" xref="S2.3.p3.18.m18.2.2.2.2.2.2.1.2.cmml">x</mi><mn id="S2.3.p3.18.m18.2.2.2.2.2.2.1.3" xref="S2.3.p3.18.m18.2.2.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S2.3.p3.18.m18.2.2.2.2.2.2.3" stretchy="false" xref="S2.3.p3.18.m18.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.3.p3.18.m18.2.2.2.3" xref="S2.3.p3.18.m18.2.2.2.3.cmml">∩</mo><mi id="S2.3.p3.18.m18.2.2.2.4" xref="S2.3.p3.18.m18.2.2.2.4.cmml">X</mi><mo id="S2.3.p3.18.m18.2.2.2.3a" xref="S2.3.p3.18.m18.2.2.2.3.cmml">∩</mo><mi id="S2.3.p3.18.m18.2.2.2.5" xref="S2.3.p3.18.m18.2.2.2.5.cmml">Y</mi></mrow><mo id="S2.3.p3.18.m18.4.4.8" xref="S2.3.p3.18.m18.4.4.8.cmml">⊆</mo><mrow id="S2.3.p3.18.m18.4.4.4" xref="S2.3.p3.18.m18.4.4.4.cmml"><mrow id="S2.3.p3.18.m18.4.4.4.2.2" xref="S2.3.p3.18.m18.4.4.4.2.3.cmml"><msub id="S2.3.p3.18.m18.3.3.3.1.1.1" xref="S2.3.p3.18.m18.3.3.3.1.1.1.cmml"><mi id="S2.3.p3.18.m18.3.3.3.1.1.1.2" xref="S2.3.p3.18.m18.3.3.3.1.1.1.2.cmml">int</mi><msup id="S2.3.p3.18.m18.3.3.3.1.1.1.3" xref="S2.3.p3.18.m18.3.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.18.m18.3.3.3.1.1.1.3.2" xref="S2.3.p3.18.m18.3.3.3.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.18.m18.3.3.3.1.1.1.3.3" xref="S2.3.p3.18.m18.3.3.3.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.18.m18.4.4.4.2.2a" xref="S2.3.p3.18.m18.4.4.4.2.3.cmml">⁡</mo><mrow id="S2.3.p3.18.m18.4.4.4.2.2.2" xref="S2.3.p3.18.m18.4.4.4.2.3.cmml"><mo id="S2.3.p3.18.m18.4.4.4.2.2.2.2" stretchy="false" xref="S2.3.p3.18.m18.4.4.4.2.3.cmml">(</mo><msub id="S2.3.p3.18.m18.4.4.4.2.2.2.1" xref="S2.3.p3.18.m18.4.4.4.2.2.2.1.cmml"><mi id="S2.3.p3.18.m18.4.4.4.2.2.2.1.2" xref="S2.3.p3.18.m18.4.4.4.2.2.2.1.2.cmml">x</mi><mn id="S2.3.p3.18.m18.4.4.4.2.2.2.1.3" xref="S2.3.p3.18.m18.4.4.4.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.3.p3.18.m18.4.4.4.2.2.2.3" stretchy="false" xref="S2.3.p3.18.m18.4.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S2.3.p3.18.m18.4.4.4.3" xref="S2.3.p3.18.m18.4.4.4.3.cmml">∩</mo><mi id="S2.3.p3.18.m18.4.4.4.4" xref="S2.3.p3.18.m18.4.4.4.4.cmml">X</mi><mo id="S2.3.p3.18.m18.4.4.4.3a" xref="S2.3.p3.18.m18.4.4.4.3.cmml">∩</mo><mi id="S2.3.p3.18.m18.4.4.4.5" xref="S2.3.p3.18.m18.4.4.4.5.cmml">Y</mi></mrow><mo id="S2.3.p3.18.m18.4.4.9" xref="S2.3.p3.18.m18.4.4.9.cmml">⊆</mo><msub id="S2.3.p3.18.m18.4.4.10" xref="S2.3.p3.18.m18.4.4.10.cmml"><mi id="S2.3.p3.18.m18.4.4.10.2" xref="S2.3.p3.18.m18.4.4.10.2.cmml">B</mi><msub id="S2.3.p3.18.m18.4.4.10.3" xref="S2.3.p3.18.m18.4.4.10.3.cmml"><mi id="S2.3.p3.18.m18.4.4.10.3.2" xref="S2.3.p3.18.m18.4.4.10.3.2.cmml">x</mi><mn id="S2.3.p3.18.m18.4.4.10.3.3" xref="S2.3.p3.18.m18.4.4.10.3.3.cmml">1</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.18.m18.4b"><apply id="S2.3.p3.18.m18.4.4.cmml" xref="S2.3.p3.18.m18.4.4"><and id="S2.3.p3.18.m18.4.4a.cmml" xref="S2.3.p3.18.m18.4.4"></and><apply id="S2.3.p3.18.m18.4.4b.cmml" xref="S2.3.p3.18.m18.4.4"><in id="S2.3.p3.18.m18.4.4.7.cmml" xref="S2.3.p3.18.m18.4.4.7"></in><ci id="S2.3.p3.18.m18.4.4.6.cmml" xref="S2.3.p3.18.m18.4.4.6">𝑣</ci><apply id="S2.3.p3.18.m18.2.2.2.cmml" xref="S2.3.p3.18.m18.2.2.2"><intersect id="S2.3.p3.18.m18.2.2.2.3.cmml" xref="S2.3.p3.18.m18.2.2.2.3"></intersect><apply id="S2.3.p3.18.m18.2.2.2.2.3.cmml" xref="S2.3.p3.18.m18.2.2.2.2.2"><apply id="S2.3.p3.18.m18.1.1.1.1.1.1.cmml" xref="S2.3.p3.18.m18.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.3.p3.18.m18.1.1.1.1.1.1.1.cmml" xref="S2.3.p3.18.m18.1.1.1.1.1.1">subscript</csymbol><ci id="S2.3.p3.18.m18.1.1.1.1.1.1.2.cmml" xref="S2.3.p3.18.m18.1.1.1.1.1.1.2">int</ci><apply id="S2.3.p3.18.m18.1.1.1.1.1.1.3.cmml" xref="S2.3.p3.18.m18.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.18.m18.1.1.1.1.1.1.3.1.cmml" xref="S2.3.p3.18.m18.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.3.p3.18.m18.1.1.1.1.1.1.3.2.cmml" xref="S2.3.p3.18.m18.1.1.1.1.1.1.3.2">𝒯</ci><ci id="S2.3.p3.18.m18.1.1.1.1.1.1.3.3.cmml" xref="S2.3.p3.18.m18.1.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S2.3.p3.18.m18.2.2.2.2.2.2.1.cmml" xref="S2.3.p3.18.m18.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S2.3.p3.18.m18.2.2.2.2.2.2.1.1.cmml" xref="S2.3.p3.18.m18.2.2.2.2.2.2.1">subscript</csymbol><ci id="S2.3.p3.18.m18.2.2.2.2.2.2.1.2.cmml" xref="S2.3.p3.18.m18.2.2.2.2.2.2.1.2">𝑥</ci><cn id="S2.3.p3.18.m18.2.2.2.2.2.2.1.3.cmml" type="integer" xref="S2.3.p3.18.m18.2.2.2.2.2.2.1.3">0</cn></apply></apply><ci id="S2.3.p3.18.m18.2.2.2.4.cmml" xref="S2.3.p3.18.m18.2.2.2.4">𝑋</ci><ci id="S2.3.p3.18.m18.2.2.2.5.cmml" xref="S2.3.p3.18.m18.2.2.2.5">𝑌</ci></apply></apply><apply id="S2.3.p3.18.m18.4.4c.cmml" xref="S2.3.p3.18.m18.4.4"><subset id="S2.3.p3.18.m18.4.4.8.cmml" xref="S2.3.p3.18.m18.4.4.8"></subset><share href="https://arxiv.org/html/2503.17112v1#S2.3.p3.18.m18.2.2.2.cmml" id="S2.3.p3.18.m18.4.4d.cmml" xref="S2.3.p3.18.m18.4.4"></share><apply id="S2.3.p3.18.m18.4.4.4.cmml" xref="S2.3.p3.18.m18.4.4.4"><intersect id="S2.3.p3.18.m18.4.4.4.3.cmml" xref="S2.3.p3.18.m18.4.4.4.3"></intersect><apply id="S2.3.p3.18.m18.4.4.4.2.3.cmml" xref="S2.3.p3.18.m18.4.4.4.2.2"><apply id="S2.3.p3.18.m18.3.3.3.1.1.1.cmml" xref="S2.3.p3.18.m18.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.3.p3.18.m18.3.3.3.1.1.1.1.cmml" xref="S2.3.p3.18.m18.3.3.3.1.1.1">subscript</csymbol><ci id="S2.3.p3.18.m18.3.3.3.1.1.1.2.cmml" xref="S2.3.p3.18.m18.3.3.3.1.1.1.2">int</ci><apply id="S2.3.p3.18.m18.3.3.3.1.1.1.3.cmml" xref="S2.3.p3.18.m18.3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.18.m18.3.3.3.1.1.1.3.1.cmml" xref="S2.3.p3.18.m18.3.3.3.1.1.1.3">superscript</csymbol><ci id="S2.3.p3.18.m18.3.3.3.1.1.1.3.2.cmml" xref="S2.3.p3.18.m18.3.3.3.1.1.1.3.2">𝒯</ci><ci id="S2.3.p3.18.m18.3.3.3.1.1.1.3.3.cmml" xref="S2.3.p3.18.m18.3.3.3.1.1.1.3.3">′</ci></apply></apply><apply id="S2.3.p3.18.m18.4.4.4.2.2.2.1.cmml" xref="S2.3.p3.18.m18.4.4.4.2.2.2.1"><csymbol cd="ambiguous" id="S2.3.p3.18.m18.4.4.4.2.2.2.1.1.cmml" xref="S2.3.p3.18.m18.4.4.4.2.2.2.1">subscript</csymbol><ci id="S2.3.p3.18.m18.4.4.4.2.2.2.1.2.cmml" xref="S2.3.p3.18.m18.4.4.4.2.2.2.1.2">𝑥</ci><cn id="S2.3.p3.18.m18.4.4.4.2.2.2.1.3.cmml" type="integer" xref="S2.3.p3.18.m18.4.4.4.2.2.2.1.3">1</cn></apply></apply><ci id="S2.3.p3.18.m18.4.4.4.4.cmml" xref="S2.3.p3.18.m18.4.4.4.4">𝑋</ci><ci id="S2.3.p3.18.m18.4.4.4.5.cmml" xref="S2.3.p3.18.m18.4.4.4.5">𝑌</ci></apply></apply><apply id="S2.3.p3.18.m18.4.4e.cmml" xref="S2.3.p3.18.m18.4.4"><subset id="S2.3.p3.18.m18.4.4.9.cmml" xref="S2.3.p3.18.m18.4.4.9"></subset><share href="https://arxiv.org/html/2503.17112v1#S2.3.p3.18.m18.4.4.4.cmml" id="S2.3.p3.18.m18.4.4f.cmml" xref="S2.3.p3.18.m18.4.4"></share><apply id="S2.3.p3.18.m18.4.4.10.cmml" xref="S2.3.p3.18.m18.4.4.10"><csymbol cd="ambiguous" id="S2.3.p3.18.m18.4.4.10.1.cmml" xref="S2.3.p3.18.m18.4.4.10">subscript</csymbol><ci id="S2.3.p3.18.m18.4.4.10.2.cmml" xref="S2.3.p3.18.m18.4.4.10.2">𝐵</ci><apply id="S2.3.p3.18.m18.4.4.10.3.cmml" xref="S2.3.p3.18.m18.4.4.10.3"><csymbol cd="ambiguous" id="S2.3.p3.18.m18.4.4.10.3.1.cmml" xref="S2.3.p3.18.m18.4.4.10.3">subscript</csymbol><ci id="S2.3.p3.18.m18.4.4.10.3.2.cmml" xref="S2.3.p3.18.m18.4.4.10.3.2">𝑥</ci><cn id="S2.3.p3.18.m18.4.4.10.3.3.cmml" type="integer" xref="S2.3.p3.18.m18.4.4.10.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.18.m18.4c">v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{0})\cap X\cap Y\subseteq% \operatorname{int}_{\mathcal{T}^{\prime}}(x_{1})\cap X\cap Y\subseteq B_{x_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.18.m18.4d">italic_v ∈ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∩ italic_X ∩ italic_Y ⊆ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∩ italic_X ∩ italic_Y ⊆ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, a contradiction. Therefore <math alttext="v\in B^{\prime}_{x_{0}}" class="ltx_Math" display="inline" id="S2.3.p3.19.m19.1"><semantics id="S2.3.p3.19.m19.1a"><mrow id="S2.3.p3.19.m19.1.1" xref="S2.3.p3.19.m19.1.1.cmml"><mi id="S2.3.p3.19.m19.1.1.2" xref="S2.3.p3.19.m19.1.1.2.cmml">v</mi><mo id="S2.3.p3.19.m19.1.1.1" xref="S2.3.p3.19.m19.1.1.1.cmml">∈</mo><msubsup id="S2.3.p3.19.m19.1.1.3" xref="S2.3.p3.19.m19.1.1.3.cmml"><mi id="S2.3.p3.19.m19.1.1.3.2.2" xref="S2.3.p3.19.m19.1.1.3.2.2.cmml">B</mi><msub id="S2.3.p3.19.m19.1.1.3.3" xref="S2.3.p3.19.m19.1.1.3.3.cmml"><mi id="S2.3.p3.19.m19.1.1.3.3.2" xref="S2.3.p3.19.m19.1.1.3.3.2.cmml">x</mi><mn id="S2.3.p3.19.m19.1.1.3.3.3" xref="S2.3.p3.19.m19.1.1.3.3.3.cmml">0</mn></msub><mo id="S2.3.p3.19.m19.1.1.3.2.3" xref="S2.3.p3.19.m19.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.19.m19.1b"><apply id="S2.3.p3.19.m19.1.1.cmml" xref="S2.3.p3.19.m19.1.1"><in id="S2.3.p3.19.m19.1.1.1.cmml" xref="S2.3.p3.19.m19.1.1.1"></in><ci id="S2.3.p3.19.m19.1.1.2.cmml" xref="S2.3.p3.19.m19.1.1.2">𝑣</ci><apply id="S2.3.p3.19.m19.1.1.3.cmml" xref="S2.3.p3.19.m19.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.19.m19.1.1.3.1.cmml" xref="S2.3.p3.19.m19.1.1.3">subscript</csymbol><apply id="S2.3.p3.19.m19.1.1.3.2.cmml" xref="S2.3.p3.19.m19.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.19.m19.1.1.3.2.1.cmml" xref="S2.3.p3.19.m19.1.1.3">superscript</csymbol><ci id="S2.3.p3.19.m19.1.1.3.2.2.cmml" xref="S2.3.p3.19.m19.1.1.3.2.2">𝐵</ci><ci id="S2.3.p3.19.m19.1.1.3.2.3.cmml" xref="S2.3.p3.19.m19.1.1.3.2.3">′</ci></apply><apply id="S2.3.p3.19.m19.1.1.3.3.cmml" xref="S2.3.p3.19.m19.1.1.3.3"><csymbol cd="ambiguous" id="S2.3.p3.19.m19.1.1.3.3.1.cmml" xref="S2.3.p3.19.m19.1.1.3.3">subscript</csymbol><ci id="S2.3.p3.19.m19.1.1.3.3.2.cmml" xref="S2.3.p3.19.m19.1.1.3.3.2">𝑥</ci><cn id="S2.3.p3.19.m19.1.1.3.3.3.cmml" type="integer" xref="S2.3.p3.19.m19.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.19.m19.1c">v\in B^{\prime}_{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.19.m19.1d">italic_v ∈ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> since <math alttext="v\in B_{x_{0}}" class="ltx_Math" display="inline" id="S2.3.p3.20.m20.1"><semantics id="S2.3.p3.20.m20.1a"><mrow id="S2.3.p3.20.m20.1.1" xref="S2.3.p3.20.m20.1.1.cmml"><mi id="S2.3.p3.20.m20.1.1.2" xref="S2.3.p3.20.m20.1.1.2.cmml">v</mi><mo id="S2.3.p3.20.m20.1.1.1" xref="S2.3.p3.20.m20.1.1.1.cmml">∈</mo><msub id="S2.3.p3.20.m20.1.1.3" xref="S2.3.p3.20.m20.1.1.3.cmml"><mi id="S2.3.p3.20.m20.1.1.3.2" xref="S2.3.p3.20.m20.1.1.3.2.cmml">B</mi><msub id="S2.3.p3.20.m20.1.1.3.3" xref="S2.3.p3.20.m20.1.1.3.3.cmml"><mi id="S2.3.p3.20.m20.1.1.3.3.2" xref="S2.3.p3.20.m20.1.1.3.3.2.cmml">x</mi><mn id="S2.3.p3.20.m20.1.1.3.3.3" xref="S2.3.p3.20.m20.1.1.3.3.3.cmml">0</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.20.m20.1b"><apply id="S2.3.p3.20.m20.1.1.cmml" xref="S2.3.p3.20.m20.1.1"><in id="S2.3.p3.20.m20.1.1.1.cmml" xref="S2.3.p3.20.m20.1.1.1"></in><ci id="S2.3.p3.20.m20.1.1.2.cmml" xref="S2.3.p3.20.m20.1.1.2">𝑣</ci><apply id="S2.3.p3.20.m20.1.1.3.cmml" xref="S2.3.p3.20.m20.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.20.m20.1.1.3.1.cmml" xref="S2.3.p3.20.m20.1.1.3">subscript</csymbol><ci id="S2.3.p3.20.m20.1.1.3.2.cmml" xref="S2.3.p3.20.m20.1.1.3.2">𝐵</ci><apply id="S2.3.p3.20.m20.1.1.3.3.cmml" xref="S2.3.p3.20.m20.1.1.3.3"><csymbol cd="ambiguous" id="S2.3.p3.20.m20.1.1.3.3.1.cmml" xref="S2.3.p3.20.m20.1.1.3.3">subscript</csymbol><ci id="S2.3.p3.20.m20.1.1.3.3.2.cmml" xref="S2.3.p3.20.m20.1.1.3.3.2">𝑥</ci><cn id="S2.3.p3.20.m20.1.1.3.3.3.cmml" type="integer" xref="S2.3.p3.20.m20.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.20.m20.1c">v\in B_{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.20.m20.1d">italic_v ∈ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Thus <math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.3.p3.21.m21.1"><semantics id="S2.3.p3.21.m21.1a"><msub id="S2.3.p3.21.m21.1.1" xref="S2.3.p3.21.m21.1.1.cmml"><mi id="S2.3.p3.21.m21.1.1.2" xref="S2.3.p3.21.m21.1.1.2.cmml">x</mi><mn id="S2.3.p3.21.m21.1.1.3" xref="S2.3.p3.21.m21.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.21.m21.1b"><apply id="S2.3.p3.21.m21.1.1.cmml" xref="S2.3.p3.21.m21.1.1"><csymbol cd="ambiguous" id="S2.3.p3.21.m21.1.1.1.cmml" xref="S2.3.p3.21.m21.1.1">subscript</csymbol><ci id="S2.3.p3.21.m21.1.1.2.cmml" xref="S2.3.p3.21.m21.1.1.2">𝑥</ci><cn id="S2.3.p3.21.m21.1.1.3.cmml" type="integer" xref="S2.3.p3.21.m21.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.21.m21.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.21.m21.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is the parent of <math alttext="x_{0}" class="ltx_Math" display="inline" id="S2.3.p3.22.m22.1"><semantics id="S2.3.p3.22.m22.1a"><msub id="S2.3.p3.22.m22.1.1" xref="S2.3.p3.22.m22.1.1.cmml"><mi id="S2.3.p3.22.m22.1.1.2" xref="S2.3.p3.22.m22.1.1.2.cmml">x</mi><mn id="S2.3.p3.22.m22.1.1.3" xref="S2.3.p3.22.m22.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.3.p3.22.m22.1b"><apply id="S2.3.p3.22.m22.1.1.cmml" xref="S2.3.p3.22.m22.1.1"><csymbol cd="ambiguous" id="S2.3.p3.22.m22.1.1.1.cmml" xref="S2.3.p3.22.m22.1.1">subscript</csymbol><ci id="S2.3.p3.22.m22.1.1.2.cmml" xref="S2.3.p3.22.m22.1.1.2">𝑥</ci><cn id="S2.3.p3.22.m22.1.1.3.cmml" type="integer" xref="S2.3.p3.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.22.m22.1c">x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.22.m22.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="v\in B^{\prime}_{x_{0}}" class="ltx_Math" display="inline" id="S2.3.p3.23.m23.1"><semantics id="S2.3.p3.23.m23.1a"><mrow id="S2.3.p3.23.m23.1.1" xref="S2.3.p3.23.m23.1.1.cmml"><mi id="S2.3.p3.23.m23.1.1.2" xref="S2.3.p3.23.m23.1.1.2.cmml">v</mi><mo id="S2.3.p3.23.m23.1.1.1" xref="S2.3.p3.23.m23.1.1.1.cmml">∈</mo><msubsup id="S2.3.p3.23.m23.1.1.3" xref="S2.3.p3.23.m23.1.1.3.cmml"><mi id="S2.3.p3.23.m23.1.1.3.2.2" xref="S2.3.p3.23.m23.1.1.3.2.2.cmml">B</mi><msub id="S2.3.p3.23.m23.1.1.3.3" xref="S2.3.p3.23.m23.1.1.3.3.cmml"><mi id="S2.3.p3.23.m23.1.1.3.3.2" xref="S2.3.p3.23.m23.1.1.3.3.2.cmml">x</mi><mn id="S2.3.p3.23.m23.1.1.3.3.3" xref="S2.3.p3.23.m23.1.1.3.3.3.cmml">0</mn></msub><mo id="S2.3.p3.23.m23.1.1.3.2.3" xref="S2.3.p3.23.m23.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.23.m23.1b"><apply id="S2.3.p3.23.m23.1.1.cmml" xref="S2.3.p3.23.m23.1.1"><in id="S2.3.p3.23.m23.1.1.1.cmml" xref="S2.3.p3.23.m23.1.1.1"></in><ci id="S2.3.p3.23.m23.1.1.2.cmml" xref="S2.3.p3.23.m23.1.1.2">𝑣</ci><apply id="S2.3.p3.23.m23.1.1.3.cmml" xref="S2.3.p3.23.m23.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.23.m23.1.1.3.1.cmml" xref="S2.3.p3.23.m23.1.1.3">subscript</csymbol><apply id="S2.3.p3.23.m23.1.1.3.2.cmml" xref="S2.3.p3.23.m23.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.23.m23.1.1.3.2.1.cmml" xref="S2.3.p3.23.m23.1.1.3">superscript</csymbol><ci id="S2.3.p3.23.m23.1.1.3.2.2.cmml" xref="S2.3.p3.23.m23.1.1.3.2.2">𝐵</ci><ci id="S2.3.p3.23.m23.1.1.3.2.3.cmml" xref="S2.3.p3.23.m23.1.1.3.2.3">′</ci></apply><apply id="S2.3.p3.23.m23.1.1.3.3.cmml" xref="S2.3.p3.23.m23.1.1.3.3"><csymbol cd="ambiguous" id="S2.3.p3.23.m23.1.1.3.3.1.cmml" xref="S2.3.p3.23.m23.1.1.3.3">subscript</csymbol><ci id="S2.3.p3.23.m23.1.1.3.3.2.cmml" xref="S2.3.p3.23.m23.1.1.3.3.2">𝑥</ci><cn id="S2.3.p3.23.m23.1.1.3.3.3.cmml" type="integer" xref="S2.3.p3.23.m23.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.23.m23.1c">v\in B^{\prime}_{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.23.m23.1d">italic_v ∈ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="v\not\in B^{\prime}_{x_{1}}" class="ltx_Math" display="inline" id="S2.3.p3.24.m24.1"><semantics id="S2.3.p3.24.m24.1a"><mrow id="S2.3.p3.24.m24.1.1" xref="S2.3.p3.24.m24.1.1.cmml"><mi id="S2.3.p3.24.m24.1.1.2" xref="S2.3.p3.24.m24.1.1.2.cmml">v</mi><mo id="S2.3.p3.24.m24.1.1.1" xref="S2.3.p3.24.m24.1.1.1.cmml">∉</mo><msubsup id="S2.3.p3.24.m24.1.1.3" xref="S2.3.p3.24.m24.1.1.3.cmml"><mi id="S2.3.p3.24.m24.1.1.3.2.2" xref="S2.3.p3.24.m24.1.1.3.2.2.cmml">B</mi><msub id="S2.3.p3.24.m24.1.1.3.3" xref="S2.3.p3.24.m24.1.1.3.3.cmml"><mi id="S2.3.p3.24.m24.1.1.3.3.2" xref="S2.3.p3.24.m24.1.1.3.3.2.cmml">x</mi><mn id="S2.3.p3.24.m24.1.1.3.3.3" xref="S2.3.p3.24.m24.1.1.3.3.3.cmml">1</mn></msub><mo id="S2.3.p3.24.m24.1.1.3.2.3" xref="S2.3.p3.24.m24.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.24.m24.1b"><apply id="S2.3.p3.24.m24.1.1.cmml" xref="S2.3.p3.24.m24.1.1"><notin id="S2.3.p3.24.m24.1.1.1.cmml" xref="S2.3.p3.24.m24.1.1.1"></notin><ci id="S2.3.p3.24.m24.1.1.2.cmml" xref="S2.3.p3.24.m24.1.1.2">𝑣</ci><apply id="S2.3.p3.24.m24.1.1.3.cmml" xref="S2.3.p3.24.m24.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.24.m24.1.1.3.1.cmml" xref="S2.3.p3.24.m24.1.1.3">subscript</csymbol><apply id="S2.3.p3.24.m24.1.1.3.2.cmml" xref="S2.3.p3.24.m24.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.24.m24.1.1.3.2.1.cmml" xref="S2.3.p3.24.m24.1.1.3">superscript</csymbol><ci id="S2.3.p3.24.m24.1.1.3.2.2.cmml" xref="S2.3.p3.24.m24.1.1.3.2.2">𝐵</ci><ci id="S2.3.p3.24.m24.1.1.3.2.3.cmml" xref="S2.3.p3.24.m24.1.1.3.2.3">′</ci></apply><apply id="S2.3.p3.24.m24.1.1.3.3.cmml" xref="S2.3.p3.24.m24.1.1.3.3"><csymbol cd="ambiguous" id="S2.3.p3.24.m24.1.1.3.3.1.cmml" xref="S2.3.p3.24.m24.1.1.3.3">subscript</csymbol><ci id="S2.3.p3.24.m24.1.1.3.3.2.cmml" xref="S2.3.p3.24.m24.1.1.3.3.2">𝑥</ci><cn id="S2.3.p3.24.m24.1.1.3.3.3.cmml" type="integer" xref="S2.3.p3.24.m24.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.24.m24.1c">v\not\in B^{\prime}_{x_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.24.m24.1d">italic_v ∉ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore <math alttext="v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{1})" class="ltx_Math" display="inline" id="S2.3.p3.25.m25.2"><semantics id="S2.3.p3.25.m25.2a"><mrow id="S2.3.p3.25.m25.2.2" xref="S2.3.p3.25.m25.2.2.cmml"><mi id="S2.3.p3.25.m25.2.2.4" xref="S2.3.p3.25.m25.2.2.4.cmml">v</mi><mo id="S2.3.p3.25.m25.2.2.3" xref="S2.3.p3.25.m25.2.2.3.cmml">∈</mo><mrow id="S2.3.p3.25.m25.2.2.2.2" xref="S2.3.p3.25.m25.2.2.2.3.cmml"><msub id="S2.3.p3.25.m25.1.1.1.1.1" xref="S2.3.p3.25.m25.1.1.1.1.1.cmml"><mi id="S2.3.p3.25.m25.1.1.1.1.1.2" xref="S2.3.p3.25.m25.1.1.1.1.1.2.cmml">int</mi><msup id="S2.3.p3.25.m25.1.1.1.1.1.3" xref="S2.3.p3.25.m25.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.25.m25.1.1.1.1.1.3.2" xref="S2.3.p3.25.m25.1.1.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.25.m25.1.1.1.1.1.3.3" xref="S2.3.p3.25.m25.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.25.m25.2.2.2.2a" xref="S2.3.p3.25.m25.2.2.2.3.cmml">⁡</mo><mrow id="S2.3.p3.25.m25.2.2.2.2.2" xref="S2.3.p3.25.m25.2.2.2.3.cmml"><mo id="S2.3.p3.25.m25.2.2.2.2.2.2" stretchy="false" xref="S2.3.p3.25.m25.2.2.2.3.cmml">(</mo><msub id="S2.3.p3.25.m25.2.2.2.2.2.1" xref="S2.3.p3.25.m25.2.2.2.2.2.1.cmml"><mi id="S2.3.p3.25.m25.2.2.2.2.2.1.2" xref="S2.3.p3.25.m25.2.2.2.2.2.1.2.cmml">x</mi><mn id="S2.3.p3.25.m25.2.2.2.2.2.1.3" xref="S2.3.p3.25.m25.2.2.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.3.p3.25.m25.2.2.2.2.2.3" stretchy="false" xref="S2.3.p3.25.m25.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.25.m25.2b"><apply id="S2.3.p3.25.m25.2.2.cmml" xref="S2.3.p3.25.m25.2.2"><in id="S2.3.p3.25.m25.2.2.3.cmml" xref="S2.3.p3.25.m25.2.2.3"></in><ci id="S2.3.p3.25.m25.2.2.4.cmml" xref="S2.3.p3.25.m25.2.2.4">𝑣</ci><apply id="S2.3.p3.25.m25.2.2.2.3.cmml" xref="S2.3.p3.25.m25.2.2.2.2"><apply id="S2.3.p3.25.m25.1.1.1.1.1.cmml" xref="S2.3.p3.25.m25.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.3.p3.25.m25.1.1.1.1.1.1.cmml" xref="S2.3.p3.25.m25.1.1.1.1.1">subscript</csymbol><ci id="S2.3.p3.25.m25.1.1.1.1.1.2.cmml" xref="S2.3.p3.25.m25.1.1.1.1.1.2">int</ci><apply id="S2.3.p3.25.m25.1.1.1.1.1.3.cmml" xref="S2.3.p3.25.m25.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.25.m25.1.1.1.1.1.3.1.cmml" xref="S2.3.p3.25.m25.1.1.1.1.1.3">superscript</csymbol><ci id="S2.3.p3.25.m25.1.1.1.1.1.3.2.cmml" xref="S2.3.p3.25.m25.1.1.1.1.1.3.2">𝒯</ci><ci id="S2.3.p3.25.m25.1.1.1.1.1.3.3.cmml" xref="S2.3.p3.25.m25.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S2.3.p3.25.m25.2.2.2.2.2.1.cmml" xref="S2.3.p3.25.m25.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S2.3.p3.25.m25.2.2.2.2.2.1.1.cmml" xref="S2.3.p3.25.m25.2.2.2.2.2.1">subscript</csymbol><ci id="S2.3.p3.25.m25.2.2.2.2.2.1.2.cmml" xref="S2.3.p3.25.m25.2.2.2.2.2.1.2">𝑥</ci><cn id="S2.3.p3.25.m25.2.2.2.2.2.1.3.cmml" type="integer" xref="S2.3.p3.25.m25.2.2.2.2.2.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.25.m25.2c">v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.25.m25.2d">italic_v ∈ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, so <math alttext="v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{1})\cap X\cap Y\subseteq B_{x% _{1}}" class="ltx_Math" display="inline" id="S2.3.p3.26.m26.2"><semantics id="S2.3.p3.26.m26.2a"><mrow id="S2.3.p3.26.m26.2.2" xref="S2.3.p3.26.m26.2.2.cmml"><mi id="S2.3.p3.26.m26.2.2.4" xref="S2.3.p3.26.m26.2.2.4.cmml">v</mi><mo id="S2.3.p3.26.m26.2.2.5" xref="S2.3.p3.26.m26.2.2.5.cmml">∈</mo><mrow id="S2.3.p3.26.m26.2.2.2" xref="S2.3.p3.26.m26.2.2.2.cmml"><mrow id="S2.3.p3.26.m26.2.2.2.2.2" xref="S2.3.p3.26.m26.2.2.2.2.3.cmml"><msub id="S2.3.p3.26.m26.1.1.1.1.1.1" xref="S2.3.p3.26.m26.1.1.1.1.1.1.cmml"><mi id="S2.3.p3.26.m26.1.1.1.1.1.1.2" xref="S2.3.p3.26.m26.1.1.1.1.1.1.2.cmml">int</mi><msup id="S2.3.p3.26.m26.1.1.1.1.1.1.3" xref="S2.3.p3.26.m26.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p3.26.m26.1.1.1.1.1.1.3.2" xref="S2.3.p3.26.m26.1.1.1.1.1.1.3.2.cmml">𝒯</mi><mo id="S2.3.p3.26.m26.1.1.1.1.1.1.3.3" xref="S2.3.p3.26.m26.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S2.3.p3.26.m26.2.2.2.2.2a" xref="S2.3.p3.26.m26.2.2.2.2.3.cmml">⁡</mo><mrow id="S2.3.p3.26.m26.2.2.2.2.2.2" xref="S2.3.p3.26.m26.2.2.2.2.3.cmml"><mo id="S2.3.p3.26.m26.2.2.2.2.2.2.2" stretchy="false" xref="S2.3.p3.26.m26.2.2.2.2.3.cmml">(</mo><msub id="S2.3.p3.26.m26.2.2.2.2.2.2.1" xref="S2.3.p3.26.m26.2.2.2.2.2.2.1.cmml"><mi id="S2.3.p3.26.m26.2.2.2.2.2.2.1.2" xref="S2.3.p3.26.m26.2.2.2.2.2.2.1.2.cmml">x</mi><mn id="S2.3.p3.26.m26.2.2.2.2.2.2.1.3" xref="S2.3.p3.26.m26.2.2.2.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.3.p3.26.m26.2.2.2.2.2.2.3" stretchy="false" xref="S2.3.p3.26.m26.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.3.p3.26.m26.2.2.2.3" xref="S2.3.p3.26.m26.2.2.2.3.cmml">∩</mo><mi id="S2.3.p3.26.m26.2.2.2.4" xref="S2.3.p3.26.m26.2.2.2.4.cmml">X</mi><mo id="S2.3.p3.26.m26.2.2.2.3a" xref="S2.3.p3.26.m26.2.2.2.3.cmml">∩</mo><mi id="S2.3.p3.26.m26.2.2.2.5" xref="S2.3.p3.26.m26.2.2.2.5.cmml">Y</mi></mrow><mo id="S2.3.p3.26.m26.2.2.6" xref="S2.3.p3.26.m26.2.2.6.cmml">⊆</mo><msub id="S2.3.p3.26.m26.2.2.7" xref="S2.3.p3.26.m26.2.2.7.cmml"><mi id="S2.3.p3.26.m26.2.2.7.2" xref="S2.3.p3.26.m26.2.2.7.2.cmml">B</mi><msub id="S2.3.p3.26.m26.2.2.7.3" xref="S2.3.p3.26.m26.2.2.7.3.cmml"><mi id="S2.3.p3.26.m26.2.2.7.3.2" xref="S2.3.p3.26.m26.2.2.7.3.2.cmml">x</mi><mn id="S2.3.p3.26.m26.2.2.7.3.3" xref="S2.3.p3.26.m26.2.2.7.3.3.cmml">1</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p3.26.m26.2b"><apply id="S2.3.p3.26.m26.2.2.cmml" xref="S2.3.p3.26.m26.2.2"><and id="S2.3.p3.26.m26.2.2a.cmml" xref="S2.3.p3.26.m26.2.2"></and><apply id="S2.3.p3.26.m26.2.2b.cmml" xref="S2.3.p3.26.m26.2.2"><in id="S2.3.p3.26.m26.2.2.5.cmml" xref="S2.3.p3.26.m26.2.2.5"></in><ci id="S2.3.p3.26.m26.2.2.4.cmml" xref="S2.3.p3.26.m26.2.2.4">𝑣</ci><apply id="S2.3.p3.26.m26.2.2.2.cmml" xref="S2.3.p3.26.m26.2.2.2"><intersect id="S2.3.p3.26.m26.2.2.2.3.cmml" xref="S2.3.p3.26.m26.2.2.2.3"></intersect><apply id="S2.3.p3.26.m26.2.2.2.2.3.cmml" xref="S2.3.p3.26.m26.2.2.2.2.2"><apply id="S2.3.p3.26.m26.1.1.1.1.1.1.cmml" xref="S2.3.p3.26.m26.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.3.p3.26.m26.1.1.1.1.1.1.1.cmml" xref="S2.3.p3.26.m26.1.1.1.1.1.1">subscript</csymbol><ci id="S2.3.p3.26.m26.1.1.1.1.1.1.2.cmml" xref="S2.3.p3.26.m26.1.1.1.1.1.1.2">int</ci><apply id="S2.3.p3.26.m26.1.1.1.1.1.1.3.cmml" xref="S2.3.p3.26.m26.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.3.p3.26.m26.1.1.1.1.1.1.3.1.cmml" xref="S2.3.p3.26.m26.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.3.p3.26.m26.1.1.1.1.1.1.3.2.cmml" xref="S2.3.p3.26.m26.1.1.1.1.1.1.3.2">𝒯</ci><ci id="S2.3.p3.26.m26.1.1.1.1.1.1.3.3.cmml" xref="S2.3.p3.26.m26.1.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S2.3.p3.26.m26.2.2.2.2.2.2.1.cmml" xref="S2.3.p3.26.m26.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S2.3.p3.26.m26.2.2.2.2.2.2.1.1.cmml" xref="S2.3.p3.26.m26.2.2.2.2.2.2.1">subscript</csymbol><ci id="S2.3.p3.26.m26.2.2.2.2.2.2.1.2.cmml" xref="S2.3.p3.26.m26.2.2.2.2.2.2.1.2">𝑥</ci><cn id="S2.3.p3.26.m26.2.2.2.2.2.2.1.3.cmml" type="integer" xref="S2.3.p3.26.m26.2.2.2.2.2.2.1.3">1</cn></apply></apply><ci id="S2.3.p3.26.m26.2.2.2.4.cmml" xref="S2.3.p3.26.m26.2.2.2.4">𝑋</ci><ci id="S2.3.p3.26.m26.2.2.2.5.cmml" xref="S2.3.p3.26.m26.2.2.2.5">𝑌</ci></apply></apply><apply id="S2.3.p3.26.m26.2.2c.cmml" xref="S2.3.p3.26.m26.2.2"><subset id="S2.3.p3.26.m26.2.2.6.cmml" xref="S2.3.p3.26.m26.2.2.6"></subset><share href="https://arxiv.org/html/2503.17112v1#S2.3.p3.26.m26.2.2.2.cmml" id="S2.3.p3.26.m26.2.2d.cmml" xref="S2.3.p3.26.m26.2.2"></share><apply id="S2.3.p3.26.m26.2.2.7.cmml" xref="S2.3.p3.26.m26.2.2.7"><csymbol cd="ambiguous" id="S2.3.p3.26.m26.2.2.7.1.cmml" xref="S2.3.p3.26.m26.2.2.7">subscript</csymbol><ci id="S2.3.p3.26.m26.2.2.7.2.cmml" xref="S2.3.p3.26.m26.2.2.7.2">𝐵</ci><apply id="S2.3.p3.26.m26.2.2.7.3.cmml" xref="S2.3.p3.26.m26.2.2.7.3"><csymbol cd="ambiguous" id="S2.3.p3.26.m26.2.2.7.3.1.cmml" xref="S2.3.p3.26.m26.2.2.7.3">subscript</csymbol><ci id="S2.3.p3.26.m26.2.2.7.3.2.cmml" xref="S2.3.p3.26.m26.2.2.7.3.2">𝑥</ci><cn id="S2.3.p3.26.m26.2.2.7.3.3.cmml" type="integer" xref="S2.3.p3.26.m26.2.2.7.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p3.26.m26.2c">v\in\operatorname{int}_{\mathcal{T}^{\prime}}(x_{1})\cap X\cap Y\subseteq B_{x% _{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p3.26.m26.2d">italic_v ∈ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∩ italic_X ∩ italic_Y ⊆ italic_B start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, also a contradiction. ∎</p> </div> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.1">The following construction of a tree decomposition using balanced separations (or variants of this construction using balanced separators) is fairly standard.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="Thmthm5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm5.1.1.1">Lemma 5</span></span><span class="ltx_text ltx_font_bold" id="Thmthm5.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm5.p1"> <p class="ltx_p" id="Thmthm5.p1.6"><span class="ltx_text ltx_font_italic" id="Thmthm5.p1.6.6">Let <math alttext="G" class="ltx_Math" display="inline" id="Thmthm5.p1.1.1.m1.1"><semantics id="Thmthm5.p1.1.1.m1.1a"><mi id="Thmthm5.p1.1.1.m1.1.1" xref="Thmthm5.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm5.p1.1.1.m1.1b"><ci id="Thmthm5.p1.1.1.m1.1.1.cmml" xref="Thmthm5.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm5.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm5.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be an <math alttext="n" class="ltx_Math" display="inline" id="Thmthm5.p1.2.2.m2.1"><semantics id="Thmthm5.p1.2.2.m2.1a"><mi id="Thmthm5.p1.2.2.m2.1.1" xref="Thmthm5.p1.2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="Thmthm5.p1.2.2.m2.1b"><ci id="Thmthm5.p1.2.2.m2.1.1.cmml" xref="Thmthm5.p1.2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm5.p1.2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="Thmthm5.p1.2.2.m2.1d">italic_n</annotation></semantics></math>-vertex graph with <math alttext="\operatorname{sn}(G)\leq a" class="ltx_Math" display="inline" id="Thmthm5.p1.3.3.m3.2"><semantics id="Thmthm5.p1.3.3.m3.2a"><mrow id="Thmthm5.p1.3.3.m3.2.3" xref="Thmthm5.p1.3.3.m3.2.3.cmml"><mrow id="Thmthm5.p1.3.3.m3.2.3.2.2" xref="Thmthm5.p1.3.3.m3.2.3.2.1.cmml"><mi id="Thmthm5.p1.3.3.m3.1.1" xref="Thmthm5.p1.3.3.m3.1.1.cmml">sn</mi><mo id="Thmthm5.p1.3.3.m3.2.3.2.2a" xref="Thmthm5.p1.3.3.m3.2.3.2.1.cmml">⁡</mo><mrow id="Thmthm5.p1.3.3.m3.2.3.2.2.1" xref="Thmthm5.p1.3.3.m3.2.3.2.1.cmml"><mo id="Thmthm5.p1.3.3.m3.2.3.2.2.1.1" stretchy="false" xref="Thmthm5.p1.3.3.m3.2.3.2.1.cmml">(</mo><mi id="Thmthm5.p1.3.3.m3.2.2" xref="Thmthm5.p1.3.3.m3.2.2.cmml">G</mi><mo id="Thmthm5.p1.3.3.m3.2.3.2.2.1.2" stretchy="false" xref="Thmthm5.p1.3.3.m3.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="Thmthm5.p1.3.3.m3.2.3.1" xref="Thmthm5.p1.3.3.m3.2.3.1.cmml">≤</mo><mi id="Thmthm5.p1.3.3.m3.2.3.3" xref="Thmthm5.p1.3.3.m3.2.3.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmthm5.p1.3.3.m3.2b"><apply id="Thmthm5.p1.3.3.m3.2.3.cmml" xref="Thmthm5.p1.3.3.m3.2.3"><leq id="Thmthm5.p1.3.3.m3.2.3.1.cmml" xref="Thmthm5.p1.3.3.m3.2.3.1"></leq><apply id="Thmthm5.p1.3.3.m3.2.3.2.1.cmml" xref="Thmthm5.p1.3.3.m3.2.3.2.2"><ci id="Thmthm5.p1.3.3.m3.1.1.cmml" xref="Thmthm5.p1.3.3.m3.1.1">sn</ci><ci id="Thmthm5.p1.3.3.m3.2.2.cmml" xref="Thmthm5.p1.3.3.m3.2.2">𝐺</ci></apply><ci id="Thmthm5.p1.3.3.m3.2.3.3.cmml" xref="Thmthm5.p1.3.3.m3.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm5.p1.3.3.m3.2c">\operatorname{sn}(G)\leq a</annotation><annotation encoding="application/x-llamapun" id="Thmthm5.p1.3.3.m3.2d">roman_sn ( italic_G ) ≤ italic_a</annotation></semantics></math>. Then, for every integer <math alttext="h\geq 0" class="ltx_Math" display="inline" id="Thmthm5.p1.4.4.m4.1"><semantics id="Thmthm5.p1.4.4.m4.1a"><mrow id="Thmthm5.p1.4.4.m4.1.1" xref="Thmthm5.p1.4.4.m4.1.1.cmml"><mi id="Thmthm5.p1.4.4.m4.1.1.2" xref="Thmthm5.p1.4.4.m4.1.1.2.cmml">h</mi><mo id="Thmthm5.p1.4.4.m4.1.1.1" xref="Thmthm5.p1.4.4.m4.1.1.1.cmml">≥</mo><mn id="Thmthm5.p1.4.4.m4.1.1.3" xref="Thmthm5.p1.4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmthm5.p1.4.4.m4.1b"><apply id="Thmthm5.p1.4.4.m4.1.1.cmml" xref="Thmthm5.p1.4.4.m4.1.1"><geq id="Thmthm5.p1.4.4.m4.1.1.1.cmml" xref="Thmthm5.p1.4.4.m4.1.1.1"></geq><ci id="Thmthm5.p1.4.4.m4.1.1.2.cmml" xref="Thmthm5.p1.4.4.m4.1.1.2">ℎ</ci><cn id="Thmthm5.p1.4.4.m4.1.1.3.cmml" type="integer" xref="Thmthm5.p1.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm5.p1.4.4.m4.1c">h\geq 0</annotation><annotation encoding="application/x-llamapun" id="Thmthm5.p1.4.4.m4.1d">italic_h ≥ 0</annotation></semantics></math>, <math alttext="G" class="ltx_Math" display="inline" id="Thmthm5.p1.5.5.m5.1"><semantics id="Thmthm5.p1.5.5.m5.1a"><mi id="Thmthm5.p1.5.5.m5.1.1" xref="Thmthm5.p1.5.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm5.p1.5.5.m5.1b"><ci id="Thmthm5.p1.5.5.m5.1.1.cmml" xref="Thmthm5.p1.5.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm5.p1.5.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm5.p1.5.5.m5.1d">italic_G</annotation></semantics></math> has a rooted tree decomposition <math alttext="\mathcal{T}:=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="Thmthm5.p1.6.6.m6.1"><semantics id="Thmthm5.p1.6.6.m6.1a"><mrow id="Thmthm5.p1.6.6.m6.1b"><mi class="ltx_font_mathcaligraphic" id="Thmthm5.p1.6.6.m6.1.1">𝒯</mi><mo id="Thmthm5.p1.6.6.m6.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="Thmthm5.p1.6.6.m6.1.3"><mo id="Thmthm5.p1.6.6.m6.1.3.1" stretchy="false">(</mo><msub id="Thmthm5.p1.6.6.m6.1.3.2"><mi id="Thmthm5.p1.6.6.m6.1.3.2.2">B</mi><mi id="Thmthm5.p1.6.6.m6.1.3.2.3">x</mi></msub><mo id="Thmthm5.p1.6.6.m6.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="Thmthm5.p1.6.6.m6.1.3.4">x</mi><mo id="Thmthm5.p1.6.6.m6.1.3.5">∈</mo><mi id="Thmthm5.p1.6.6.m6.1.3.6">V</mi><mrow id="Thmthm5.p1.6.6.m6.1.3.7"><mo id="Thmthm5.p1.6.6.m6.1.3.7.1" stretchy="false">(</mo><mi id="Thmthm5.p1.6.6.m6.1.3.7.2">T</mi><mo id="Thmthm5.p1.6.6.m6.1.3.7.3" stretchy="false">)</mo></mrow><mo id="Thmthm5.p1.6.6.m6.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmthm5.p1.6.6.m6.1c">\mathcal{T}:=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="Thmthm5.p1.6.6.m6.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> such that</span></p> <ol class="ltx_enumerate" id="S2.I2"> <li class="ltx_item" id="S2.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S2.I2.i1.p1"> <p class="ltx_p" id="S2.I2.i1.p1.1"><math alttext="\operatorname{height}(T)\leq h" class="ltx_Math" display="inline" id="S2.I2.i1.p1.1.m1.2"><semantics id="S2.I2.i1.p1.1.m1.2a"><mrow id="S2.I2.i1.p1.1.m1.2.3" xref="S2.I2.i1.p1.1.m1.2.3.cmml"><mrow id="S2.I2.i1.p1.1.m1.2.3.2.2" xref="S2.I2.i1.p1.1.m1.2.3.2.1.cmml"><mi id="S2.I2.i1.p1.1.m1.1.1" xref="S2.I2.i1.p1.1.m1.1.1.cmml">height</mi><mo id="S2.I2.i1.p1.1.m1.2.3.2.2a" xref="S2.I2.i1.p1.1.m1.2.3.2.1.cmml">⁡</mo><mrow id="S2.I2.i1.p1.1.m1.2.3.2.2.1" xref="S2.I2.i1.p1.1.m1.2.3.2.1.cmml"><mo id="S2.I2.i1.p1.1.m1.2.3.2.2.1.1" stretchy="false" xref="S2.I2.i1.p1.1.m1.2.3.2.1.cmml">(</mo><mi id="S2.I2.i1.p1.1.m1.2.2" xref="S2.I2.i1.p1.1.m1.2.2.cmml">T</mi><mo id="S2.I2.i1.p1.1.m1.2.3.2.2.1.2" stretchy="false" xref="S2.I2.i1.p1.1.m1.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S2.I2.i1.p1.1.m1.2.3.1" xref="S2.I2.i1.p1.1.m1.2.3.1.cmml">≤</mo><mi id="S2.I2.i1.p1.1.m1.2.3.3" xref="S2.I2.i1.p1.1.m1.2.3.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i1.p1.1.m1.2b"><apply id="S2.I2.i1.p1.1.m1.2.3.cmml" xref="S2.I2.i1.p1.1.m1.2.3"><leq id="S2.I2.i1.p1.1.m1.2.3.1.cmml" xref="S2.I2.i1.p1.1.m1.2.3.1"></leq><apply id="S2.I2.i1.p1.1.m1.2.3.2.1.cmml" xref="S2.I2.i1.p1.1.m1.2.3.2.2"><ci id="S2.I2.i1.p1.1.m1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1">height</ci><ci id="S2.I2.i1.p1.1.m1.2.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2">𝑇</ci></apply><ci id="S2.I2.i1.p1.1.m1.2.3.3.cmml" xref="S2.I2.i1.p1.1.m1.2.3.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i1.p1.1.m1.2c">\operatorname{height}(T)\leq h</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i1.p1.1.m1.2d">roman_height ( italic_T ) ≤ italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i1.p1.1.1">;</span></p> </div> </li> <li class="ltx_item" id="S2.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S2.I2.i2.p1"> <p class="ltx_p" id="S2.I2.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.3.1">for each </span><math alttext="x\in V(T)" class="ltx_Math" display="inline" id="S2.I2.i2.p1.1.m1.1"><semantics id="S2.I2.i2.p1.1.m1.1a"><mrow id="S2.I2.i2.p1.1.m1.1.2" xref="S2.I2.i2.p1.1.m1.1.2.cmml"><mi id="S2.I2.i2.p1.1.m1.1.2.2" xref="S2.I2.i2.p1.1.m1.1.2.2.cmml">x</mi><mo id="S2.I2.i2.p1.1.m1.1.2.1" xref="S2.I2.i2.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S2.I2.i2.p1.1.m1.1.2.3" xref="S2.I2.i2.p1.1.m1.1.2.3.cmml"><mi id="S2.I2.i2.p1.1.m1.1.2.3.2" xref="S2.I2.i2.p1.1.m1.1.2.3.2.cmml">V</mi><mo id="S2.I2.i2.p1.1.m1.1.2.3.1" xref="S2.I2.i2.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.I2.i2.p1.1.m1.1.2.3.3.2" xref="S2.I2.i2.p1.1.m1.1.2.3.cmml"><mo 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encoding="application/x-llamapun" id="S2.I2.i2.p1.1.m1.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.3.2">, </span><math alttext="|\operatorname{int}_{\mathcal{T}}(x)|\leq n\cdot(\tfrac{2}{3})^{\operatorname{% depth}_{T}(x)}" class="ltx_Math" display="inline" id="S2.I2.i2.p1.2.m2.5"><semantics id="S2.I2.i2.p1.2.m2.5a"><mrow id="S2.I2.i2.p1.2.m2.5.5" xref="S2.I2.i2.p1.2.m2.5.5.cmml"><mrow id="S2.I2.i2.p1.2.m2.5.5.1.1" xref="S2.I2.i2.p1.2.m2.5.5.1.2.cmml"><mo id="S2.I2.i2.p1.2.m2.5.5.1.1.2" stretchy="false" xref="S2.I2.i2.p1.2.m2.5.5.1.2.1.cmml">|</mo><mrow id="S2.I2.i2.p1.2.m2.5.5.1.1.1.1" xref="S2.I2.i2.p1.2.m2.5.5.1.1.1.2.cmml"><msub id="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1" xref="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1.cmml"><mi id="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1.2" xref="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1.3" 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xref="S2.I2.i2.p1.2.m2.5.5.3.3.cmml"><mrow id="S2.I2.i2.p1.2.m2.5.5.3.3.2.2" xref="S2.I2.i2.p1.2.m2.4.4.cmml"><mo id="S2.I2.i2.p1.2.m2.5.5.3.3.2.2.1" stretchy="false" xref="S2.I2.i2.p1.2.m2.4.4.cmml">(</mo><mfrac id="S2.I2.i2.p1.2.m2.4.4" xref="S2.I2.i2.p1.2.m2.4.4.cmml"><mn id="S2.I2.i2.p1.2.m2.4.4.2" xref="S2.I2.i2.p1.2.m2.4.4.2.cmml">2</mn><mn id="S2.I2.i2.p1.2.m2.4.4.3" xref="S2.I2.i2.p1.2.m2.4.4.3.cmml">3</mn></mfrac><mo id="S2.I2.i2.p1.2.m2.5.5.3.3.2.2.2" stretchy="false" xref="S2.I2.i2.p1.2.m2.4.4.cmml">)</mo></mrow><mrow id="S2.I2.i2.p1.2.m2.2.2.2.2" xref="S2.I2.i2.p1.2.m2.2.2.2.3.cmml"><msub id="S2.I2.i2.p1.2.m2.2.2.2.2.1" xref="S2.I2.i2.p1.2.m2.2.2.2.2.1.cmml"><mi id="S2.I2.i2.p1.2.m2.2.2.2.2.1.2" xref="S2.I2.i2.p1.2.m2.2.2.2.2.1.2.cmml">depth</mi><mi id="S2.I2.i2.p1.2.m2.2.2.2.2.1.3" xref="S2.I2.i2.p1.2.m2.2.2.2.2.1.3.cmml">T</mi></msub><mo id="S2.I2.i2.p1.2.m2.2.2.2.2a" xref="S2.I2.i2.p1.2.m2.2.2.2.3.cmml">⁡</mo><mrow id="S2.I2.i2.p1.2.m2.2.2.2.2.2" 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xref="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1">subscript</csymbol><ci id="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1.2.cmml" xref="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1.2">int</ci><ci id="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1.3.cmml" xref="S2.I2.i2.p1.2.m2.5.5.1.1.1.1.1.3">𝒯</ci></apply><ci id="S2.I2.i2.p1.2.m2.3.3.cmml" xref="S2.I2.i2.p1.2.m2.3.3">𝑥</ci></apply></apply><apply id="S2.I2.i2.p1.2.m2.5.5.3.cmml" xref="S2.I2.i2.p1.2.m2.5.5.3"><ci id="S2.I2.i2.p1.2.m2.5.5.3.1.cmml" xref="S2.I2.i2.p1.2.m2.5.5.3.1">⋅</ci><ci id="S2.I2.i2.p1.2.m2.5.5.3.2.cmml" xref="S2.I2.i2.p1.2.m2.5.5.3.2">𝑛</ci><apply id="S2.I2.i2.p1.2.m2.5.5.3.3.cmml" xref="S2.I2.i2.p1.2.m2.5.5.3.3"><csymbol cd="ambiguous" id="S2.I2.i2.p1.2.m2.5.5.3.3.1.cmml" xref="S2.I2.i2.p1.2.m2.5.5.3.3">superscript</csymbol><apply id="S2.I2.i2.p1.2.m2.4.4.cmml" xref="S2.I2.i2.p1.2.m2.5.5.3.3.2.2"><divide id="S2.I2.i2.p1.2.m2.4.4.1.cmml" xref="S2.I2.i2.p1.2.m2.5.5.3.3.2.2"></divide><cn id="S2.I2.i2.p1.2.m2.4.4.2.cmml" type="integer" 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end_POSTSUBSCRIPT ( italic_x ) | ≤ italic_n ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT roman_depth start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x ) end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.3.3"> and </span><math alttext="|\operatorname{\partial}(x)|\leq\operatorname{depth}_{T}(x)\cdot a" class="ltx_Math" display="inline" id="S2.I2.i2.p1.3.m3.5"><semantics id="S2.I2.i2.p1.3.m3.5a"><mrow id="S2.I2.i2.p1.3.m3.5.5" xref="S2.I2.i2.p1.3.m3.5.5.cmml"><mrow id="S2.I2.i2.p1.3.m3.4.4.1.1" xref="S2.I2.i2.p1.3.m3.4.4.1.2.cmml"><mo id="S2.I2.i2.p1.3.m3.4.4.1.1.2" stretchy="false" xref="S2.I2.i2.p1.3.m3.4.4.1.2.1.cmml">|</mo><mrow id="S2.I2.i2.p1.3.m3.4.4.1.1.1.2" xref="S2.I2.i2.p1.3.m3.4.4.1.1.1.1.cmml"><mi id="S2.I2.i2.p1.3.m3.1.1" mathvariant="normal" xref="S2.I2.i2.p1.3.m3.1.1.cmml">∂</mi><mo id="S2.I2.i2.p1.3.m3.4.4.1.1.1.2a" xref="S2.I2.i2.p1.3.m3.4.4.1.1.1.1.cmml">⁡</mo><mrow 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xref="S2.I2.i2.p1.3.m3.4.4.1.1.2"></abs><apply id="S2.I2.i2.p1.3.m3.4.4.1.1.1.1.cmml" xref="S2.I2.i2.p1.3.m3.4.4.1.1.1.2"><partialdiff id="S2.I2.i2.p1.3.m3.1.1.cmml" xref="S2.I2.i2.p1.3.m3.1.1"></partialdiff><ci id="S2.I2.i2.p1.3.m3.2.2.cmml" xref="S2.I2.i2.p1.3.m3.2.2">𝑥</ci></apply></apply><apply id="S2.I2.i2.p1.3.m3.5.5.2.cmml" xref="S2.I2.i2.p1.3.m3.5.5.2"><ci id="S2.I2.i2.p1.3.m3.5.5.2.2.cmml" xref="S2.I2.i2.p1.3.m3.5.5.2.2">⋅</ci><apply id="S2.I2.i2.p1.3.m3.5.5.2.1.2.cmml" xref="S2.I2.i2.p1.3.m3.5.5.2.1.1"><apply id="S2.I2.i2.p1.3.m3.5.5.2.1.1.1.cmml" xref="S2.I2.i2.p1.3.m3.5.5.2.1.1.1"><csymbol cd="ambiguous" id="S2.I2.i2.p1.3.m3.5.5.2.1.1.1.1.cmml" xref="S2.I2.i2.p1.3.m3.5.5.2.1.1.1">subscript</csymbol><ci id="S2.I2.i2.p1.3.m3.5.5.2.1.1.1.2.cmml" xref="S2.I2.i2.p1.3.m3.5.5.2.1.1.1.2">depth</ci><ci id="S2.I2.i2.p1.3.m3.5.5.2.1.1.1.3.cmml" xref="S2.I2.i2.p1.3.m3.5.5.2.1.1.1.3">𝑇</ci></apply><ci id="S2.I2.i2.p1.3.m3.3.3.cmml" xref="S2.I2.i2.p1.3.m3.3.3">𝑥</ci></apply><ci id="S2.I2.i2.p1.3.m3.5.5.2.3.cmml" xref="S2.I2.i2.p1.3.m3.5.5.2.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i2.p1.3.m3.5c">|\operatorname{\partial}(x)|\leq\operatorname{depth}_{T}(x)\cdot a</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i2.p1.3.m3.5d">| ∂ ( italic_x ) | ≤ roman_depth start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x ) ⋅ italic_a</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.3.4">;</span></p> </div> </li> <li class="ltx_item" id="S2.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S2.I2.i3.p1"> <p class="ltx_p" id="S2.I2.i3.p1.3"><span class="ltx_text ltx_font_italic" id="S2.I2.i3.p1.3.1">for each leaf </span><math alttext="y" class="ltx_Math" display="inline" id="S2.I2.i3.p1.1.m1.1"><semantics id="S2.I2.i3.p1.1.m1.1a"><mi id="S2.I2.i3.p1.1.m1.1.1" xref="S2.I2.i3.p1.1.m1.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.1.m1.1b"><ci id="S2.I2.i3.p1.1.m1.1.1.cmml" xref="S2.I2.i3.p1.1.m1.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.1.m1.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.1.m1.1d">italic_y</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i3.p1.3.2"> of </span><math alttext="T" class="ltx_Math" display="inline" id="S2.I2.i3.p1.2.m2.1"><semantics id="S2.I2.i3.p1.2.m2.1a"><mi id="S2.I2.i3.p1.2.m2.1.1" xref="S2.I2.i3.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.2.m2.1b"><ci id="S2.I2.i3.p1.2.m2.1.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.2.m2.1d">italic_T</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i3.p1.3.3">, </span><math alttext="|\operatorname{int}_{\mathcal{T}}(y)|\leq n\cdot(\tfrac{2}{3})^{h}" class="ltx_Math" display="inline" id="S2.I2.i3.p1.3.m3.3"><semantics id="S2.I2.i3.p1.3.m3.3a"><mrow id="S2.I2.i3.p1.3.m3.3.3" xref="S2.I2.i3.p1.3.m3.3.3.cmml"><mrow id="S2.I2.i3.p1.3.m3.3.3.1.1" xref="S2.I2.i3.p1.3.m3.3.3.1.2.cmml"><mo id="S2.I2.i3.p1.3.m3.3.3.1.1.2" stretchy="false" xref="S2.I2.i3.p1.3.m3.3.3.1.2.1.cmml">|</mo><mrow id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.2.cmml"><msub id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.cmml"><mi id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.2" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.3" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1a" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.2.cmml">⁡</mo><mrow id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.2" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.2.cmml"><mo id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.2.1" stretchy="false" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.2.cmml">(</mo><mi id="S2.I2.i3.p1.3.m3.1.1" xref="S2.I2.i3.p1.3.m3.1.1.cmml">y</mi><mo id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.2.2" stretchy="false" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.I2.i3.p1.3.m3.3.3.1.1.3" stretchy="false" xref="S2.I2.i3.p1.3.m3.3.3.1.2.1.cmml">|</mo></mrow><mo id="S2.I2.i3.p1.3.m3.3.3.2" xref="S2.I2.i3.p1.3.m3.3.3.2.cmml">≤</mo><mrow id="S2.I2.i3.p1.3.m3.3.3.3" xref="S2.I2.i3.p1.3.m3.3.3.3.cmml"><mi id="S2.I2.i3.p1.3.m3.3.3.3.2" xref="S2.I2.i3.p1.3.m3.3.3.3.2.cmml">n</mi><mo id="S2.I2.i3.p1.3.m3.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.I2.i3.p1.3.m3.3.3.3.1.cmml">⋅</mo><msup id="S2.I2.i3.p1.3.m3.3.3.3.3" xref="S2.I2.i3.p1.3.m3.3.3.3.3.cmml"><mrow id="S2.I2.i3.p1.3.m3.3.3.3.3.2.2" xref="S2.I2.i3.p1.3.m3.2.2.cmml"><mo id="S2.I2.i3.p1.3.m3.3.3.3.3.2.2.1" stretchy="false" xref="S2.I2.i3.p1.3.m3.2.2.cmml">(</mo><mfrac id="S2.I2.i3.p1.3.m3.2.2" xref="S2.I2.i3.p1.3.m3.2.2.cmml"><mn id="S2.I2.i3.p1.3.m3.2.2.2" xref="S2.I2.i3.p1.3.m3.2.2.2.cmml">2</mn><mn id="S2.I2.i3.p1.3.m3.2.2.3" xref="S2.I2.i3.p1.3.m3.2.2.3.cmml">3</mn></mfrac><mo id="S2.I2.i3.p1.3.m3.3.3.3.3.2.2.2" stretchy="false" xref="S2.I2.i3.p1.3.m3.2.2.cmml">)</mo></mrow><mi id="S2.I2.i3.p1.3.m3.3.3.3.3.3" xref="S2.I2.i3.p1.3.m3.3.3.3.3.3.cmml">h</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.3.m3.3b"><apply id="S2.I2.i3.p1.3.m3.3.3.cmml" xref="S2.I2.i3.p1.3.m3.3.3"><leq id="S2.I2.i3.p1.3.m3.3.3.2.cmml" xref="S2.I2.i3.p1.3.m3.3.3.2"></leq><apply id="S2.I2.i3.p1.3.m3.3.3.1.2.cmml" xref="S2.I2.i3.p1.3.m3.3.3.1.1"><abs id="S2.I2.i3.p1.3.m3.3.3.1.2.1.cmml" xref="S2.I2.i3.p1.3.m3.3.3.1.1.2"></abs><apply id="S2.I2.i3.p1.3.m3.3.3.1.1.1.2.cmml" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.1"><apply id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.cmml" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.1.cmml" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1">subscript</csymbol><ci id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.2.cmml" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.2">int</ci><ci id="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.3.cmml" xref="S2.I2.i3.p1.3.m3.3.3.1.1.1.1.1.3">𝒯</ci></apply><ci id="S2.I2.i3.p1.3.m3.1.1.cmml" xref="S2.I2.i3.p1.3.m3.1.1">𝑦</ci></apply></apply><apply id="S2.I2.i3.p1.3.m3.3.3.3.cmml" xref="S2.I2.i3.p1.3.m3.3.3.3"><ci id="S2.I2.i3.p1.3.m3.3.3.3.1.cmml" xref="S2.I2.i3.p1.3.m3.3.3.3.1">⋅</ci><ci id="S2.I2.i3.p1.3.m3.3.3.3.2.cmml" xref="S2.I2.i3.p1.3.m3.3.3.3.2">𝑛</ci><apply id="S2.I2.i3.p1.3.m3.3.3.3.3.cmml" xref="S2.I2.i3.p1.3.m3.3.3.3.3"><csymbol cd="ambiguous" id="S2.I2.i3.p1.3.m3.3.3.3.3.1.cmml" xref="S2.I2.i3.p1.3.m3.3.3.3.3">superscript</csymbol><apply id="S2.I2.i3.p1.3.m3.2.2.cmml" xref="S2.I2.i3.p1.3.m3.3.3.3.3.2.2"><divide id="S2.I2.i3.p1.3.m3.2.2.1.cmml" xref="S2.I2.i3.p1.3.m3.3.3.3.3.2.2"></divide><cn id="S2.I2.i3.p1.3.m3.2.2.2.cmml" type="integer" xref="S2.I2.i3.p1.3.m3.2.2.2">2</cn><cn id="S2.I2.i3.p1.3.m3.2.2.3.cmml" type="integer" xref="S2.I2.i3.p1.3.m3.2.2.3">3</cn></apply><ci id="S2.I2.i3.p1.3.m3.3.3.3.3.3.cmml" xref="S2.I2.i3.p1.3.m3.3.3.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.3.m3.3c">|\operatorname{int}_{\mathcal{T}}(y)|\leq n\cdot(\tfrac{2}{3})^{h}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.3.m3.3d">| roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_y ) | ≤ italic_n ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i3.p1.3.4">.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S2.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.4.p1"> <p class="ltx_p" id="S2.4.p1.21">The tree decomposition <math alttext="\mathcal{T}=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="S2.4.p1.1.m1.1"><semantics id="S2.4.p1.1.m1.1a"><mrow id="S2.4.p1.1.m1.1b"><mi class="ltx_font_mathcaligraphic" id="S2.4.p1.1.m1.1.1">𝒯</mi><mo id="S2.4.p1.1.m1.1.2">=</mo><mrow id="S2.4.p1.1.m1.1.3"><mo id="S2.4.p1.1.m1.1.3.1" stretchy="false">(</mo><msub id="S2.4.p1.1.m1.1.3.2"><mi id="S2.4.p1.1.m1.1.3.2.2">B</mi><mi id="S2.4.p1.1.m1.1.3.2.3">x</mi></msub><mo id="S2.4.p1.1.m1.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.4.p1.1.m1.1.3.4">x</mi><mo id="S2.4.p1.1.m1.1.3.5">∈</mo><mi id="S2.4.p1.1.m1.1.3.6">V</mi><mrow id="S2.4.p1.1.m1.1.3.7"><mo id="S2.4.p1.1.m1.1.3.7.1" stretchy="false">(</mo><mi id="S2.4.p1.1.m1.1.3.7.2">T</mi><mo id="S2.4.p1.1.m1.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S2.4.p1.1.m1.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.4.p1.1.m1.1c">\mathcal{T}=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.1.m1.1d">caligraphic_T = ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> and its supporting tree <math alttext="T" class="ltx_Math" display="inline" id="S2.4.p1.2.m2.1"><semantics id="S2.4.p1.2.m2.1a"><mi id="S2.4.p1.2.m2.1.1" xref="S2.4.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.2.m2.1b"><ci id="S2.4.p1.2.m2.1.1.cmml" xref="S2.4.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.2.m2.1d">italic_T</annotation></semantics></math> is constructed recursively, as follows: Fix a global value <math alttext="N:=n\cdot(\tfrac{2}{3})^{h}" class="ltx_Math" display="inline" id="S2.4.p1.3.m3.1"><semantics id="S2.4.p1.3.m3.1a"><mrow id="S2.4.p1.3.m3.1.2" xref="S2.4.p1.3.m3.1.2.cmml"><mi id="S2.4.p1.3.m3.1.2.2" xref="S2.4.p1.3.m3.1.2.2.cmml">N</mi><mo id="S2.4.p1.3.m3.1.2.1" lspace="0.278em" rspace="0.278em" xref="S2.4.p1.3.m3.1.2.1.cmml">:=</mo><mrow id="S2.4.p1.3.m3.1.2.3" xref="S2.4.p1.3.m3.1.2.3.cmml"><mi id="S2.4.p1.3.m3.1.2.3.2" xref="S2.4.p1.3.m3.1.2.3.2.cmml">n</mi><mo id="S2.4.p1.3.m3.1.2.3.1" lspace="0.222em" rspace="0.222em" xref="S2.4.p1.3.m3.1.2.3.1.cmml">⋅</mo><msup id="S2.4.p1.3.m3.1.2.3.3" xref="S2.4.p1.3.m3.1.2.3.3.cmml"><mrow id="S2.4.p1.3.m3.1.2.3.3.2.2" xref="S2.4.p1.3.m3.1.1.cmml"><mo id="S2.4.p1.3.m3.1.2.3.3.2.2.1" stretchy="false" xref="S2.4.p1.3.m3.1.1.cmml">(</mo><mfrac id="S2.4.p1.3.m3.1.1" xref="S2.4.p1.3.m3.1.1.cmml"><mn id="S2.4.p1.3.m3.1.1.2" xref="S2.4.p1.3.m3.1.1.2.cmml">2</mn><mn id="S2.4.p1.3.m3.1.1.3" xref="S2.4.p1.3.m3.1.1.3.cmml">3</mn></mfrac><mo id="S2.4.p1.3.m3.1.2.3.3.2.2.2" stretchy="false" xref="S2.4.p1.3.m3.1.1.cmml">)</mo></mrow><mi id="S2.4.p1.3.m3.1.2.3.3.3" xref="S2.4.p1.3.m3.1.2.3.3.3.cmml">h</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.3.m3.1b"><apply id="S2.4.p1.3.m3.1.2.cmml" xref="S2.4.p1.3.m3.1.2"><csymbol cd="latexml" id="S2.4.p1.3.m3.1.2.1.cmml" xref="S2.4.p1.3.m3.1.2.1">assign</csymbol><ci id="S2.4.p1.3.m3.1.2.2.cmml" xref="S2.4.p1.3.m3.1.2.2">𝑁</ci><apply id="S2.4.p1.3.m3.1.2.3.cmml" xref="S2.4.p1.3.m3.1.2.3"><ci id="S2.4.p1.3.m3.1.2.3.1.cmml" xref="S2.4.p1.3.m3.1.2.3.1">⋅</ci><ci id="S2.4.p1.3.m3.1.2.3.2.cmml" xref="S2.4.p1.3.m3.1.2.3.2">𝑛</ci><apply id="S2.4.p1.3.m3.1.2.3.3.cmml" xref="S2.4.p1.3.m3.1.2.3.3"><csymbol cd="ambiguous" id="S2.4.p1.3.m3.1.2.3.3.1.cmml" xref="S2.4.p1.3.m3.1.2.3.3">superscript</csymbol><apply id="S2.4.p1.3.m3.1.1.cmml" xref="S2.4.p1.3.m3.1.2.3.3.2.2"><divide id="S2.4.p1.3.m3.1.1.1.cmml" xref="S2.4.p1.3.m3.1.2.3.3.2.2"></divide><cn id="S2.4.p1.3.m3.1.1.2.cmml" type="integer" xref="S2.4.p1.3.m3.1.1.2">2</cn><cn id="S2.4.p1.3.m3.1.1.3.cmml" type="integer" xref="S2.4.p1.3.m3.1.1.3">3</cn></apply><ci id="S2.4.p1.3.m3.1.2.3.3.3.cmml" xref="S2.4.p1.3.m3.1.2.3.3.3">ℎ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.3.m3.1c">N:=n\cdot(\tfrac{2}{3})^{h}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.3.m3.1d">italic_N := italic_n ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> that does not change during recursive invocations. Each recursive invocation takes a pair <math alttext="(G^{\prime},\partial^{\prime})" class="ltx_Math" display="inline" id="S2.4.p1.4.m4.2"><semantics id="S2.4.p1.4.m4.2a"><mrow id="S2.4.p1.4.m4.2.2.2" xref="S2.4.p1.4.m4.2.2.3.cmml"><mo id="S2.4.p1.4.m4.2.2.2.3" stretchy="false" xref="S2.4.p1.4.m4.2.2.3.cmml">(</mo><msup id="S2.4.p1.4.m4.1.1.1.1" xref="S2.4.p1.4.m4.1.1.1.1.cmml"><mi id="S2.4.p1.4.m4.1.1.1.1.2" xref="S2.4.p1.4.m4.1.1.1.1.2.cmml">G</mi><mo id="S2.4.p1.4.m4.1.1.1.1.3" xref="S2.4.p1.4.m4.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.4.p1.4.m4.2.2.2.4" xref="S2.4.p1.4.m4.2.2.3.cmml">,</mo><msup id="S2.4.p1.4.m4.2.2.2.2" xref="S2.4.p1.4.m4.2.2.2.2.cmml"><mo id="S2.4.p1.4.m4.2.2.2.2.2" lspace="0em" rspace="0em" xref="S2.4.p1.4.m4.2.2.2.2.2.cmml">∂</mo><mo id="S2.4.p1.4.m4.2.2.2.2.3" xref="S2.4.p1.4.m4.2.2.2.2.3.cmml">′</mo></msup><mo id="S2.4.p1.4.m4.2.2.2.5" stretchy="false" xref="S2.4.p1.4.m4.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.4.m4.2b"><interval closure="open" id="S2.4.p1.4.m4.2.2.3.cmml" xref="S2.4.p1.4.m4.2.2.2"><apply id="S2.4.p1.4.m4.1.1.1.1.cmml" xref="S2.4.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S2.4.p1.4.m4.1.1.1.1.1.cmml" xref="S2.4.p1.4.m4.1.1.1.1">superscript</csymbol><ci id="S2.4.p1.4.m4.1.1.1.1.2.cmml" xref="S2.4.p1.4.m4.1.1.1.1.2">𝐺</ci><ci id="S2.4.p1.4.m4.1.1.1.1.3.cmml" xref="S2.4.p1.4.m4.1.1.1.1.3">′</ci></apply><apply id="S2.4.p1.4.m4.2.2.2.2.cmml" xref="S2.4.p1.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S2.4.p1.4.m4.2.2.2.2.1.cmml" xref="S2.4.p1.4.m4.2.2.2.2">superscript</csymbol><partialdiff id="S2.4.p1.4.m4.2.2.2.2.2.cmml" xref="S2.4.p1.4.m4.2.2.2.2.2"></partialdiff><ci id="S2.4.p1.4.m4.2.2.2.2.3.cmml" xref="S2.4.p1.4.m4.2.2.2.2.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.4.m4.2c">(G^{\prime},\partial^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.4.m4.2d">( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , ∂ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and the initial invocation is on the pair <math alttext="(G,\emptyset)" class="ltx_Math" display="inline" id="S2.4.p1.5.m5.2"><semantics id="S2.4.p1.5.m5.2a"><mrow id="S2.4.p1.5.m5.2.3.2" xref="S2.4.p1.5.m5.2.3.1.cmml"><mo id="S2.4.p1.5.m5.2.3.2.1" stretchy="false" xref="S2.4.p1.5.m5.2.3.1.cmml">(</mo><mi id="S2.4.p1.5.m5.1.1" xref="S2.4.p1.5.m5.1.1.cmml">G</mi><mo id="S2.4.p1.5.m5.2.3.2.2" xref="S2.4.p1.5.m5.2.3.1.cmml">,</mo><mi id="S2.4.p1.5.m5.2.2" mathvariant="normal" xref="S2.4.p1.5.m5.2.2.cmml">∅</mi><mo id="S2.4.p1.5.m5.2.3.2.3" stretchy="false" xref="S2.4.p1.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.5.m5.2b"><interval closure="open" id="S2.4.p1.5.m5.2.3.1.cmml" xref="S2.4.p1.5.m5.2.3.2"><ci id="S2.4.p1.5.m5.1.1.cmml" xref="S2.4.p1.5.m5.1.1">𝐺</ci><emptyset id="S2.4.p1.5.m5.2.2.cmml" xref="S2.4.p1.5.m5.2.2"></emptyset></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.5.m5.2c">(G,\emptyset)</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.5.m5.2d">( italic_G , ∅ )</annotation></semantics></math>. When recursing on <math alttext="(G^{\prime},\partial^{\prime})" class="ltx_Math" display="inline" id="S2.4.p1.6.m6.2"><semantics id="S2.4.p1.6.m6.2a"><mrow id="S2.4.p1.6.m6.2.2.2" xref="S2.4.p1.6.m6.2.2.3.cmml"><mo id="S2.4.p1.6.m6.2.2.2.3" stretchy="false" xref="S2.4.p1.6.m6.2.2.3.cmml">(</mo><msup id="S2.4.p1.6.m6.1.1.1.1" xref="S2.4.p1.6.m6.1.1.1.1.cmml"><mi id="S2.4.p1.6.m6.1.1.1.1.2" xref="S2.4.p1.6.m6.1.1.1.1.2.cmml">G</mi><mo id="S2.4.p1.6.m6.1.1.1.1.3" xref="S2.4.p1.6.m6.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.4.p1.6.m6.2.2.2.4" xref="S2.4.p1.6.m6.2.2.3.cmml">,</mo><msup id="S2.4.p1.6.m6.2.2.2.2" xref="S2.4.p1.6.m6.2.2.2.2.cmml"><mo id="S2.4.p1.6.m6.2.2.2.2.2" lspace="0em" rspace="0em" xref="S2.4.p1.6.m6.2.2.2.2.2.cmml">∂</mo><mo id="S2.4.p1.6.m6.2.2.2.2.3" xref="S2.4.p1.6.m6.2.2.2.2.3.cmml">′</mo></msup><mo id="S2.4.p1.6.m6.2.2.2.5" stretchy="false" xref="S2.4.p1.6.m6.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.6.m6.2b"><interval closure="open" id="S2.4.p1.6.m6.2.2.3.cmml" xref="S2.4.p1.6.m6.2.2.2"><apply id="S2.4.p1.6.m6.1.1.1.1.cmml" xref="S2.4.p1.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S2.4.p1.6.m6.1.1.1.1.1.cmml" xref="S2.4.p1.6.m6.1.1.1.1">superscript</csymbol><ci id="S2.4.p1.6.m6.1.1.1.1.2.cmml" xref="S2.4.p1.6.m6.1.1.1.1.2">𝐺</ci><ci id="S2.4.p1.6.m6.1.1.1.1.3.cmml" xref="S2.4.p1.6.m6.1.1.1.1.3">′</ci></apply><apply id="S2.4.p1.6.m6.2.2.2.2.cmml" xref="S2.4.p1.6.m6.2.2.2.2"><csymbol cd="ambiguous" id="S2.4.p1.6.m6.2.2.2.2.1.cmml" xref="S2.4.p1.6.m6.2.2.2.2">superscript</csymbol><partialdiff id="S2.4.p1.6.m6.2.2.2.2.2.cmml" xref="S2.4.p1.6.m6.2.2.2.2.2"></partialdiff><ci id="S2.4.p1.6.m6.2.2.2.2.3.cmml" xref="S2.4.p1.6.m6.2.2.2.2.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.6.m6.2c">(G^{\prime},\partial^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.6.m6.2d">( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , ∂ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> to construct a subtree <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S2.4.p1.7.m7.1"><semantics id="S2.4.p1.7.m7.1a"><msup id="S2.4.p1.7.m7.1.1" xref="S2.4.p1.7.m7.1.1.cmml"><mi id="S2.4.p1.7.m7.1.1.2" xref="S2.4.p1.7.m7.1.1.2.cmml">T</mi><mo id="S2.4.p1.7.m7.1.1.3" xref="S2.4.p1.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.4.p1.7.m7.1b"><apply id="S2.4.p1.7.m7.1.1.cmml" xref="S2.4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.4.p1.7.m7.1.1.1.cmml" xref="S2.4.p1.7.m7.1.1">superscript</csymbol><ci id="S2.4.p1.7.m7.1.1.2.cmml" xref="S2.4.p1.7.m7.1.1.2">𝑇</ci><ci id="S2.4.p1.7.m7.1.1.3.cmml" xref="S2.4.p1.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.7.m7.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.7.m7.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> we apply the following rule: If <math alttext="|V(G^{\prime})\setminus\partial^{\prime}|\leq N" class="ltx_Math" display="inline" id="S2.4.p1.8.m8.1"><semantics id="S2.4.p1.8.m8.1a"><mrow id="S2.4.p1.8.m8.1.1" xref="S2.4.p1.8.m8.1.1.cmml"><mrow id="S2.4.p1.8.m8.1.1.1.1" xref="S2.4.p1.8.m8.1.1.1.2.cmml"><mo id="S2.4.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S2.4.p1.8.m8.1.1.1.2.1.cmml">|</mo><mrow id="S2.4.p1.8.m8.1.1.1.1.1" xref="S2.4.p1.8.m8.1.1.1.1.1.cmml"><mrow id="S2.4.p1.8.m8.1.1.1.1.1.1" xref="S2.4.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="S2.4.p1.8.m8.1.1.1.1.1.1.3" xref="S2.4.p1.8.m8.1.1.1.1.1.1.3.cmml">V</mi><mo id="S2.4.p1.8.m8.1.1.1.1.1.1.2" xref="S2.4.p1.8.m8.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.2" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.2.cmml">G</mi><mo id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.3" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.4.p1.8.m8.1.1.1.1.1.2" xref="S2.4.p1.8.m8.1.1.1.1.1.2.cmml">∖</mo><msup id="S2.4.p1.8.m8.1.1.1.1.1.3" xref="S2.4.p1.8.m8.1.1.1.1.1.3.cmml"><mo id="S2.4.p1.8.m8.1.1.1.1.1.3.2" lspace="0em" rspace="0em" xref="S2.4.p1.8.m8.1.1.1.1.1.3.2.cmml">∂</mo><mo id="S2.4.p1.8.m8.1.1.1.1.1.3.3" xref="S2.4.p1.8.m8.1.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S2.4.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S2.4.p1.8.m8.1.1.1.2.1.cmml">|</mo></mrow><mo id="S2.4.p1.8.m8.1.1.2" xref="S2.4.p1.8.m8.1.1.2.cmml">≤</mo><mi id="S2.4.p1.8.m8.1.1.3" xref="S2.4.p1.8.m8.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.8.m8.1b"><apply id="S2.4.p1.8.m8.1.1.cmml" xref="S2.4.p1.8.m8.1.1"><leq id="S2.4.p1.8.m8.1.1.2.cmml" xref="S2.4.p1.8.m8.1.1.2"></leq><apply id="S2.4.p1.8.m8.1.1.1.2.cmml" xref="S2.4.p1.8.m8.1.1.1.1"><abs id="S2.4.p1.8.m8.1.1.1.2.1.cmml" xref="S2.4.p1.8.m8.1.1.1.1.2"></abs><apply id="S2.4.p1.8.m8.1.1.1.1.1.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1"><setdiff id="S2.4.p1.8.m8.1.1.1.1.1.2.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.2"></setdiff><apply id="S2.4.p1.8.m8.1.1.1.1.1.1.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.1"><times id="S2.4.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.1.2"></times><ci id="S2.4.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.1.3">𝑉</ci><apply id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S2.4.p1.8.m8.1.1.1.1.1.3.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.8.m8.1.1.1.1.1.3.1.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.3">superscript</csymbol><partialdiff id="S2.4.p1.8.m8.1.1.1.1.1.3.2.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.3.2"></partialdiff><ci id="S2.4.p1.8.m8.1.1.1.1.1.3.3.cmml" xref="S2.4.p1.8.m8.1.1.1.1.1.3.3">′</ci></apply></apply></apply><ci id="S2.4.p1.8.m8.1.1.3.cmml" xref="S2.4.p1.8.m8.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.8.m8.1c">|V(G^{\prime})\setminus\partial^{\prime}|\leq N</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.8.m8.1d">| italic_V ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∖ ∂ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≤ italic_N</annotation></semantics></math>, then <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S2.4.p1.9.m9.1"><semantics id="S2.4.p1.9.m9.1a"><msup id="S2.4.p1.9.m9.1.1" xref="S2.4.p1.9.m9.1.1.cmml"><mi id="S2.4.p1.9.m9.1.1.2" xref="S2.4.p1.9.m9.1.1.2.cmml">T</mi><mo id="S2.4.p1.9.m9.1.1.3" xref="S2.4.p1.9.m9.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.4.p1.9.m9.1b"><apply id="S2.4.p1.9.m9.1.1.cmml" xref="S2.4.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.4.p1.9.m9.1.1.1.cmml" xref="S2.4.p1.9.m9.1.1">superscript</csymbol><ci id="S2.4.p1.9.m9.1.1.2.cmml" xref="S2.4.p1.9.m9.1.1.2">𝑇</ci><ci id="S2.4.p1.9.m9.1.1.3.cmml" xref="S2.4.p1.9.m9.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.9.m9.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.9.m9.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> consists of a single node <math alttext="x" class="ltx_Math" display="inline" id="S2.4.p1.10.m10.1"><semantics id="S2.4.p1.10.m10.1a"><mi id="S2.4.p1.10.m10.1.1" xref="S2.4.p1.10.m10.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.10.m10.1b"><ci id="S2.4.p1.10.m10.1.1.cmml" xref="S2.4.p1.10.m10.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.10.m10.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.10.m10.1d">italic_x</annotation></semantics></math> with <math alttext="B_{x}:=V(G^{\prime})" class="ltx_Math" display="inline" id="S2.4.p1.11.m11.1"><semantics id="S2.4.p1.11.m11.1a"><mrow id="S2.4.p1.11.m11.1.1" xref="S2.4.p1.11.m11.1.1.cmml"><msub id="S2.4.p1.11.m11.1.1.3" xref="S2.4.p1.11.m11.1.1.3.cmml"><mi id="S2.4.p1.11.m11.1.1.3.2" xref="S2.4.p1.11.m11.1.1.3.2.cmml">B</mi><mi id="S2.4.p1.11.m11.1.1.3.3" xref="S2.4.p1.11.m11.1.1.3.3.cmml">x</mi></msub><mo id="S2.4.p1.11.m11.1.1.2" lspace="0.278em" rspace="0.278em" xref="S2.4.p1.11.m11.1.1.2.cmml">:=</mo><mrow id="S2.4.p1.11.m11.1.1.1" xref="S2.4.p1.11.m11.1.1.1.cmml"><mi id="S2.4.p1.11.m11.1.1.1.3" xref="S2.4.p1.11.m11.1.1.1.3.cmml">V</mi><mo id="S2.4.p1.11.m11.1.1.1.2" xref="S2.4.p1.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="S2.4.p1.11.m11.1.1.1.1.1" xref="S2.4.p1.11.m11.1.1.1.1.1.1.cmml"><mo id="S2.4.p1.11.m11.1.1.1.1.1.2" stretchy="false" xref="S2.4.p1.11.m11.1.1.1.1.1.1.cmml">(</mo><msup id="S2.4.p1.11.m11.1.1.1.1.1.1" xref="S2.4.p1.11.m11.1.1.1.1.1.1.cmml"><mi id="S2.4.p1.11.m11.1.1.1.1.1.1.2" xref="S2.4.p1.11.m11.1.1.1.1.1.1.2.cmml">G</mi><mo id="S2.4.p1.11.m11.1.1.1.1.1.1.3" xref="S2.4.p1.11.m11.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.4.p1.11.m11.1.1.1.1.1.3" stretchy="false" xref="S2.4.p1.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.11.m11.1b"><apply id="S2.4.p1.11.m11.1.1.cmml" xref="S2.4.p1.11.m11.1.1"><csymbol cd="latexml" id="S2.4.p1.11.m11.1.1.2.cmml" xref="S2.4.p1.11.m11.1.1.2">assign</csymbol><apply id="S2.4.p1.11.m11.1.1.3.cmml" xref="S2.4.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.11.m11.1.1.3.1.cmml" xref="S2.4.p1.11.m11.1.1.3">subscript</csymbol><ci id="S2.4.p1.11.m11.1.1.3.2.cmml" xref="S2.4.p1.11.m11.1.1.3.2">𝐵</ci><ci id="S2.4.p1.11.m11.1.1.3.3.cmml" xref="S2.4.p1.11.m11.1.1.3.3">𝑥</ci></apply><apply id="S2.4.p1.11.m11.1.1.1.cmml" xref="S2.4.p1.11.m11.1.1.1"><times id="S2.4.p1.11.m11.1.1.1.2.cmml" xref="S2.4.p1.11.m11.1.1.1.2"></times><ci id="S2.4.p1.11.m11.1.1.1.3.cmml" xref="S2.4.p1.11.m11.1.1.1.3">𝑉</ci><apply id="S2.4.p1.11.m11.1.1.1.1.1.1.cmml" xref="S2.4.p1.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.4.p1.11.m11.1.1.1.1.1.1.1.cmml" xref="S2.4.p1.11.m11.1.1.1.1.1">superscript</csymbol><ci id="S2.4.p1.11.m11.1.1.1.1.1.1.2.cmml" xref="S2.4.p1.11.m11.1.1.1.1.1.1.2">𝐺</ci><ci id="S2.4.p1.11.m11.1.1.1.1.1.1.3.cmml" xref="S2.4.p1.11.m11.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.11.m11.1c">B_{x}:=V(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.11.m11.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT := italic_V ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Otherwise, let <math alttext="(A^{\prime},B^{\prime})" class="ltx_Math" display="inline" id="S2.4.p1.12.m12.2"><semantics id="S2.4.p1.12.m12.2a"><mrow id="S2.4.p1.12.m12.2.2.2" xref="S2.4.p1.12.m12.2.2.3.cmml"><mo id="S2.4.p1.12.m12.2.2.2.3" stretchy="false" xref="S2.4.p1.12.m12.2.2.3.cmml">(</mo><msup id="S2.4.p1.12.m12.1.1.1.1" xref="S2.4.p1.12.m12.1.1.1.1.cmml"><mi id="S2.4.p1.12.m12.1.1.1.1.2" xref="S2.4.p1.12.m12.1.1.1.1.2.cmml">A</mi><mo id="S2.4.p1.12.m12.1.1.1.1.3" xref="S2.4.p1.12.m12.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.4.p1.12.m12.2.2.2.4" xref="S2.4.p1.12.m12.2.2.3.cmml">,</mo><msup id="S2.4.p1.12.m12.2.2.2.2" xref="S2.4.p1.12.m12.2.2.2.2.cmml"><mi id="S2.4.p1.12.m12.2.2.2.2.2" xref="S2.4.p1.12.m12.2.2.2.2.2.cmml">B</mi><mo id="S2.4.p1.12.m12.2.2.2.2.3" xref="S2.4.p1.12.m12.2.2.2.2.3.cmml">′</mo></msup><mo id="S2.4.p1.12.m12.2.2.2.5" stretchy="false" xref="S2.4.p1.12.m12.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.12.m12.2b"><interval closure="open" id="S2.4.p1.12.m12.2.2.3.cmml" xref="S2.4.p1.12.m12.2.2.2"><apply id="S2.4.p1.12.m12.1.1.1.1.cmml" xref="S2.4.p1.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S2.4.p1.12.m12.1.1.1.1.1.cmml" xref="S2.4.p1.12.m12.1.1.1.1">superscript</csymbol><ci id="S2.4.p1.12.m12.1.1.1.1.2.cmml" xref="S2.4.p1.12.m12.1.1.1.1.2">𝐴</ci><ci id="S2.4.p1.12.m12.1.1.1.1.3.cmml" xref="S2.4.p1.12.m12.1.1.1.1.3">′</ci></apply><apply id="S2.4.p1.12.m12.2.2.2.2.cmml" xref="S2.4.p1.12.m12.2.2.2.2"><csymbol cd="ambiguous" id="S2.4.p1.12.m12.2.2.2.2.1.cmml" xref="S2.4.p1.12.m12.2.2.2.2">superscript</csymbol><ci id="S2.4.p1.12.m12.2.2.2.2.2.cmml" xref="S2.4.p1.12.m12.2.2.2.2.2">𝐵</ci><ci id="S2.4.p1.12.m12.2.2.2.2.3.cmml" xref="S2.4.p1.12.m12.2.2.2.2.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.12.m12.2c">(A^{\prime},B^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.12.m12.2d">( italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> be a balanced separation of <math alttext="G^{\prime}-\partial^{\prime}" class="ltx_Math" display="inline" id="S2.4.p1.13.m13.1"><semantics id="S2.4.p1.13.m13.1a"><mrow id="S2.4.p1.13.m13.1.1" xref="S2.4.p1.13.m13.1.1.cmml"><msup id="S2.4.p1.13.m13.1.1.2" xref="S2.4.p1.13.m13.1.1.2.cmml"><mi id="S2.4.p1.13.m13.1.1.2.2" xref="S2.4.p1.13.m13.1.1.2.2.cmml">G</mi><mo id="S2.4.p1.13.m13.1.1.2.3" xref="S2.4.p1.13.m13.1.1.2.3.cmml">′</mo></msup><mo id="S2.4.p1.13.m13.1.1.1" xref="S2.4.p1.13.m13.1.1.1.cmml">−</mo><msup id="S2.4.p1.13.m13.1.1.3" xref="S2.4.p1.13.m13.1.1.3.cmml"><mo id="S2.4.p1.13.m13.1.1.3.2" lspace="0em" xref="S2.4.p1.13.m13.1.1.3.2.cmml">∂</mo><mo id="S2.4.p1.13.m13.1.1.3.3" xref="S2.4.p1.13.m13.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.13.m13.1b"><apply id="S2.4.p1.13.m13.1.1.cmml" xref="S2.4.p1.13.m13.1.1"><minus id="S2.4.p1.13.m13.1.1.1.cmml" xref="S2.4.p1.13.m13.1.1.1"></minus><apply id="S2.4.p1.13.m13.1.1.2.cmml" xref="S2.4.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S2.4.p1.13.m13.1.1.2.1.cmml" xref="S2.4.p1.13.m13.1.1.2">superscript</csymbol><ci id="S2.4.p1.13.m13.1.1.2.2.cmml" xref="S2.4.p1.13.m13.1.1.2.2">𝐺</ci><ci id="S2.4.p1.13.m13.1.1.2.3.cmml" xref="S2.4.p1.13.m13.1.1.2.3">′</ci></apply><apply id="S2.4.p1.13.m13.1.1.3.cmml" xref="S2.4.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.13.m13.1.1.3.1.cmml" xref="S2.4.p1.13.m13.1.1.3">superscript</csymbol><partialdiff id="S2.4.p1.13.m13.1.1.3.2.cmml" xref="S2.4.p1.13.m13.1.1.3.2"></partialdiff><ci id="S2.4.p1.13.m13.1.1.3.3.cmml" xref="S2.4.p1.13.m13.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.13.m13.1c">G^{\prime}-\partial^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.13.m13.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - ∂ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> of order at most <math alttext="a" class="ltx_Math" display="inline" id="S2.4.p1.14.m14.1"><semantics id="S2.4.p1.14.m14.1a"><mi id="S2.4.p1.14.m14.1.1" xref="S2.4.p1.14.m14.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.14.m14.1b"><ci id="S2.4.p1.14.m14.1.1.cmml" xref="S2.4.p1.14.m14.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.14.m14.1c">a</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.14.m14.1d">italic_a</annotation></semantics></math>. The root <math alttext="x" class="ltx_Math" display="inline" id="S2.4.p1.15.m15.1"><semantics id="S2.4.p1.15.m15.1a"><mi id="S2.4.p1.15.m15.1.1" xref="S2.4.p1.15.m15.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.15.m15.1b"><ci id="S2.4.p1.15.m15.1.1.cmml" xref="S2.4.p1.15.m15.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.15.m15.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.15.m15.1d">italic_x</annotation></semantics></math> of <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S2.4.p1.16.m16.1"><semantics id="S2.4.p1.16.m16.1a"><msup id="S2.4.p1.16.m16.1.1" xref="S2.4.p1.16.m16.1.1.cmml"><mi id="S2.4.p1.16.m16.1.1.2" xref="S2.4.p1.16.m16.1.1.2.cmml">T</mi><mo id="S2.4.p1.16.m16.1.1.3" xref="S2.4.p1.16.m16.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.4.p1.16.m16.1b"><apply id="S2.4.p1.16.m16.1.1.cmml" xref="S2.4.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S2.4.p1.16.m16.1.1.1.cmml" xref="S2.4.p1.16.m16.1.1">superscript</csymbol><ci id="S2.4.p1.16.m16.1.1.2.cmml" xref="S2.4.p1.16.m16.1.1.2">𝑇</ci><ci id="S2.4.p1.16.m16.1.1.3.cmml" xref="S2.4.p1.16.m16.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.16.m16.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.16.m16.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> has <math alttext="B_{x}:=\partial^{\prime}\cup(A^{\prime}\cap B^{\prime})" class="ltx_Math" display="inline" id="S2.4.p1.17.m17.1"><semantics id="S2.4.p1.17.m17.1a"><mrow id="S2.4.p1.17.m17.1.1" xref="S2.4.p1.17.m17.1.1.cmml"><msub id="S2.4.p1.17.m17.1.1.3" xref="S2.4.p1.17.m17.1.1.3.cmml"><mi id="S2.4.p1.17.m17.1.1.3.2" xref="S2.4.p1.17.m17.1.1.3.2.cmml">B</mi><mi id="S2.4.p1.17.m17.1.1.3.3" xref="S2.4.p1.17.m17.1.1.3.3.cmml">x</mi></msub><mo id="S2.4.p1.17.m17.1.1.2" lspace="0.278em" xref="S2.4.p1.17.m17.1.1.2.cmml">:=</mo><mrow id="S2.4.p1.17.m17.1.1.1" xref="S2.4.p1.17.m17.1.1.1.cmml"><msup id="S2.4.p1.17.m17.1.1.1.3" xref="S2.4.p1.17.m17.1.1.1.3.cmml"><mo id="S2.4.p1.17.m17.1.1.1.3.2" lspace="0.111em" rspace="0em" xref="S2.4.p1.17.m17.1.1.1.3.2.cmml">∂</mo><mo id="S2.4.p1.17.m17.1.1.1.3.3" xref="S2.4.p1.17.m17.1.1.1.3.3.cmml">′</mo></msup><mo id="S2.4.p1.17.m17.1.1.1.2" xref="S2.4.p1.17.m17.1.1.1.2.cmml">∪</mo><mrow id="S2.4.p1.17.m17.1.1.1.1.1" xref="S2.4.p1.17.m17.1.1.1.1.1.1.cmml"><mo id="S2.4.p1.17.m17.1.1.1.1.1.2" stretchy="false" xref="S2.4.p1.17.m17.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.4.p1.17.m17.1.1.1.1.1.1" xref="S2.4.p1.17.m17.1.1.1.1.1.1.cmml"><msup id="S2.4.p1.17.m17.1.1.1.1.1.1.2" xref="S2.4.p1.17.m17.1.1.1.1.1.1.2.cmml"><mi id="S2.4.p1.17.m17.1.1.1.1.1.1.2.2" xref="S2.4.p1.17.m17.1.1.1.1.1.1.2.2.cmml">A</mi><mo id="S2.4.p1.17.m17.1.1.1.1.1.1.2.3" xref="S2.4.p1.17.m17.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S2.4.p1.17.m17.1.1.1.1.1.1.1" xref="S2.4.p1.17.m17.1.1.1.1.1.1.1.cmml">∩</mo><msup id="S2.4.p1.17.m17.1.1.1.1.1.1.3" xref="S2.4.p1.17.m17.1.1.1.1.1.1.3.cmml"><mi id="S2.4.p1.17.m17.1.1.1.1.1.1.3.2" xref="S2.4.p1.17.m17.1.1.1.1.1.1.3.2.cmml">B</mi><mo id="S2.4.p1.17.m17.1.1.1.1.1.1.3.3" xref="S2.4.p1.17.m17.1.1.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S2.4.p1.17.m17.1.1.1.1.1.3" stretchy="false" xref="S2.4.p1.17.m17.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.17.m17.1b"><apply id="S2.4.p1.17.m17.1.1.cmml" xref="S2.4.p1.17.m17.1.1"><csymbol cd="latexml" id="S2.4.p1.17.m17.1.1.2.cmml" xref="S2.4.p1.17.m17.1.1.2">assign</csymbol><apply id="S2.4.p1.17.m17.1.1.3.cmml" xref="S2.4.p1.17.m17.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.17.m17.1.1.3.1.cmml" xref="S2.4.p1.17.m17.1.1.3">subscript</csymbol><ci id="S2.4.p1.17.m17.1.1.3.2.cmml" xref="S2.4.p1.17.m17.1.1.3.2">𝐵</ci><ci id="S2.4.p1.17.m17.1.1.3.3.cmml" xref="S2.4.p1.17.m17.1.1.3.3">𝑥</ci></apply><apply id="S2.4.p1.17.m17.1.1.1.cmml" xref="S2.4.p1.17.m17.1.1.1"><union id="S2.4.p1.17.m17.1.1.1.2.cmml" xref="S2.4.p1.17.m17.1.1.1.2"></union><apply id="S2.4.p1.17.m17.1.1.1.3.cmml" xref="S2.4.p1.17.m17.1.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.17.m17.1.1.1.3.1.cmml" xref="S2.4.p1.17.m17.1.1.1.3">superscript</csymbol><partialdiff id="S2.4.p1.17.m17.1.1.1.3.2.cmml" xref="S2.4.p1.17.m17.1.1.1.3.2"></partialdiff><ci id="S2.4.p1.17.m17.1.1.1.3.3.cmml" xref="S2.4.p1.17.m17.1.1.1.3.3">′</ci></apply><apply id="S2.4.p1.17.m17.1.1.1.1.1.1.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1"><intersect id="S2.4.p1.17.m17.1.1.1.1.1.1.1.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.1"></intersect><apply id="S2.4.p1.17.m17.1.1.1.1.1.1.2.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.4.p1.17.m17.1.1.1.1.1.1.2.1.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.4.p1.17.m17.1.1.1.1.1.1.2.2.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.2.2">𝐴</ci><ci id="S2.4.p1.17.m17.1.1.1.1.1.1.2.3.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.2.3">′</ci></apply><apply id="S2.4.p1.17.m17.1.1.1.1.1.1.3.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.17.m17.1.1.1.1.1.1.3.1.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.4.p1.17.m17.1.1.1.1.1.1.3.2.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.3.2">𝐵</ci><ci id="S2.4.p1.17.m17.1.1.1.1.1.1.3.3.cmml" xref="S2.4.p1.17.m17.1.1.1.1.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.17.m17.1c">B_{x}:=\partial^{\prime}\cup(A^{\prime}\cap B^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.17.m17.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT := ∂ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ ( italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. The left child of <math alttext="x" class="ltx_Math" display="inline" id="S2.4.p1.18.m18.1"><semantics id="S2.4.p1.18.m18.1a"><mi id="S2.4.p1.18.m18.1.1" xref="S2.4.p1.18.m18.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.18.m18.1b"><ci id="S2.4.p1.18.m18.1.1.cmml" xref="S2.4.p1.18.m18.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.18.m18.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.18.m18.1d">italic_x</annotation></semantics></math> is the root of the tree obtained by recursing on <math alttext="(G^{\prime}[A^{\prime}],A^{\prime}\cap(\partial^{\prime}\cup B^{\prime}))" class="ltx_Math" display="inline" id="S2.4.p1.19.m19.2"><semantics id="S2.4.p1.19.m19.2a"><mrow id="S2.4.p1.19.m19.2.2.2" xref="S2.4.p1.19.m19.2.2.3.cmml"><mo id="S2.4.p1.19.m19.2.2.2.3" stretchy="false" xref="S2.4.p1.19.m19.2.2.3.cmml">(</mo><mrow id="S2.4.p1.19.m19.1.1.1.1" xref="S2.4.p1.19.m19.1.1.1.1.cmml"><msup id="S2.4.p1.19.m19.1.1.1.1.3" xref="S2.4.p1.19.m19.1.1.1.1.3.cmml"><mi id="S2.4.p1.19.m19.1.1.1.1.3.2" xref="S2.4.p1.19.m19.1.1.1.1.3.2.cmml">G</mi><mo id="S2.4.p1.19.m19.1.1.1.1.3.3" xref="S2.4.p1.19.m19.1.1.1.1.3.3.cmml">′</mo></msup><mo id="S2.4.p1.19.m19.1.1.1.1.2" xref="S2.4.p1.19.m19.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.4.p1.19.m19.1.1.1.1.1.1" xref="S2.4.p1.19.m19.1.1.1.1.1.2.cmml"><mo id="S2.4.p1.19.m19.1.1.1.1.1.1.2" stretchy="false" xref="S2.4.p1.19.m19.1.1.1.1.1.2.1.cmml">[</mo><msup id="S2.4.p1.19.m19.1.1.1.1.1.1.1" xref="S2.4.p1.19.m19.1.1.1.1.1.1.1.cmml"><mi id="S2.4.p1.19.m19.1.1.1.1.1.1.1.2" xref="S2.4.p1.19.m19.1.1.1.1.1.1.1.2.cmml">A</mi><mo id="S2.4.p1.19.m19.1.1.1.1.1.1.1.3" xref="S2.4.p1.19.m19.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.4.p1.19.m19.1.1.1.1.1.1.3" stretchy="false" xref="S2.4.p1.19.m19.1.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.4.p1.19.m19.2.2.2.4" xref="S2.4.p1.19.m19.2.2.3.cmml">,</mo><mrow id="S2.4.p1.19.m19.2.2.2.2" xref="S2.4.p1.19.m19.2.2.2.2.cmml"><msup id="S2.4.p1.19.m19.2.2.2.2.3" xref="S2.4.p1.19.m19.2.2.2.2.3.cmml"><mi id="S2.4.p1.19.m19.2.2.2.2.3.2" xref="S2.4.p1.19.m19.2.2.2.2.3.2.cmml">A</mi><mo id="S2.4.p1.19.m19.2.2.2.2.3.3" xref="S2.4.p1.19.m19.2.2.2.2.3.3.cmml">′</mo></msup><mo id="S2.4.p1.19.m19.2.2.2.2.2" xref="S2.4.p1.19.m19.2.2.2.2.2.cmml">∩</mo><mrow id="S2.4.p1.19.m19.2.2.2.2.1.1" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.cmml"><mo id="S2.4.p1.19.m19.2.2.2.2.1.1.2" stretchy="false" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.4.p1.19.m19.2.2.2.2.1.1.1" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.cmml"><msup id="S2.4.p1.19.m19.2.2.2.2.1.1.1.2" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.cmml"><mo id="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.2" lspace="0em" rspace="0em" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.2.cmml">∂</mo><mo id="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.3" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="S2.4.p1.19.m19.2.2.2.2.1.1.1.1" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.1.cmml">∪</mo><msup id="S2.4.p1.19.m19.2.2.2.2.1.1.1.3" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.cmml"><mi id="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.2" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.2.cmml">B</mi><mo id="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.3" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S2.4.p1.19.m19.2.2.2.2.1.1.3" stretchy="false" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.4.p1.19.m19.2.2.2.5" stretchy="false" xref="S2.4.p1.19.m19.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.19.m19.2b"><interval closure="open" id="S2.4.p1.19.m19.2.2.3.cmml" xref="S2.4.p1.19.m19.2.2.2"><apply id="S2.4.p1.19.m19.1.1.1.1.cmml" xref="S2.4.p1.19.m19.1.1.1.1"><times id="S2.4.p1.19.m19.1.1.1.1.2.cmml" xref="S2.4.p1.19.m19.1.1.1.1.2"></times><apply id="S2.4.p1.19.m19.1.1.1.1.3.cmml" xref="S2.4.p1.19.m19.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.19.m19.1.1.1.1.3.1.cmml" xref="S2.4.p1.19.m19.1.1.1.1.3">superscript</csymbol><ci id="S2.4.p1.19.m19.1.1.1.1.3.2.cmml" xref="S2.4.p1.19.m19.1.1.1.1.3.2">𝐺</ci><ci id="S2.4.p1.19.m19.1.1.1.1.3.3.cmml" xref="S2.4.p1.19.m19.1.1.1.1.3.3">′</ci></apply><apply id="S2.4.p1.19.m19.1.1.1.1.1.2.cmml" xref="S2.4.p1.19.m19.1.1.1.1.1.1"><csymbol cd="latexml" id="S2.4.p1.19.m19.1.1.1.1.1.2.1.cmml" xref="S2.4.p1.19.m19.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.4.p1.19.m19.1.1.1.1.1.1.1.cmml" xref="S2.4.p1.19.m19.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.4.p1.19.m19.1.1.1.1.1.1.1.1.cmml" xref="S2.4.p1.19.m19.1.1.1.1.1.1.1">superscript</csymbol><ci id="S2.4.p1.19.m19.1.1.1.1.1.1.1.2.cmml" xref="S2.4.p1.19.m19.1.1.1.1.1.1.1.2">𝐴</ci><ci id="S2.4.p1.19.m19.1.1.1.1.1.1.1.3.cmml" xref="S2.4.p1.19.m19.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S2.4.p1.19.m19.2.2.2.2.cmml" xref="S2.4.p1.19.m19.2.2.2.2"><intersect id="S2.4.p1.19.m19.2.2.2.2.2.cmml" xref="S2.4.p1.19.m19.2.2.2.2.2"></intersect><apply id="S2.4.p1.19.m19.2.2.2.2.3.cmml" xref="S2.4.p1.19.m19.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.4.p1.19.m19.2.2.2.2.3.1.cmml" xref="S2.4.p1.19.m19.2.2.2.2.3">superscript</csymbol><ci id="S2.4.p1.19.m19.2.2.2.2.3.2.cmml" xref="S2.4.p1.19.m19.2.2.2.2.3.2">𝐴</ci><ci id="S2.4.p1.19.m19.2.2.2.2.3.3.cmml" xref="S2.4.p1.19.m19.2.2.2.2.3.3">′</ci></apply><apply id="S2.4.p1.19.m19.2.2.2.2.1.1.1.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1"><union id="S2.4.p1.19.m19.2.2.2.2.1.1.1.1.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.1"></union><apply id="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.1.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.2">superscript</csymbol><partialdiff id="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.2.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.2"></partialdiff><ci id="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.3.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.2.3">′</ci></apply><apply id="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.1.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.3">superscript</csymbol><ci id="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.2.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.2">𝐵</ci><ci id="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.3.cmml" xref="S2.4.p1.19.m19.2.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.19.m19.2c">(G^{\prime}[A^{\prime}],A^{\prime}\cap(\partial^{\prime}\cup B^{\prime}))</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.19.m19.2d">( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT [ italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] , italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ ( ∂ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math> and the right child of <math alttext="x" class="ltx_Math" display="inline" id="S2.4.p1.20.m20.1"><semantics id="S2.4.p1.20.m20.1a"><mi id="S2.4.p1.20.m20.1.1" xref="S2.4.p1.20.m20.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.20.m20.1b"><ci id="S2.4.p1.20.m20.1.1.cmml" xref="S2.4.p1.20.m20.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.20.m20.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.20.m20.1d">italic_x</annotation></semantics></math> is the root of the tree obtained by recursing on <math alttext="(G[B^{\prime}],B^{\prime}\cap(\partial^{\prime}\cup A^{\prime}))" class="ltx_Math" display="inline" id="S2.4.p1.21.m21.2"><semantics id="S2.4.p1.21.m21.2a"><mrow id="S2.4.p1.21.m21.2.2.2" xref="S2.4.p1.21.m21.2.2.3.cmml"><mo id="S2.4.p1.21.m21.2.2.2.3" stretchy="false" xref="S2.4.p1.21.m21.2.2.3.cmml">(</mo><mrow id="S2.4.p1.21.m21.1.1.1.1" xref="S2.4.p1.21.m21.1.1.1.1.cmml"><mi id="S2.4.p1.21.m21.1.1.1.1.3" xref="S2.4.p1.21.m21.1.1.1.1.3.cmml">G</mi><mo id="S2.4.p1.21.m21.1.1.1.1.2" xref="S2.4.p1.21.m21.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.4.p1.21.m21.1.1.1.1.1.1" xref="S2.4.p1.21.m21.1.1.1.1.1.2.cmml"><mo id="S2.4.p1.21.m21.1.1.1.1.1.1.2" stretchy="false" xref="S2.4.p1.21.m21.1.1.1.1.1.2.1.cmml">[</mo><msup id="S2.4.p1.21.m21.1.1.1.1.1.1.1" xref="S2.4.p1.21.m21.1.1.1.1.1.1.1.cmml"><mi id="S2.4.p1.21.m21.1.1.1.1.1.1.1.2" xref="S2.4.p1.21.m21.1.1.1.1.1.1.1.2.cmml">B</mi><mo id="S2.4.p1.21.m21.1.1.1.1.1.1.1.3" xref="S2.4.p1.21.m21.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.4.p1.21.m21.1.1.1.1.1.1.3" stretchy="false" xref="S2.4.p1.21.m21.1.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.4.p1.21.m21.2.2.2.4" xref="S2.4.p1.21.m21.2.2.3.cmml">,</mo><mrow id="S2.4.p1.21.m21.2.2.2.2" xref="S2.4.p1.21.m21.2.2.2.2.cmml"><msup id="S2.4.p1.21.m21.2.2.2.2.3" xref="S2.4.p1.21.m21.2.2.2.2.3.cmml"><mi id="S2.4.p1.21.m21.2.2.2.2.3.2" xref="S2.4.p1.21.m21.2.2.2.2.3.2.cmml">B</mi><mo id="S2.4.p1.21.m21.2.2.2.2.3.3" xref="S2.4.p1.21.m21.2.2.2.2.3.3.cmml">′</mo></msup><mo id="S2.4.p1.21.m21.2.2.2.2.2" xref="S2.4.p1.21.m21.2.2.2.2.2.cmml">∩</mo><mrow id="S2.4.p1.21.m21.2.2.2.2.1.1" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.cmml"><mo id="S2.4.p1.21.m21.2.2.2.2.1.1.2" stretchy="false" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.4.p1.21.m21.2.2.2.2.1.1.1" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.cmml"><msup id="S2.4.p1.21.m21.2.2.2.2.1.1.1.2" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.cmml"><mo id="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.2" lspace="0em" rspace="0em" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.2.cmml">∂</mo><mo id="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.3" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="S2.4.p1.21.m21.2.2.2.2.1.1.1.1" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.1.cmml">∪</mo><msup id="S2.4.p1.21.m21.2.2.2.2.1.1.1.3" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.cmml"><mi id="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.2" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.2.cmml">A</mi><mo id="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.3" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S2.4.p1.21.m21.2.2.2.2.1.1.3" stretchy="false" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.4.p1.21.m21.2.2.2.5" stretchy="false" xref="S2.4.p1.21.m21.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.21.m21.2b"><interval closure="open" id="S2.4.p1.21.m21.2.2.3.cmml" xref="S2.4.p1.21.m21.2.2.2"><apply id="S2.4.p1.21.m21.1.1.1.1.cmml" xref="S2.4.p1.21.m21.1.1.1.1"><times id="S2.4.p1.21.m21.1.1.1.1.2.cmml" xref="S2.4.p1.21.m21.1.1.1.1.2"></times><ci id="S2.4.p1.21.m21.1.1.1.1.3.cmml" xref="S2.4.p1.21.m21.1.1.1.1.3">𝐺</ci><apply id="S2.4.p1.21.m21.1.1.1.1.1.2.cmml" xref="S2.4.p1.21.m21.1.1.1.1.1.1"><csymbol cd="latexml" id="S2.4.p1.21.m21.1.1.1.1.1.2.1.cmml" xref="S2.4.p1.21.m21.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.4.p1.21.m21.1.1.1.1.1.1.1.cmml" xref="S2.4.p1.21.m21.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.4.p1.21.m21.1.1.1.1.1.1.1.1.cmml" xref="S2.4.p1.21.m21.1.1.1.1.1.1.1">superscript</csymbol><ci id="S2.4.p1.21.m21.1.1.1.1.1.1.1.2.cmml" xref="S2.4.p1.21.m21.1.1.1.1.1.1.1.2">𝐵</ci><ci id="S2.4.p1.21.m21.1.1.1.1.1.1.1.3.cmml" xref="S2.4.p1.21.m21.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S2.4.p1.21.m21.2.2.2.2.cmml" xref="S2.4.p1.21.m21.2.2.2.2"><intersect id="S2.4.p1.21.m21.2.2.2.2.2.cmml" xref="S2.4.p1.21.m21.2.2.2.2.2"></intersect><apply id="S2.4.p1.21.m21.2.2.2.2.3.cmml" xref="S2.4.p1.21.m21.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.4.p1.21.m21.2.2.2.2.3.1.cmml" xref="S2.4.p1.21.m21.2.2.2.2.3">superscript</csymbol><ci id="S2.4.p1.21.m21.2.2.2.2.3.2.cmml" xref="S2.4.p1.21.m21.2.2.2.2.3.2">𝐵</ci><ci id="S2.4.p1.21.m21.2.2.2.2.3.3.cmml" xref="S2.4.p1.21.m21.2.2.2.2.3.3">′</ci></apply><apply id="S2.4.p1.21.m21.2.2.2.2.1.1.1.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1"><union id="S2.4.p1.21.m21.2.2.2.2.1.1.1.1.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.1"></union><apply id="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.1.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.2">superscript</csymbol><partialdiff id="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.2.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.2"></partialdiff><ci id="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.3.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.2.3">′</ci></apply><apply id="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.1.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.3">superscript</csymbol><ci id="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.2.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.2">𝐴</ci><ci id="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.3.cmml" xref="S2.4.p1.21.m21.2.2.2.2.1.1.1.3.3">′</ci></apply></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.21.m21.2c">(G[B^{\prime}],B^{\prime}\cap(\partial^{\prime}\cup A^{\prime}))</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.21.m21.2d">( italic_G [ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ ( ∂ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.5.p2"> <p class="ltx_p" id="S2.5.p2.31">We now show that <math alttext="\operatorname{\partial}_{\mathcal{T}}(x)=\partial_{x}" class="ltx_Math" display="inline" id="S2.5.p2.1.m1.2"><semantics id="S2.5.p2.1.m1.2a"><mrow id="S2.5.p2.1.m1.2.2" xref="S2.5.p2.1.m1.2.2.cmml"><mrow id="S2.5.p2.1.m1.2.2.1.1" xref="S2.5.p2.1.m1.2.2.1.2.cmml"><msub id="S2.5.p2.1.m1.2.2.1.1.1" xref="S2.5.p2.1.m1.2.2.1.1.1.cmml"><mi id="S2.5.p2.1.m1.2.2.1.1.1.2" mathvariant="normal" xref="S2.5.p2.1.m1.2.2.1.1.1.2.cmml">∂</mi><mi class="ltx_font_mathcaligraphic" id="S2.5.p2.1.m1.2.2.1.1.1.3" xref="S2.5.p2.1.m1.2.2.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.5.p2.1.m1.2.2.1.1a" xref="S2.5.p2.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="S2.5.p2.1.m1.2.2.1.1.2" xref="S2.5.p2.1.m1.2.2.1.2.cmml"><mo id="S2.5.p2.1.m1.2.2.1.1.2.1" stretchy="false" xref="S2.5.p2.1.m1.2.2.1.2.cmml">(</mo><mi id="S2.5.p2.1.m1.1.1" xref="S2.5.p2.1.m1.1.1.cmml">x</mi><mo id="S2.5.p2.1.m1.2.2.1.1.2.2" stretchy="false" xref="S2.5.p2.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.1.m1.2.2.2" rspace="0.1389em" xref="S2.5.p2.1.m1.2.2.2.cmml">=</mo><msub id="S2.5.p2.1.m1.2.2.3" xref="S2.5.p2.1.m1.2.2.3.cmml"><mo id="S2.5.p2.1.m1.2.2.3.2" lspace="0.1389em" xref="S2.5.p2.1.m1.2.2.3.2.cmml">∂</mo><mi id="S2.5.p2.1.m1.2.2.3.3" xref="S2.5.p2.1.m1.2.2.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.1.m1.2b"><apply id="S2.5.p2.1.m1.2.2.cmml" xref="S2.5.p2.1.m1.2.2"><eq id="S2.5.p2.1.m1.2.2.2.cmml" xref="S2.5.p2.1.m1.2.2.2"></eq><apply id="S2.5.p2.1.m1.2.2.1.2.cmml" xref="S2.5.p2.1.m1.2.2.1.1"><apply id="S2.5.p2.1.m1.2.2.1.1.1.cmml" xref="S2.5.p2.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.1.m1.2.2.1.1.1.1.cmml" xref="S2.5.p2.1.m1.2.2.1.1.1">subscript</csymbol><partialdiff id="S2.5.p2.1.m1.2.2.1.1.1.2.cmml" xref="S2.5.p2.1.m1.2.2.1.1.1.2"></partialdiff><ci id="S2.5.p2.1.m1.2.2.1.1.1.3.cmml" xref="S2.5.p2.1.m1.2.2.1.1.1.3">𝒯</ci></apply><ci id="S2.5.p2.1.m1.1.1.cmml" xref="S2.5.p2.1.m1.1.1">𝑥</ci></apply><apply id="S2.5.p2.1.m1.2.2.3.cmml" xref="S2.5.p2.1.m1.2.2.3"><csymbol cd="ambiguous" id="S2.5.p2.1.m1.2.2.3.1.cmml" xref="S2.5.p2.1.m1.2.2.3">subscript</csymbol><partialdiff id="S2.5.p2.1.m1.2.2.3.2.cmml" xref="S2.5.p2.1.m1.2.2.3.2"></partialdiff><ci id="S2.5.p2.1.m1.2.2.3.3.cmml" xref="S2.5.p2.1.m1.2.2.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.1.m1.2c">\operatorname{\partial}_{\mathcal{T}}(x)=\partial_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.1.m1.2d">∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) = ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, for each subtree <math alttext="T_{x}" class="ltx_Math" display="inline" id="S2.5.p2.2.m2.1"><semantics id="S2.5.p2.2.m2.1a"><msub id="S2.5.p2.2.m2.1.1" xref="S2.5.p2.2.m2.1.1.cmml"><mi id="S2.5.p2.2.m2.1.1.2" xref="S2.5.p2.2.m2.1.1.2.cmml">T</mi><mi id="S2.5.p2.2.m2.1.1.3" xref="S2.5.p2.2.m2.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S2.5.p2.2.m2.1b"><apply id="S2.5.p2.2.m2.1.1.cmml" xref="S2.5.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.5.p2.2.m2.1.1.1.cmml" xref="S2.5.p2.2.m2.1.1">subscript</csymbol><ci id="S2.5.p2.2.m2.1.1.2.cmml" xref="S2.5.p2.2.m2.1.1.2">𝑇</ci><ci id="S2.5.p2.2.m2.1.1.3.cmml" xref="S2.5.p2.2.m2.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.2.m2.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> rooted at <math alttext="x\in V(T)" class="ltx_Math" display="inline" id="S2.5.p2.3.m3.1"><semantics id="S2.5.p2.3.m3.1a"><mrow id="S2.5.p2.3.m3.1.2" xref="S2.5.p2.3.m3.1.2.cmml"><mi id="S2.5.p2.3.m3.1.2.2" xref="S2.5.p2.3.m3.1.2.2.cmml">x</mi><mo id="S2.5.p2.3.m3.1.2.1" xref="S2.5.p2.3.m3.1.2.1.cmml">∈</mo><mrow id="S2.5.p2.3.m3.1.2.3" xref="S2.5.p2.3.m3.1.2.3.cmml"><mi id="S2.5.p2.3.m3.1.2.3.2" xref="S2.5.p2.3.m3.1.2.3.2.cmml">V</mi><mo id="S2.5.p2.3.m3.1.2.3.1" xref="S2.5.p2.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S2.5.p2.3.m3.1.2.3.3.2" xref="S2.5.p2.3.m3.1.2.3.cmml"><mo id="S2.5.p2.3.m3.1.2.3.3.2.1" stretchy="false" xref="S2.5.p2.3.m3.1.2.3.cmml">(</mo><mi id="S2.5.p2.3.m3.1.1" xref="S2.5.p2.3.m3.1.1.cmml">T</mi><mo id="S2.5.p2.3.m3.1.2.3.3.2.2" stretchy="false" xref="S2.5.p2.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.3.m3.1b"><apply id="S2.5.p2.3.m3.1.2.cmml" xref="S2.5.p2.3.m3.1.2"><in id="S2.5.p2.3.m3.1.2.1.cmml" xref="S2.5.p2.3.m3.1.2.1"></in><ci id="S2.5.p2.3.m3.1.2.2.cmml" xref="S2.5.p2.3.m3.1.2.2">𝑥</ci><apply id="S2.5.p2.3.m3.1.2.3.cmml" xref="S2.5.p2.3.m3.1.2.3"><times id="S2.5.p2.3.m3.1.2.3.1.cmml" xref="S2.5.p2.3.m3.1.2.3.1"></times><ci id="S2.5.p2.3.m3.1.2.3.2.cmml" xref="S2.5.p2.3.m3.1.2.3.2">𝑉</ci><ci id="S2.5.p2.3.m3.1.1.cmml" xref="S2.5.p2.3.m3.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.3.m3.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.3.m3.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math> that was constructed by a recursive invocation on <math alttext="(G_{x},\partial_{x})" class="ltx_Math" display="inline" id="S2.5.p2.4.m4.2"><semantics id="S2.5.p2.4.m4.2a"><mrow id="S2.5.p2.4.m4.2.2.2" xref="S2.5.p2.4.m4.2.2.3.cmml"><mo id="S2.5.p2.4.m4.2.2.2.3" stretchy="false" xref="S2.5.p2.4.m4.2.2.3.cmml">(</mo><msub id="S2.5.p2.4.m4.1.1.1.1" xref="S2.5.p2.4.m4.1.1.1.1.cmml"><mi id="S2.5.p2.4.m4.1.1.1.1.2" xref="S2.5.p2.4.m4.1.1.1.1.2.cmml">G</mi><mi id="S2.5.p2.4.m4.1.1.1.1.3" xref="S2.5.p2.4.m4.1.1.1.1.3.cmml">x</mi></msub><mo id="S2.5.p2.4.m4.2.2.2.4" xref="S2.5.p2.4.m4.2.2.3.cmml">,</mo><msub id="S2.5.p2.4.m4.2.2.2.2" xref="S2.5.p2.4.m4.2.2.2.2.cmml"><mo id="S2.5.p2.4.m4.2.2.2.2.2" lspace="0em" rspace="0em" xref="S2.5.p2.4.m4.2.2.2.2.2.cmml">∂</mo><mi id="S2.5.p2.4.m4.2.2.2.2.3" xref="S2.5.p2.4.m4.2.2.2.2.3.cmml">x</mi></msub><mo id="S2.5.p2.4.m4.2.2.2.5" stretchy="false" xref="S2.5.p2.4.m4.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.4.m4.2b"><interval closure="open" id="S2.5.p2.4.m4.2.2.3.cmml" xref="S2.5.p2.4.m4.2.2.2"><apply id="S2.5.p2.4.m4.1.1.1.1.cmml" xref="S2.5.p2.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.4.m4.1.1.1.1.1.cmml" xref="S2.5.p2.4.m4.1.1.1.1">subscript</csymbol><ci id="S2.5.p2.4.m4.1.1.1.1.2.cmml" xref="S2.5.p2.4.m4.1.1.1.1.2">𝐺</ci><ci id="S2.5.p2.4.m4.1.1.1.1.3.cmml" xref="S2.5.p2.4.m4.1.1.1.1.3">𝑥</ci></apply><apply id="S2.5.p2.4.m4.2.2.2.2.cmml" xref="S2.5.p2.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S2.5.p2.4.m4.2.2.2.2.1.cmml" xref="S2.5.p2.4.m4.2.2.2.2">subscript</csymbol><partialdiff id="S2.5.p2.4.m4.2.2.2.2.2.cmml" xref="S2.5.p2.4.m4.2.2.2.2.2"></partialdiff><ci id="S2.5.p2.4.m4.2.2.2.2.3.cmml" xref="S2.5.p2.4.m4.2.2.2.2.3">𝑥</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.4.m4.2c">(G_{x},\partial_{x})</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.4.m4.2d">( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math>. If <math alttext="\operatorname{depth}_{T}(x)=0" class="ltx_Math" display="inline" id="S2.5.p2.5.m5.2"><semantics id="S2.5.p2.5.m5.2a"><mrow id="S2.5.p2.5.m5.2.2" xref="S2.5.p2.5.m5.2.2.cmml"><mrow id="S2.5.p2.5.m5.2.2.1.1" xref="S2.5.p2.5.m5.2.2.1.2.cmml"><msub id="S2.5.p2.5.m5.2.2.1.1.1" xref="S2.5.p2.5.m5.2.2.1.1.1.cmml"><mi id="S2.5.p2.5.m5.2.2.1.1.1.2" xref="S2.5.p2.5.m5.2.2.1.1.1.2.cmml">depth</mi><mi id="S2.5.p2.5.m5.2.2.1.1.1.3" xref="S2.5.p2.5.m5.2.2.1.1.1.3.cmml">T</mi></msub><mo id="S2.5.p2.5.m5.2.2.1.1a" xref="S2.5.p2.5.m5.2.2.1.2.cmml">⁡</mo><mrow id="S2.5.p2.5.m5.2.2.1.1.2" xref="S2.5.p2.5.m5.2.2.1.2.cmml"><mo id="S2.5.p2.5.m5.2.2.1.1.2.1" stretchy="false" xref="S2.5.p2.5.m5.2.2.1.2.cmml">(</mo><mi id="S2.5.p2.5.m5.1.1" xref="S2.5.p2.5.m5.1.1.cmml">x</mi><mo id="S2.5.p2.5.m5.2.2.1.1.2.2" stretchy="false" xref="S2.5.p2.5.m5.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.5.m5.2.2.2" xref="S2.5.p2.5.m5.2.2.2.cmml">=</mo><mn id="S2.5.p2.5.m5.2.2.3" xref="S2.5.p2.5.m5.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.5.m5.2b"><apply id="S2.5.p2.5.m5.2.2.cmml" xref="S2.5.p2.5.m5.2.2"><eq id="S2.5.p2.5.m5.2.2.2.cmml" xref="S2.5.p2.5.m5.2.2.2"></eq><apply id="S2.5.p2.5.m5.2.2.1.2.cmml" xref="S2.5.p2.5.m5.2.2.1.1"><apply id="S2.5.p2.5.m5.2.2.1.1.1.cmml" xref="S2.5.p2.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.5.m5.2.2.1.1.1.1.cmml" xref="S2.5.p2.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S2.5.p2.5.m5.2.2.1.1.1.2.cmml" xref="S2.5.p2.5.m5.2.2.1.1.1.2">depth</ci><ci id="S2.5.p2.5.m5.2.2.1.1.1.3.cmml" xref="S2.5.p2.5.m5.2.2.1.1.1.3">𝑇</ci></apply><ci id="S2.5.p2.5.m5.1.1.cmml" xref="S2.5.p2.5.m5.1.1">𝑥</ci></apply><cn id="S2.5.p2.5.m5.2.2.3.cmml" type="integer" xref="S2.5.p2.5.m5.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.5.m5.2c">\operatorname{depth}_{T}(x)=0</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.5.m5.2d">roman_depth start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x ) = 0</annotation></semantics></math> then <math alttext="x" class="ltx_Math" display="inline" id="S2.5.p2.6.m6.1"><semantics id="S2.5.p2.6.m6.1a"><mi id="S2.5.p2.6.m6.1.1" xref="S2.5.p2.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.6.m6.1b"><ci id="S2.5.p2.6.m6.1.1.cmml" xref="S2.5.p2.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.6.m6.1d">italic_x</annotation></semantics></math> is the root of <math alttext="T" class="ltx_Math" display="inline" id="S2.5.p2.7.m7.1"><semantics id="S2.5.p2.7.m7.1a"><mi id="S2.5.p2.7.m7.1.1" xref="S2.5.p2.7.m7.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.7.m7.1b"><ci id="S2.5.p2.7.m7.1.1.cmml" xref="S2.5.p2.7.m7.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.7.m7.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.7.m7.1d">italic_T</annotation></semantics></math> and <math alttext="\operatorname{\partial}_{\mathcal{T}}(x)=\emptyset=\partial_{x}" class="ltx_Math" display="inline" id="S2.5.p2.8.m8.2"><semantics id="S2.5.p2.8.m8.2a"><mrow id="S2.5.p2.8.m8.2.2" xref="S2.5.p2.8.m8.2.2.cmml"><mrow id="S2.5.p2.8.m8.2.2.1.1" xref="S2.5.p2.8.m8.2.2.1.2.cmml"><msub id="S2.5.p2.8.m8.2.2.1.1.1" xref="S2.5.p2.8.m8.2.2.1.1.1.cmml"><mi id="S2.5.p2.8.m8.2.2.1.1.1.2" mathvariant="normal" xref="S2.5.p2.8.m8.2.2.1.1.1.2.cmml">∂</mi><mi class="ltx_font_mathcaligraphic" id="S2.5.p2.8.m8.2.2.1.1.1.3" xref="S2.5.p2.8.m8.2.2.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.5.p2.8.m8.2.2.1.1a" xref="S2.5.p2.8.m8.2.2.1.2.cmml">⁡</mo><mrow id="S2.5.p2.8.m8.2.2.1.1.2" xref="S2.5.p2.8.m8.2.2.1.2.cmml"><mo id="S2.5.p2.8.m8.2.2.1.1.2.1" stretchy="false" xref="S2.5.p2.8.m8.2.2.1.2.cmml">(</mo><mi id="S2.5.p2.8.m8.1.1" xref="S2.5.p2.8.m8.1.1.cmml">x</mi><mo id="S2.5.p2.8.m8.2.2.1.1.2.2" stretchy="false" xref="S2.5.p2.8.m8.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.8.m8.2.2.3" xref="S2.5.p2.8.m8.2.2.3.cmml">=</mo><mi id="S2.5.p2.8.m8.2.2.4" mathvariant="normal" xref="S2.5.p2.8.m8.2.2.4.cmml">∅</mi><mo id="S2.5.p2.8.m8.2.2.5" rspace="0.1389em" xref="S2.5.p2.8.m8.2.2.5.cmml">=</mo><msub id="S2.5.p2.8.m8.2.2.6" xref="S2.5.p2.8.m8.2.2.6.cmml"><mo id="S2.5.p2.8.m8.2.2.6.2" lspace="0.1389em" xref="S2.5.p2.8.m8.2.2.6.2.cmml">∂</mo><mi id="S2.5.p2.8.m8.2.2.6.3" xref="S2.5.p2.8.m8.2.2.6.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.8.m8.2b"><apply id="S2.5.p2.8.m8.2.2.cmml" xref="S2.5.p2.8.m8.2.2"><and id="S2.5.p2.8.m8.2.2a.cmml" xref="S2.5.p2.8.m8.2.2"></and><apply id="S2.5.p2.8.m8.2.2b.cmml" xref="S2.5.p2.8.m8.2.2"><eq id="S2.5.p2.8.m8.2.2.3.cmml" xref="S2.5.p2.8.m8.2.2.3"></eq><apply id="S2.5.p2.8.m8.2.2.1.2.cmml" xref="S2.5.p2.8.m8.2.2.1.1"><apply id="S2.5.p2.8.m8.2.2.1.1.1.cmml" xref="S2.5.p2.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.8.m8.2.2.1.1.1.1.cmml" xref="S2.5.p2.8.m8.2.2.1.1.1">subscript</csymbol><partialdiff id="S2.5.p2.8.m8.2.2.1.1.1.2.cmml" xref="S2.5.p2.8.m8.2.2.1.1.1.2"></partialdiff><ci id="S2.5.p2.8.m8.2.2.1.1.1.3.cmml" xref="S2.5.p2.8.m8.2.2.1.1.1.3">𝒯</ci></apply><ci id="S2.5.p2.8.m8.1.1.cmml" xref="S2.5.p2.8.m8.1.1">𝑥</ci></apply><emptyset id="S2.5.p2.8.m8.2.2.4.cmml" xref="S2.5.p2.8.m8.2.2.4"></emptyset></apply><apply id="S2.5.p2.8.m8.2.2c.cmml" xref="S2.5.p2.8.m8.2.2"><eq id="S2.5.p2.8.m8.2.2.5.cmml" xref="S2.5.p2.8.m8.2.2.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.5.p2.8.m8.2.2.4.cmml" id="S2.5.p2.8.m8.2.2d.cmml" xref="S2.5.p2.8.m8.2.2"></share><apply id="S2.5.p2.8.m8.2.2.6.cmml" xref="S2.5.p2.8.m8.2.2.6"><csymbol cd="ambiguous" id="S2.5.p2.8.m8.2.2.6.1.cmml" xref="S2.5.p2.8.m8.2.2.6">subscript</csymbol><partialdiff id="S2.5.p2.8.m8.2.2.6.2.cmml" xref="S2.5.p2.8.m8.2.2.6.2"></partialdiff><ci id="S2.5.p2.8.m8.2.2.6.3.cmml" xref="S2.5.p2.8.m8.2.2.6.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.8.m8.2c">\operatorname{\partial}_{\mathcal{T}}(x)=\emptyset=\partial_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.8.m8.2d">∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) = ∅ = ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, by definition. Now suppose <math alttext="\operatorname{depth}_{T}(x)\geq 1" class="ltx_Math" display="inline" id="S2.5.p2.9.m9.2"><semantics id="S2.5.p2.9.m9.2a"><mrow id="S2.5.p2.9.m9.2.2" xref="S2.5.p2.9.m9.2.2.cmml"><mrow id="S2.5.p2.9.m9.2.2.1.1" xref="S2.5.p2.9.m9.2.2.1.2.cmml"><msub id="S2.5.p2.9.m9.2.2.1.1.1" xref="S2.5.p2.9.m9.2.2.1.1.1.cmml"><mi id="S2.5.p2.9.m9.2.2.1.1.1.2" xref="S2.5.p2.9.m9.2.2.1.1.1.2.cmml">depth</mi><mi id="S2.5.p2.9.m9.2.2.1.1.1.3" xref="S2.5.p2.9.m9.2.2.1.1.1.3.cmml">T</mi></msub><mo id="S2.5.p2.9.m9.2.2.1.1a" xref="S2.5.p2.9.m9.2.2.1.2.cmml">⁡</mo><mrow id="S2.5.p2.9.m9.2.2.1.1.2" xref="S2.5.p2.9.m9.2.2.1.2.cmml"><mo id="S2.5.p2.9.m9.2.2.1.1.2.1" stretchy="false" xref="S2.5.p2.9.m9.2.2.1.2.cmml">(</mo><mi id="S2.5.p2.9.m9.1.1" xref="S2.5.p2.9.m9.1.1.cmml">x</mi><mo id="S2.5.p2.9.m9.2.2.1.1.2.2" stretchy="false" xref="S2.5.p2.9.m9.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.9.m9.2.2.2" xref="S2.5.p2.9.m9.2.2.2.cmml">≥</mo><mn id="S2.5.p2.9.m9.2.2.3" xref="S2.5.p2.9.m9.2.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.9.m9.2b"><apply id="S2.5.p2.9.m9.2.2.cmml" xref="S2.5.p2.9.m9.2.2"><geq id="S2.5.p2.9.m9.2.2.2.cmml" xref="S2.5.p2.9.m9.2.2.2"></geq><apply id="S2.5.p2.9.m9.2.2.1.2.cmml" xref="S2.5.p2.9.m9.2.2.1.1"><apply id="S2.5.p2.9.m9.2.2.1.1.1.cmml" xref="S2.5.p2.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.9.m9.2.2.1.1.1.1.cmml" xref="S2.5.p2.9.m9.2.2.1.1.1">subscript</csymbol><ci id="S2.5.p2.9.m9.2.2.1.1.1.2.cmml" xref="S2.5.p2.9.m9.2.2.1.1.1.2">depth</ci><ci id="S2.5.p2.9.m9.2.2.1.1.1.3.cmml" xref="S2.5.p2.9.m9.2.2.1.1.1.3">𝑇</ci></apply><ci id="S2.5.p2.9.m9.1.1.cmml" xref="S2.5.p2.9.m9.1.1">𝑥</ci></apply><cn id="S2.5.p2.9.m9.2.2.3.cmml" type="integer" xref="S2.5.p2.9.m9.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.9.m9.2c">\operatorname{depth}_{T}(x)\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.9.m9.2d">roman_depth start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x ) ≥ 1</annotation></semantics></math>, the parent of <math alttext="x" class="ltx_Math" display="inline" id="S2.5.p2.10.m10.1"><semantics id="S2.5.p2.10.m10.1a"><mi id="S2.5.p2.10.m10.1.1" xref="S2.5.p2.10.m10.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.10.m10.1b"><ci id="S2.5.p2.10.m10.1.1.cmml" xref="S2.5.p2.10.m10.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.10.m10.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.10.m10.1d">italic_x</annotation></semantics></math> is <math alttext="y" class="ltx_Math" display="inline" id="S2.5.p2.11.m11.1"><semantics id="S2.5.p2.11.m11.1a"><mi id="S2.5.p2.11.m11.1.1" xref="S2.5.p2.11.m11.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.11.m11.1b"><ci id="S2.5.p2.11.m11.1.1.cmml" xref="S2.5.p2.11.m11.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.11.m11.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.11.m11.1d">italic_y</annotation></semantics></math> and <math alttext="T_{y}" class="ltx_Math" display="inline" id="S2.5.p2.12.m12.1"><semantics id="S2.5.p2.12.m12.1a"><msub id="S2.5.p2.12.m12.1.1" xref="S2.5.p2.12.m12.1.1.cmml"><mi id="S2.5.p2.12.m12.1.1.2" xref="S2.5.p2.12.m12.1.1.2.cmml">T</mi><mi id="S2.5.p2.12.m12.1.1.3" xref="S2.5.p2.12.m12.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S2.5.p2.12.m12.1b"><apply id="S2.5.p2.12.m12.1.1.cmml" xref="S2.5.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S2.5.p2.12.m12.1.1.1.cmml" xref="S2.5.p2.12.m12.1.1">subscript</csymbol><ci id="S2.5.p2.12.m12.1.1.2.cmml" xref="S2.5.p2.12.m12.1.1.2">𝑇</ci><ci id="S2.5.p2.12.m12.1.1.3.cmml" xref="S2.5.p2.12.m12.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.12.m12.1c">T_{y}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.12.m12.1d">italic_T start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> is the result of a recursive invocation on <math alttext="(G_{y},\partial_{y})" class="ltx_Math" display="inline" id="S2.5.p2.13.m13.2"><semantics id="S2.5.p2.13.m13.2a"><mrow id="S2.5.p2.13.m13.2.2.2" xref="S2.5.p2.13.m13.2.2.3.cmml"><mo id="S2.5.p2.13.m13.2.2.2.3" stretchy="false" xref="S2.5.p2.13.m13.2.2.3.cmml">(</mo><msub id="S2.5.p2.13.m13.1.1.1.1" xref="S2.5.p2.13.m13.1.1.1.1.cmml"><mi id="S2.5.p2.13.m13.1.1.1.1.2" xref="S2.5.p2.13.m13.1.1.1.1.2.cmml">G</mi><mi id="S2.5.p2.13.m13.1.1.1.1.3" xref="S2.5.p2.13.m13.1.1.1.1.3.cmml">y</mi></msub><mo id="S2.5.p2.13.m13.2.2.2.4" xref="S2.5.p2.13.m13.2.2.3.cmml">,</mo><msub id="S2.5.p2.13.m13.2.2.2.2" xref="S2.5.p2.13.m13.2.2.2.2.cmml"><mo id="S2.5.p2.13.m13.2.2.2.2.2" lspace="0em" rspace="0em" xref="S2.5.p2.13.m13.2.2.2.2.2.cmml">∂</mo><mi id="S2.5.p2.13.m13.2.2.2.2.3" xref="S2.5.p2.13.m13.2.2.2.2.3.cmml">y</mi></msub><mo id="S2.5.p2.13.m13.2.2.2.5" stretchy="false" xref="S2.5.p2.13.m13.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.13.m13.2b"><interval closure="open" id="S2.5.p2.13.m13.2.2.3.cmml" xref="S2.5.p2.13.m13.2.2.2"><apply id="S2.5.p2.13.m13.1.1.1.1.cmml" xref="S2.5.p2.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.13.m13.1.1.1.1.1.cmml" xref="S2.5.p2.13.m13.1.1.1.1">subscript</csymbol><ci id="S2.5.p2.13.m13.1.1.1.1.2.cmml" xref="S2.5.p2.13.m13.1.1.1.1.2">𝐺</ci><ci id="S2.5.p2.13.m13.1.1.1.1.3.cmml" xref="S2.5.p2.13.m13.1.1.1.1.3">𝑦</ci></apply><apply id="S2.5.p2.13.m13.2.2.2.2.cmml" xref="S2.5.p2.13.m13.2.2.2.2"><csymbol cd="ambiguous" id="S2.5.p2.13.m13.2.2.2.2.1.cmml" xref="S2.5.p2.13.m13.2.2.2.2">subscript</csymbol><partialdiff id="S2.5.p2.13.m13.2.2.2.2.2.cmml" xref="S2.5.p2.13.m13.2.2.2.2.2"></partialdiff><ci id="S2.5.p2.13.m13.2.2.2.2.3.cmml" xref="S2.5.p2.13.m13.2.2.2.2.3">𝑦</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.13.m13.2c">(G_{y},\partial_{y})</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.13.m13.2d">( italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , ∂ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT )</annotation></semantics></math>. Without loss of generality, <math alttext="G_{x}=G[A^{y}]" class="ltx_Math" display="inline" id="S2.5.p2.14.m14.1"><semantics id="S2.5.p2.14.m14.1a"><mrow id="S2.5.p2.14.m14.1.1" xref="S2.5.p2.14.m14.1.1.cmml"><msub id="S2.5.p2.14.m14.1.1.3" xref="S2.5.p2.14.m14.1.1.3.cmml"><mi id="S2.5.p2.14.m14.1.1.3.2" xref="S2.5.p2.14.m14.1.1.3.2.cmml">G</mi><mi id="S2.5.p2.14.m14.1.1.3.3" xref="S2.5.p2.14.m14.1.1.3.3.cmml">x</mi></msub><mo id="S2.5.p2.14.m14.1.1.2" xref="S2.5.p2.14.m14.1.1.2.cmml">=</mo><mrow id="S2.5.p2.14.m14.1.1.1" xref="S2.5.p2.14.m14.1.1.1.cmml"><mi id="S2.5.p2.14.m14.1.1.1.3" xref="S2.5.p2.14.m14.1.1.1.3.cmml">G</mi><mo id="S2.5.p2.14.m14.1.1.1.2" xref="S2.5.p2.14.m14.1.1.1.2.cmml">⁢</mo><mrow id="S2.5.p2.14.m14.1.1.1.1.1" xref="S2.5.p2.14.m14.1.1.1.1.2.cmml"><mo id="S2.5.p2.14.m14.1.1.1.1.1.2" stretchy="false" xref="S2.5.p2.14.m14.1.1.1.1.2.1.cmml">[</mo><msup id="S2.5.p2.14.m14.1.1.1.1.1.1" xref="S2.5.p2.14.m14.1.1.1.1.1.1.cmml"><mi id="S2.5.p2.14.m14.1.1.1.1.1.1.2" xref="S2.5.p2.14.m14.1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.5.p2.14.m14.1.1.1.1.1.1.3" xref="S2.5.p2.14.m14.1.1.1.1.1.1.3.cmml">y</mi></msup><mo id="S2.5.p2.14.m14.1.1.1.1.1.3" stretchy="false" xref="S2.5.p2.14.m14.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.14.m14.1b"><apply id="S2.5.p2.14.m14.1.1.cmml" xref="S2.5.p2.14.m14.1.1"><eq id="S2.5.p2.14.m14.1.1.2.cmml" xref="S2.5.p2.14.m14.1.1.2"></eq><apply id="S2.5.p2.14.m14.1.1.3.cmml" xref="S2.5.p2.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.14.m14.1.1.3.1.cmml" xref="S2.5.p2.14.m14.1.1.3">subscript</csymbol><ci id="S2.5.p2.14.m14.1.1.3.2.cmml" xref="S2.5.p2.14.m14.1.1.3.2">𝐺</ci><ci id="S2.5.p2.14.m14.1.1.3.3.cmml" xref="S2.5.p2.14.m14.1.1.3.3">𝑥</ci></apply><apply id="S2.5.p2.14.m14.1.1.1.cmml" xref="S2.5.p2.14.m14.1.1.1"><times id="S2.5.p2.14.m14.1.1.1.2.cmml" xref="S2.5.p2.14.m14.1.1.1.2"></times><ci id="S2.5.p2.14.m14.1.1.1.3.cmml" xref="S2.5.p2.14.m14.1.1.1.3">𝐺</ci><apply id="S2.5.p2.14.m14.1.1.1.1.2.cmml" xref="S2.5.p2.14.m14.1.1.1.1.1"><csymbol cd="latexml" id="S2.5.p2.14.m14.1.1.1.1.2.1.cmml" xref="S2.5.p2.14.m14.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.5.p2.14.m14.1.1.1.1.1.1.cmml" xref="S2.5.p2.14.m14.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.14.m14.1.1.1.1.1.1.1.cmml" xref="S2.5.p2.14.m14.1.1.1.1.1.1">superscript</csymbol><ci id="S2.5.p2.14.m14.1.1.1.1.1.1.2.cmml" xref="S2.5.p2.14.m14.1.1.1.1.1.1.2">𝐴</ci><ci id="S2.5.p2.14.m14.1.1.1.1.1.1.3.cmml" xref="S2.5.p2.14.m14.1.1.1.1.1.1.3">𝑦</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.14.m14.1c">G_{x}=G[A^{y}]</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.14.m14.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_G [ italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ]</annotation></semantics></math> where <math alttext="(A^{y},B^{y})" class="ltx_Math" display="inline" id="S2.5.p2.15.m15.2"><semantics id="S2.5.p2.15.m15.2a"><mrow id="S2.5.p2.15.m15.2.2.2" xref="S2.5.p2.15.m15.2.2.3.cmml"><mo id="S2.5.p2.15.m15.2.2.2.3" stretchy="false" xref="S2.5.p2.15.m15.2.2.3.cmml">(</mo><msup id="S2.5.p2.15.m15.1.1.1.1" xref="S2.5.p2.15.m15.1.1.1.1.cmml"><mi id="S2.5.p2.15.m15.1.1.1.1.2" xref="S2.5.p2.15.m15.1.1.1.1.2.cmml">A</mi><mi id="S2.5.p2.15.m15.1.1.1.1.3" xref="S2.5.p2.15.m15.1.1.1.1.3.cmml">y</mi></msup><mo id="S2.5.p2.15.m15.2.2.2.4" xref="S2.5.p2.15.m15.2.2.3.cmml">,</mo><msup id="S2.5.p2.15.m15.2.2.2.2" xref="S2.5.p2.15.m15.2.2.2.2.cmml"><mi id="S2.5.p2.15.m15.2.2.2.2.2" xref="S2.5.p2.15.m15.2.2.2.2.2.cmml">B</mi><mi id="S2.5.p2.15.m15.2.2.2.2.3" xref="S2.5.p2.15.m15.2.2.2.2.3.cmml">y</mi></msup><mo id="S2.5.p2.15.m15.2.2.2.5" stretchy="false" xref="S2.5.p2.15.m15.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.15.m15.2b"><interval closure="open" id="S2.5.p2.15.m15.2.2.3.cmml" xref="S2.5.p2.15.m15.2.2.2"><apply id="S2.5.p2.15.m15.1.1.1.1.cmml" xref="S2.5.p2.15.m15.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.15.m15.1.1.1.1.1.cmml" xref="S2.5.p2.15.m15.1.1.1.1">superscript</csymbol><ci id="S2.5.p2.15.m15.1.1.1.1.2.cmml" xref="S2.5.p2.15.m15.1.1.1.1.2">𝐴</ci><ci id="S2.5.p2.15.m15.1.1.1.1.3.cmml" xref="S2.5.p2.15.m15.1.1.1.1.3">𝑦</ci></apply><apply id="S2.5.p2.15.m15.2.2.2.2.cmml" xref="S2.5.p2.15.m15.2.2.2.2"><csymbol cd="ambiguous" id="S2.5.p2.15.m15.2.2.2.2.1.cmml" xref="S2.5.p2.15.m15.2.2.2.2">superscript</csymbol><ci id="S2.5.p2.15.m15.2.2.2.2.2.cmml" xref="S2.5.p2.15.m15.2.2.2.2.2">𝐵</ci><ci id="S2.5.p2.15.m15.2.2.2.2.3.cmml" xref="S2.5.p2.15.m15.2.2.2.2.3">𝑦</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.15.m15.2c">(A^{y},B^{y})</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.15.m15.2d">( italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT , italic_B start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT )</annotation></semantics></math> is a separation of <math alttext="G_{y}-\partial_{y}" class="ltx_Math" display="inline" id="S2.5.p2.16.m16.1"><semantics id="S2.5.p2.16.m16.1a"><mrow id="S2.5.p2.16.m16.1.1" xref="S2.5.p2.16.m16.1.1.cmml"><msub id="S2.5.p2.16.m16.1.1.2" xref="S2.5.p2.16.m16.1.1.2.cmml"><mi id="S2.5.p2.16.m16.1.1.2.2" xref="S2.5.p2.16.m16.1.1.2.2.cmml">G</mi><mi id="S2.5.p2.16.m16.1.1.2.3" xref="S2.5.p2.16.m16.1.1.2.3.cmml">y</mi></msub><mo id="S2.5.p2.16.m16.1.1.1" xref="S2.5.p2.16.m16.1.1.1.cmml">−</mo><msub id="S2.5.p2.16.m16.1.1.3" xref="S2.5.p2.16.m16.1.1.3.cmml"><mo id="S2.5.p2.16.m16.1.1.3.2" lspace="0em" xref="S2.5.p2.16.m16.1.1.3.2.cmml">∂</mo><mi id="S2.5.p2.16.m16.1.1.3.3" xref="S2.5.p2.16.m16.1.1.3.3.cmml">y</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.16.m16.1b"><apply id="S2.5.p2.16.m16.1.1.cmml" xref="S2.5.p2.16.m16.1.1"><minus id="S2.5.p2.16.m16.1.1.1.cmml" xref="S2.5.p2.16.m16.1.1.1"></minus><apply id="S2.5.p2.16.m16.1.1.2.cmml" xref="S2.5.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="S2.5.p2.16.m16.1.1.2.1.cmml" xref="S2.5.p2.16.m16.1.1.2">subscript</csymbol><ci id="S2.5.p2.16.m16.1.1.2.2.cmml" xref="S2.5.p2.16.m16.1.1.2.2">𝐺</ci><ci id="S2.5.p2.16.m16.1.1.2.3.cmml" xref="S2.5.p2.16.m16.1.1.2.3">𝑦</ci></apply><apply id="S2.5.p2.16.m16.1.1.3.cmml" xref="S2.5.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.16.m16.1.1.3.1.cmml" xref="S2.5.p2.16.m16.1.1.3">subscript</csymbol><partialdiff id="S2.5.p2.16.m16.1.1.3.2.cmml" xref="S2.5.p2.16.m16.1.1.3.2"></partialdiff><ci id="S2.5.p2.16.m16.1.1.3.3.cmml" xref="S2.5.p2.16.m16.1.1.3.3">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.16.m16.1c">G_{y}-\partial_{y}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.16.m16.1d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - ∂ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\partial_{x}=(A^{y}\cap(\partial_{y}\cup B^{y})=(A^{y}\cap(\partial_{y}\cup(A^% {y}\cap B^{y}))=A^{y}\cap B_{y}" class="ltx_math_unparsed" display="inline" id="S2.5.p2.17.m17.1"><semantics id="S2.5.p2.17.m17.1a"><mrow id="S2.5.p2.17.m17.1b"><msub id="S2.5.p2.17.m17.1.1"><mo id="S2.5.p2.17.m17.1.1.2">∂</mo><mi id="S2.5.p2.17.m17.1.1.3">x</mi></msub><mo id="S2.5.p2.17.m17.1.2" lspace="0.278em">=</mo><mrow id="S2.5.p2.17.m17.1.3"><mo id="S2.5.p2.17.m17.1.3.1" stretchy="false">(</mo><msup id="S2.5.p2.17.m17.1.3.2"><mi id="S2.5.p2.17.m17.1.3.2.2">A</mi><mi id="S2.5.p2.17.m17.1.3.2.3">y</mi></msup><mo id="S2.5.p2.17.m17.1.3.3">∩</mo><mrow id="S2.5.p2.17.m17.1.3.4"><mo id="S2.5.p2.17.m17.1.3.4.1" stretchy="false">(</mo><msub id="S2.5.p2.17.m17.1.3.4.2"><mo id="S2.5.p2.17.m17.1.3.4.2.2" lspace="0em" rspace="0em">∂</mo><mi id="S2.5.p2.17.m17.1.3.4.2.3">y</mi></msub><mo id="S2.5.p2.17.m17.1.3.4.3">∪</mo><msup id="S2.5.p2.17.m17.1.3.4.4"><mi id="S2.5.p2.17.m17.1.3.4.4.2">B</mi><mi id="S2.5.p2.17.m17.1.3.4.4.3">y</mi></msup><mo id="S2.5.p2.17.m17.1.3.4.5" stretchy="false">)</mo></mrow><mo id="S2.5.p2.17.m17.1.3.5">=</mo><mrow id="S2.5.p2.17.m17.1.3.6"><mo id="S2.5.p2.17.m17.1.3.6.1" stretchy="false">(</mo><msup id="S2.5.p2.17.m17.1.3.6.2"><mi id="S2.5.p2.17.m17.1.3.6.2.2">A</mi><mi id="S2.5.p2.17.m17.1.3.6.2.3">y</mi></msup><mo id="S2.5.p2.17.m17.1.3.6.3">∩</mo><mrow id="S2.5.p2.17.m17.1.3.6.4"><mo id="S2.5.p2.17.m17.1.3.6.4.1" stretchy="false">(</mo><msub id="S2.5.p2.17.m17.1.3.6.4.2"><mo id="S2.5.p2.17.m17.1.3.6.4.2.2" lspace="0em" rspace="0em">∂</mo><mi id="S2.5.p2.17.m17.1.3.6.4.2.3">y</mi></msub><mo id="S2.5.p2.17.m17.1.3.6.4.3">∪</mo><mrow id="S2.5.p2.17.m17.1.3.6.4.4"><mo id="S2.5.p2.17.m17.1.3.6.4.4.1" stretchy="false">(</mo><msup id="S2.5.p2.17.m17.1.3.6.4.4.2"><mi id="S2.5.p2.17.m17.1.3.6.4.4.2.2">A</mi><mi id="S2.5.p2.17.m17.1.3.6.4.4.2.3">y</mi></msup><mo id="S2.5.p2.17.m17.1.3.6.4.4.3">∩</mo><msup id="S2.5.p2.17.m17.1.3.6.4.4.4"><mi id="S2.5.p2.17.m17.1.3.6.4.4.4.2">B</mi><mi id="S2.5.p2.17.m17.1.3.6.4.4.4.3">y</mi></msup><mo id="S2.5.p2.17.m17.1.3.6.4.4.5" stretchy="false">)</mo></mrow><mo id="S2.5.p2.17.m17.1.3.6.4.5" stretchy="false">)</mo></mrow><mo id="S2.5.p2.17.m17.1.3.6.5">=</mo><msup id="S2.5.p2.17.m17.1.3.6.6"><mi id="S2.5.p2.17.m17.1.3.6.6.2">A</mi><mi id="S2.5.p2.17.m17.1.3.6.6.3">y</mi></msup><mo id="S2.5.p2.17.m17.1.3.6.7">∩</mo><msub id="S2.5.p2.17.m17.1.3.6.8"><mi id="S2.5.p2.17.m17.1.3.6.8.2">B</mi><mi id="S2.5.p2.17.m17.1.3.6.8.3">y</mi></msub></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S2.5.p2.17.m17.1c">\partial_{x}=(A^{y}\cap(\partial_{y}\cup B^{y})=(A^{y}\cap(\partial_{y}\cup(A^% {y}\cap B^{y}))=A^{y}\cap B_{y}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.17.m17.1d">∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = ( italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ ( ∂ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∪ italic_B start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ) = ( italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ ( ∂ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∪ ( italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ) ) = italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. If <math alttext="x" class="ltx_Math" display="inline" id="S2.5.p2.18.m18.1"><semantics id="S2.5.p2.18.m18.1a"><mi id="S2.5.p2.18.m18.1.1" xref="S2.5.p2.18.m18.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.18.m18.1b"><ci id="S2.5.p2.18.m18.1.1.cmml" xref="S2.5.p2.18.m18.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.18.m18.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.18.m18.1d">italic_x</annotation></semantics></math> is a leaf of <math alttext="T" class="ltx_Math" display="inline" id="S2.5.p2.19.m19.1"><semantics id="S2.5.p2.19.m19.1a"><mi id="S2.5.p2.19.m19.1.1" xref="S2.5.p2.19.m19.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.19.m19.1b"><ci id="S2.5.p2.19.m19.1.1.cmml" xref="S2.5.p2.19.m19.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.19.m19.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.19.m19.1d">italic_T</annotation></semantics></math> then <math alttext="B_{x}=V(G_{x})=A^{y}" class="ltx_Math" display="inline" id="S2.5.p2.20.m20.1"><semantics id="S2.5.p2.20.m20.1a"><mrow id="S2.5.p2.20.m20.1.1" xref="S2.5.p2.20.m20.1.1.cmml"><msub id="S2.5.p2.20.m20.1.1.3" xref="S2.5.p2.20.m20.1.1.3.cmml"><mi id="S2.5.p2.20.m20.1.1.3.2" xref="S2.5.p2.20.m20.1.1.3.2.cmml">B</mi><mi id="S2.5.p2.20.m20.1.1.3.3" xref="S2.5.p2.20.m20.1.1.3.3.cmml">x</mi></msub><mo id="S2.5.p2.20.m20.1.1.4" xref="S2.5.p2.20.m20.1.1.4.cmml">=</mo><mrow id="S2.5.p2.20.m20.1.1.1" xref="S2.5.p2.20.m20.1.1.1.cmml"><mi id="S2.5.p2.20.m20.1.1.1.3" xref="S2.5.p2.20.m20.1.1.1.3.cmml">V</mi><mo id="S2.5.p2.20.m20.1.1.1.2" xref="S2.5.p2.20.m20.1.1.1.2.cmml">⁢</mo><mrow id="S2.5.p2.20.m20.1.1.1.1.1" xref="S2.5.p2.20.m20.1.1.1.1.1.1.cmml"><mo id="S2.5.p2.20.m20.1.1.1.1.1.2" stretchy="false" xref="S2.5.p2.20.m20.1.1.1.1.1.1.cmml">(</mo><msub id="S2.5.p2.20.m20.1.1.1.1.1.1" xref="S2.5.p2.20.m20.1.1.1.1.1.1.cmml"><mi id="S2.5.p2.20.m20.1.1.1.1.1.1.2" xref="S2.5.p2.20.m20.1.1.1.1.1.1.2.cmml">G</mi><mi id="S2.5.p2.20.m20.1.1.1.1.1.1.3" xref="S2.5.p2.20.m20.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S2.5.p2.20.m20.1.1.1.1.1.3" stretchy="false" xref="S2.5.p2.20.m20.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.20.m20.1.1.5" xref="S2.5.p2.20.m20.1.1.5.cmml">=</mo><msup id="S2.5.p2.20.m20.1.1.6" xref="S2.5.p2.20.m20.1.1.6.cmml"><mi id="S2.5.p2.20.m20.1.1.6.2" xref="S2.5.p2.20.m20.1.1.6.2.cmml">A</mi><mi id="S2.5.p2.20.m20.1.1.6.3" xref="S2.5.p2.20.m20.1.1.6.3.cmml">y</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.20.m20.1b"><apply id="S2.5.p2.20.m20.1.1.cmml" xref="S2.5.p2.20.m20.1.1"><and id="S2.5.p2.20.m20.1.1a.cmml" xref="S2.5.p2.20.m20.1.1"></and><apply id="S2.5.p2.20.m20.1.1b.cmml" xref="S2.5.p2.20.m20.1.1"><eq id="S2.5.p2.20.m20.1.1.4.cmml" xref="S2.5.p2.20.m20.1.1.4"></eq><apply id="S2.5.p2.20.m20.1.1.3.cmml" xref="S2.5.p2.20.m20.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.20.m20.1.1.3.1.cmml" xref="S2.5.p2.20.m20.1.1.3">subscript</csymbol><ci id="S2.5.p2.20.m20.1.1.3.2.cmml" xref="S2.5.p2.20.m20.1.1.3.2">𝐵</ci><ci id="S2.5.p2.20.m20.1.1.3.3.cmml" xref="S2.5.p2.20.m20.1.1.3.3">𝑥</ci></apply><apply id="S2.5.p2.20.m20.1.1.1.cmml" xref="S2.5.p2.20.m20.1.1.1"><times id="S2.5.p2.20.m20.1.1.1.2.cmml" xref="S2.5.p2.20.m20.1.1.1.2"></times><ci id="S2.5.p2.20.m20.1.1.1.3.cmml" xref="S2.5.p2.20.m20.1.1.1.3">𝑉</ci><apply id="S2.5.p2.20.m20.1.1.1.1.1.1.cmml" xref="S2.5.p2.20.m20.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.20.m20.1.1.1.1.1.1.1.cmml" xref="S2.5.p2.20.m20.1.1.1.1.1">subscript</csymbol><ci id="S2.5.p2.20.m20.1.1.1.1.1.1.2.cmml" xref="S2.5.p2.20.m20.1.1.1.1.1.1.2">𝐺</ci><ci id="S2.5.p2.20.m20.1.1.1.1.1.1.3.cmml" xref="S2.5.p2.20.m20.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply><apply id="S2.5.p2.20.m20.1.1c.cmml" xref="S2.5.p2.20.m20.1.1"><eq id="S2.5.p2.20.m20.1.1.5.cmml" xref="S2.5.p2.20.m20.1.1.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.5.p2.20.m20.1.1.1.cmml" id="S2.5.p2.20.m20.1.1d.cmml" xref="S2.5.p2.20.m20.1.1"></share><apply id="S2.5.p2.20.m20.1.1.6.cmml" xref="S2.5.p2.20.m20.1.1.6"><csymbol cd="ambiguous" id="S2.5.p2.20.m20.1.1.6.1.cmml" xref="S2.5.p2.20.m20.1.1.6">superscript</csymbol><ci id="S2.5.p2.20.m20.1.1.6.2.cmml" xref="S2.5.p2.20.m20.1.1.6.2">𝐴</ci><ci id="S2.5.p2.20.m20.1.1.6.3.cmml" xref="S2.5.p2.20.m20.1.1.6.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.20.m20.1c">B_{x}=V(G_{x})=A^{y}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.20.m20.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) = italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT</annotation></semantics></math>, so <math alttext="B_{x}\cap B_{y}=A^{y}\cap B_{y}=\partial_{x}" class="ltx_Math" display="inline" id="S2.5.p2.21.m21.1"><semantics id="S2.5.p2.21.m21.1a"><mrow id="S2.5.p2.21.m21.1.1" xref="S2.5.p2.21.m21.1.1.cmml"><mrow id="S2.5.p2.21.m21.1.1.2" xref="S2.5.p2.21.m21.1.1.2.cmml"><msub id="S2.5.p2.21.m21.1.1.2.2" xref="S2.5.p2.21.m21.1.1.2.2.cmml"><mi id="S2.5.p2.21.m21.1.1.2.2.2" xref="S2.5.p2.21.m21.1.1.2.2.2.cmml">B</mi><mi id="S2.5.p2.21.m21.1.1.2.2.3" xref="S2.5.p2.21.m21.1.1.2.2.3.cmml">x</mi></msub><mo id="S2.5.p2.21.m21.1.1.2.1" xref="S2.5.p2.21.m21.1.1.2.1.cmml">∩</mo><msub id="S2.5.p2.21.m21.1.1.2.3" xref="S2.5.p2.21.m21.1.1.2.3.cmml"><mi id="S2.5.p2.21.m21.1.1.2.3.2" xref="S2.5.p2.21.m21.1.1.2.3.2.cmml">B</mi><mi id="S2.5.p2.21.m21.1.1.2.3.3" xref="S2.5.p2.21.m21.1.1.2.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.21.m21.1.1.3" xref="S2.5.p2.21.m21.1.1.3.cmml">=</mo><mrow id="S2.5.p2.21.m21.1.1.4" xref="S2.5.p2.21.m21.1.1.4.cmml"><msup id="S2.5.p2.21.m21.1.1.4.2" xref="S2.5.p2.21.m21.1.1.4.2.cmml"><mi id="S2.5.p2.21.m21.1.1.4.2.2" xref="S2.5.p2.21.m21.1.1.4.2.2.cmml">A</mi><mi id="S2.5.p2.21.m21.1.1.4.2.3" xref="S2.5.p2.21.m21.1.1.4.2.3.cmml">y</mi></msup><mo id="S2.5.p2.21.m21.1.1.4.1" xref="S2.5.p2.21.m21.1.1.4.1.cmml">∩</mo><msub id="S2.5.p2.21.m21.1.1.4.3" xref="S2.5.p2.21.m21.1.1.4.3.cmml"><mi id="S2.5.p2.21.m21.1.1.4.3.2" xref="S2.5.p2.21.m21.1.1.4.3.2.cmml">B</mi><mi id="S2.5.p2.21.m21.1.1.4.3.3" xref="S2.5.p2.21.m21.1.1.4.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.21.m21.1.1.5" rspace="0.1389em" xref="S2.5.p2.21.m21.1.1.5.cmml">=</mo><msub id="S2.5.p2.21.m21.1.1.6" xref="S2.5.p2.21.m21.1.1.6.cmml"><mo id="S2.5.p2.21.m21.1.1.6.2" lspace="0.1389em" xref="S2.5.p2.21.m21.1.1.6.2.cmml">∂</mo><mi id="S2.5.p2.21.m21.1.1.6.3" xref="S2.5.p2.21.m21.1.1.6.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.21.m21.1b"><apply id="S2.5.p2.21.m21.1.1.cmml" xref="S2.5.p2.21.m21.1.1"><and id="S2.5.p2.21.m21.1.1a.cmml" xref="S2.5.p2.21.m21.1.1"></and><apply id="S2.5.p2.21.m21.1.1b.cmml" xref="S2.5.p2.21.m21.1.1"><eq id="S2.5.p2.21.m21.1.1.3.cmml" xref="S2.5.p2.21.m21.1.1.3"></eq><apply id="S2.5.p2.21.m21.1.1.2.cmml" xref="S2.5.p2.21.m21.1.1.2"><intersect id="S2.5.p2.21.m21.1.1.2.1.cmml" xref="S2.5.p2.21.m21.1.1.2.1"></intersect><apply id="S2.5.p2.21.m21.1.1.2.2.cmml" xref="S2.5.p2.21.m21.1.1.2.2"><csymbol cd="ambiguous" id="S2.5.p2.21.m21.1.1.2.2.1.cmml" xref="S2.5.p2.21.m21.1.1.2.2">subscript</csymbol><ci id="S2.5.p2.21.m21.1.1.2.2.2.cmml" xref="S2.5.p2.21.m21.1.1.2.2.2">𝐵</ci><ci id="S2.5.p2.21.m21.1.1.2.2.3.cmml" xref="S2.5.p2.21.m21.1.1.2.2.3">𝑥</ci></apply><apply id="S2.5.p2.21.m21.1.1.2.3.cmml" xref="S2.5.p2.21.m21.1.1.2.3"><csymbol cd="ambiguous" id="S2.5.p2.21.m21.1.1.2.3.1.cmml" xref="S2.5.p2.21.m21.1.1.2.3">subscript</csymbol><ci id="S2.5.p2.21.m21.1.1.2.3.2.cmml" xref="S2.5.p2.21.m21.1.1.2.3.2">𝐵</ci><ci id="S2.5.p2.21.m21.1.1.2.3.3.cmml" xref="S2.5.p2.21.m21.1.1.2.3.3">𝑦</ci></apply></apply><apply id="S2.5.p2.21.m21.1.1.4.cmml" xref="S2.5.p2.21.m21.1.1.4"><intersect id="S2.5.p2.21.m21.1.1.4.1.cmml" xref="S2.5.p2.21.m21.1.1.4.1"></intersect><apply id="S2.5.p2.21.m21.1.1.4.2.cmml" xref="S2.5.p2.21.m21.1.1.4.2"><csymbol cd="ambiguous" id="S2.5.p2.21.m21.1.1.4.2.1.cmml" xref="S2.5.p2.21.m21.1.1.4.2">superscript</csymbol><ci id="S2.5.p2.21.m21.1.1.4.2.2.cmml" xref="S2.5.p2.21.m21.1.1.4.2.2">𝐴</ci><ci id="S2.5.p2.21.m21.1.1.4.2.3.cmml" xref="S2.5.p2.21.m21.1.1.4.2.3">𝑦</ci></apply><apply id="S2.5.p2.21.m21.1.1.4.3.cmml" xref="S2.5.p2.21.m21.1.1.4.3"><csymbol cd="ambiguous" id="S2.5.p2.21.m21.1.1.4.3.1.cmml" xref="S2.5.p2.21.m21.1.1.4.3">subscript</csymbol><ci id="S2.5.p2.21.m21.1.1.4.3.2.cmml" xref="S2.5.p2.21.m21.1.1.4.3.2">𝐵</ci><ci id="S2.5.p2.21.m21.1.1.4.3.3.cmml" xref="S2.5.p2.21.m21.1.1.4.3.3">𝑦</ci></apply></apply></apply><apply id="S2.5.p2.21.m21.1.1c.cmml" xref="S2.5.p2.21.m21.1.1"><eq id="S2.5.p2.21.m21.1.1.5.cmml" xref="S2.5.p2.21.m21.1.1.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.5.p2.21.m21.1.1.4.cmml" id="S2.5.p2.21.m21.1.1d.cmml" xref="S2.5.p2.21.m21.1.1"></share><apply id="S2.5.p2.21.m21.1.1.6.cmml" xref="S2.5.p2.21.m21.1.1.6"><csymbol cd="ambiguous" id="S2.5.p2.21.m21.1.1.6.1.cmml" xref="S2.5.p2.21.m21.1.1.6">subscript</csymbol><partialdiff id="S2.5.p2.21.m21.1.1.6.2.cmml" xref="S2.5.p2.21.m21.1.1.6.2"></partialdiff><ci id="S2.5.p2.21.m21.1.1.6.3.cmml" xref="S2.5.p2.21.m21.1.1.6.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.21.m21.1c">B_{x}\cap B_{y}=A^{y}\cap B_{y}=\partial_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.21.m21.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. If <math alttext="x" class="ltx_Math" display="inline" id="S2.5.p2.22.m22.1"><semantics id="S2.5.p2.22.m22.1a"><mi id="S2.5.p2.22.m22.1.1" xref="S2.5.p2.22.m22.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.22.m22.1b"><ci id="S2.5.p2.22.m22.1.1.cmml" xref="S2.5.p2.22.m22.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.22.m22.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.22.m22.1d">italic_x</annotation></semantics></math> is not a leaf of <math alttext="T" class="ltx_Math" display="inline" id="S2.5.p2.23.m23.1"><semantics id="S2.5.p2.23.m23.1a"><mi id="S2.5.p2.23.m23.1.1" xref="S2.5.p2.23.m23.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.23.m23.1b"><ci id="S2.5.p2.23.m23.1.1.cmml" xref="S2.5.p2.23.m23.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.23.m23.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.23.m23.1d">italic_T</annotation></semantics></math> then <math alttext="B_{x}=\partial_{x}\cup(A^{x}\cap B^{x})" class="ltx_Math" display="inline" id="S2.5.p2.24.m24.1"><semantics id="S2.5.p2.24.m24.1a"><mrow id="S2.5.p2.24.m24.1.1" xref="S2.5.p2.24.m24.1.1.cmml"><msub id="S2.5.p2.24.m24.1.1.3" xref="S2.5.p2.24.m24.1.1.3.cmml"><mi id="S2.5.p2.24.m24.1.1.3.2" xref="S2.5.p2.24.m24.1.1.3.2.cmml">B</mi><mi id="S2.5.p2.24.m24.1.1.3.3" xref="S2.5.p2.24.m24.1.1.3.3.cmml">x</mi></msub><mo id="S2.5.p2.24.m24.1.1.2" rspace="0.1389em" xref="S2.5.p2.24.m24.1.1.2.cmml">=</mo><mrow id="S2.5.p2.24.m24.1.1.1" xref="S2.5.p2.24.m24.1.1.1.cmml"><msub id="S2.5.p2.24.m24.1.1.1.3" xref="S2.5.p2.24.m24.1.1.1.3.cmml"><mo id="S2.5.p2.24.m24.1.1.1.3.2" lspace="0.1389em" rspace="0em" xref="S2.5.p2.24.m24.1.1.1.3.2.cmml">∂</mo><mi id="S2.5.p2.24.m24.1.1.1.3.3" xref="S2.5.p2.24.m24.1.1.1.3.3.cmml">x</mi></msub><mo id="S2.5.p2.24.m24.1.1.1.2" xref="S2.5.p2.24.m24.1.1.1.2.cmml">∪</mo><mrow id="S2.5.p2.24.m24.1.1.1.1.1" xref="S2.5.p2.24.m24.1.1.1.1.1.1.cmml"><mo id="S2.5.p2.24.m24.1.1.1.1.1.2" stretchy="false" xref="S2.5.p2.24.m24.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.5.p2.24.m24.1.1.1.1.1.1" xref="S2.5.p2.24.m24.1.1.1.1.1.1.cmml"><msup id="S2.5.p2.24.m24.1.1.1.1.1.1.2" xref="S2.5.p2.24.m24.1.1.1.1.1.1.2.cmml"><mi id="S2.5.p2.24.m24.1.1.1.1.1.1.2.2" xref="S2.5.p2.24.m24.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S2.5.p2.24.m24.1.1.1.1.1.1.2.3" xref="S2.5.p2.24.m24.1.1.1.1.1.1.2.3.cmml">x</mi></msup><mo id="S2.5.p2.24.m24.1.1.1.1.1.1.1" xref="S2.5.p2.24.m24.1.1.1.1.1.1.1.cmml">∩</mo><msup id="S2.5.p2.24.m24.1.1.1.1.1.1.3" xref="S2.5.p2.24.m24.1.1.1.1.1.1.3.cmml"><mi id="S2.5.p2.24.m24.1.1.1.1.1.1.3.2" xref="S2.5.p2.24.m24.1.1.1.1.1.1.3.2.cmml">B</mi><mi id="S2.5.p2.24.m24.1.1.1.1.1.1.3.3" xref="S2.5.p2.24.m24.1.1.1.1.1.1.3.3.cmml">x</mi></msup></mrow><mo id="S2.5.p2.24.m24.1.1.1.1.1.3" stretchy="false" xref="S2.5.p2.24.m24.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.24.m24.1b"><apply id="S2.5.p2.24.m24.1.1.cmml" xref="S2.5.p2.24.m24.1.1"><eq id="S2.5.p2.24.m24.1.1.2.cmml" xref="S2.5.p2.24.m24.1.1.2"></eq><apply id="S2.5.p2.24.m24.1.1.3.cmml" xref="S2.5.p2.24.m24.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.24.m24.1.1.3.1.cmml" xref="S2.5.p2.24.m24.1.1.3">subscript</csymbol><ci id="S2.5.p2.24.m24.1.1.3.2.cmml" xref="S2.5.p2.24.m24.1.1.3.2">𝐵</ci><ci id="S2.5.p2.24.m24.1.1.3.3.cmml" xref="S2.5.p2.24.m24.1.1.3.3">𝑥</ci></apply><apply id="S2.5.p2.24.m24.1.1.1.cmml" xref="S2.5.p2.24.m24.1.1.1"><union id="S2.5.p2.24.m24.1.1.1.2.cmml" xref="S2.5.p2.24.m24.1.1.1.2"></union><apply id="S2.5.p2.24.m24.1.1.1.3.cmml" xref="S2.5.p2.24.m24.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.24.m24.1.1.1.3.1.cmml" xref="S2.5.p2.24.m24.1.1.1.3">subscript</csymbol><partialdiff id="S2.5.p2.24.m24.1.1.1.3.2.cmml" xref="S2.5.p2.24.m24.1.1.1.3.2"></partialdiff><ci id="S2.5.p2.24.m24.1.1.1.3.3.cmml" xref="S2.5.p2.24.m24.1.1.1.3.3">𝑥</ci></apply><apply id="S2.5.p2.24.m24.1.1.1.1.1.1.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1"><intersect id="S2.5.p2.24.m24.1.1.1.1.1.1.1.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.1"></intersect><apply id="S2.5.p2.24.m24.1.1.1.1.1.1.2.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.5.p2.24.m24.1.1.1.1.1.1.2.1.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.5.p2.24.m24.1.1.1.1.1.1.2.2.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.2.2">𝐴</ci><ci id="S2.5.p2.24.m24.1.1.1.1.1.1.2.3.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.2.3">𝑥</ci></apply><apply id="S2.5.p2.24.m24.1.1.1.1.1.1.3.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.24.m24.1.1.1.1.1.1.3.1.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.5.p2.24.m24.1.1.1.1.1.1.3.2.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.3.2">𝐵</ci><ci id="S2.5.p2.24.m24.1.1.1.1.1.1.3.3.cmml" xref="S2.5.p2.24.m24.1.1.1.1.1.1.3.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.24.m24.1c">B_{x}=\partial_{x}\cup(A^{x}\cap B^{x})</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.24.m24.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∪ ( italic_A start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT )</annotation></semantics></math> where <math alttext="(A^{x},B^{x})" class="ltx_Math" display="inline" id="S2.5.p2.25.m25.2"><semantics id="S2.5.p2.25.m25.2a"><mrow id="S2.5.p2.25.m25.2.2.2" xref="S2.5.p2.25.m25.2.2.3.cmml"><mo id="S2.5.p2.25.m25.2.2.2.3" stretchy="false" xref="S2.5.p2.25.m25.2.2.3.cmml">(</mo><msup id="S2.5.p2.25.m25.1.1.1.1" xref="S2.5.p2.25.m25.1.1.1.1.cmml"><mi id="S2.5.p2.25.m25.1.1.1.1.2" xref="S2.5.p2.25.m25.1.1.1.1.2.cmml">A</mi><mi id="S2.5.p2.25.m25.1.1.1.1.3" xref="S2.5.p2.25.m25.1.1.1.1.3.cmml">x</mi></msup><mo id="S2.5.p2.25.m25.2.2.2.4" xref="S2.5.p2.25.m25.2.2.3.cmml">,</mo><msup id="S2.5.p2.25.m25.2.2.2.2" xref="S2.5.p2.25.m25.2.2.2.2.cmml"><mi id="S2.5.p2.25.m25.2.2.2.2.2" xref="S2.5.p2.25.m25.2.2.2.2.2.cmml">B</mi><mi id="S2.5.p2.25.m25.2.2.2.2.3" xref="S2.5.p2.25.m25.2.2.2.2.3.cmml">x</mi></msup><mo id="S2.5.p2.25.m25.2.2.2.5" stretchy="false" xref="S2.5.p2.25.m25.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.25.m25.2b"><interval closure="open" id="S2.5.p2.25.m25.2.2.3.cmml" xref="S2.5.p2.25.m25.2.2.2"><apply id="S2.5.p2.25.m25.1.1.1.1.cmml" xref="S2.5.p2.25.m25.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.25.m25.1.1.1.1.1.cmml" xref="S2.5.p2.25.m25.1.1.1.1">superscript</csymbol><ci id="S2.5.p2.25.m25.1.1.1.1.2.cmml" xref="S2.5.p2.25.m25.1.1.1.1.2">𝐴</ci><ci id="S2.5.p2.25.m25.1.1.1.1.3.cmml" xref="S2.5.p2.25.m25.1.1.1.1.3">𝑥</ci></apply><apply id="S2.5.p2.25.m25.2.2.2.2.cmml" xref="S2.5.p2.25.m25.2.2.2.2"><csymbol cd="ambiguous" id="S2.5.p2.25.m25.2.2.2.2.1.cmml" xref="S2.5.p2.25.m25.2.2.2.2">superscript</csymbol><ci id="S2.5.p2.25.m25.2.2.2.2.2.cmml" xref="S2.5.p2.25.m25.2.2.2.2.2">𝐵</ci><ci id="S2.5.p2.25.m25.2.2.2.2.3.cmml" xref="S2.5.p2.25.m25.2.2.2.2.3">𝑥</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.25.m25.2c">(A^{x},B^{x})</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.25.m25.2d">( italic_A start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT , italic_B start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT )</annotation></semantics></math> is a separation of <math alttext="G_{x}-\partial_{x}=G[A^{y}\setminus(A^{y}\cap B_{y})]=G[A^{y}\setminus B_{y}]" class="ltx_Math" display="inline" id="S2.5.p2.26.m26.2"><semantics id="S2.5.p2.26.m26.2a"><mrow id="S2.5.p2.26.m26.2.2" xref="S2.5.p2.26.m26.2.2.cmml"><mrow id="S2.5.p2.26.m26.2.2.4" xref="S2.5.p2.26.m26.2.2.4.cmml"><msub id="S2.5.p2.26.m26.2.2.4.2" xref="S2.5.p2.26.m26.2.2.4.2.cmml"><mi id="S2.5.p2.26.m26.2.2.4.2.2" xref="S2.5.p2.26.m26.2.2.4.2.2.cmml">G</mi><mi id="S2.5.p2.26.m26.2.2.4.2.3" xref="S2.5.p2.26.m26.2.2.4.2.3.cmml">x</mi></msub><mo id="S2.5.p2.26.m26.2.2.4.1" xref="S2.5.p2.26.m26.2.2.4.1.cmml">−</mo><msub id="S2.5.p2.26.m26.2.2.4.3" xref="S2.5.p2.26.m26.2.2.4.3.cmml"><mo id="S2.5.p2.26.m26.2.2.4.3.2" lspace="0em" rspace="0.1389em" xref="S2.5.p2.26.m26.2.2.4.3.2.cmml">∂</mo><mi id="S2.5.p2.26.m26.2.2.4.3.3" xref="S2.5.p2.26.m26.2.2.4.3.3.cmml">x</mi></msub></mrow><mo id="S2.5.p2.26.m26.2.2.5" lspace="0.1389em" xref="S2.5.p2.26.m26.2.2.5.cmml">=</mo><mrow id="S2.5.p2.26.m26.1.1.1" xref="S2.5.p2.26.m26.1.1.1.cmml"><mi id="S2.5.p2.26.m26.1.1.1.3" xref="S2.5.p2.26.m26.1.1.1.3.cmml">G</mi><mo id="S2.5.p2.26.m26.1.1.1.2" xref="S2.5.p2.26.m26.1.1.1.2.cmml">⁢</mo><mrow id="S2.5.p2.26.m26.1.1.1.1.1" xref="S2.5.p2.26.m26.1.1.1.1.2.cmml"><mo id="S2.5.p2.26.m26.1.1.1.1.1.2" stretchy="false" xref="S2.5.p2.26.m26.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.5.p2.26.m26.1.1.1.1.1.1" xref="S2.5.p2.26.m26.1.1.1.1.1.1.cmml"><msup id="S2.5.p2.26.m26.1.1.1.1.1.1.3" xref="S2.5.p2.26.m26.1.1.1.1.1.1.3.cmml"><mi id="S2.5.p2.26.m26.1.1.1.1.1.1.3.2" xref="S2.5.p2.26.m26.1.1.1.1.1.1.3.2.cmml">A</mi><mi id="S2.5.p2.26.m26.1.1.1.1.1.1.3.3" xref="S2.5.p2.26.m26.1.1.1.1.1.1.3.3.cmml">y</mi></msup><mo id="S2.5.p2.26.m26.1.1.1.1.1.1.2" xref="S2.5.p2.26.m26.1.1.1.1.1.1.2.cmml">∖</mo><mrow id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.cmml"><msup id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.2" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.3" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.3.cmml">y</mi></msup><mo id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.1" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.1.cmml">∩</mo><msub id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.2" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.2.cmml">B</mi><mi id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.3" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.26.m26.1.1.1.1.1.3" stretchy="false" xref="S2.5.p2.26.m26.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.5.p2.26.m26.2.2.6" xref="S2.5.p2.26.m26.2.2.6.cmml">=</mo><mrow id="S2.5.p2.26.m26.2.2.2" xref="S2.5.p2.26.m26.2.2.2.cmml"><mi id="S2.5.p2.26.m26.2.2.2.3" xref="S2.5.p2.26.m26.2.2.2.3.cmml">G</mi><mo id="S2.5.p2.26.m26.2.2.2.2" xref="S2.5.p2.26.m26.2.2.2.2.cmml">⁢</mo><mrow id="S2.5.p2.26.m26.2.2.2.1.1" xref="S2.5.p2.26.m26.2.2.2.1.2.cmml"><mo id="S2.5.p2.26.m26.2.2.2.1.1.2" stretchy="false" xref="S2.5.p2.26.m26.2.2.2.1.2.1.cmml">[</mo><mrow id="S2.5.p2.26.m26.2.2.2.1.1.1" xref="S2.5.p2.26.m26.2.2.2.1.1.1.cmml"><msup id="S2.5.p2.26.m26.2.2.2.1.1.1.2" xref="S2.5.p2.26.m26.2.2.2.1.1.1.2.cmml"><mi id="S2.5.p2.26.m26.2.2.2.1.1.1.2.2" xref="S2.5.p2.26.m26.2.2.2.1.1.1.2.2.cmml">A</mi><mi id="S2.5.p2.26.m26.2.2.2.1.1.1.2.3" xref="S2.5.p2.26.m26.2.2.2.1.1.1.2.3.cmml">y</mi></msup><mo id="S2.5.p2.26.m26.2.2.2.1.1.1.1" xref="S2.5.p2.26.m26.2.2.2.1.1.1.1.cmml">∖</mo><msub id="S2.5.p2.26.m26.2.2.2.1.1.1.3" xref="S2.5.p2.26.m26.2.2.2.1.1.1.3.cmml"><mi id="S2.5.p2.26.m26.2.2.2.1.1.1.3.2" xref="S2.5.p2.26.m26.2.2.2.1.1.1.3.2.cmml">B</mi><mi id="S2.5.p2.26.m26.2.2.2.1.1.1.3.3" xref="S2.5.p2.26.m26.2.2.2.1.1.1.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.26.m26.2.2.2.1.1.3" stretchy="false" xref="S2.5.p2.26.m26.2.2.2.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.26.m26.2b"><apply id="S2.5.p2.26.m26.2.2.cmml" xref="S2.5.p2.26.m26.2.2"><and id="S2.5.p2.26.m26.2.2a.cmml" xref="S2.5.p2.26.m26.2.2"></and><apply id="S2.5.p2.26.m26.2.2b.cmml" xref="S2.5.p2.26.m26.2.2"><eq id="S2.5.p2.26.m26.2.2.5.cmml" xref="S2.5.p2.26.m26.2.2.5"></eq><apply id="S2.5.p2.26.m26.2.2.4.cmml" xref="S2.5.p2.26.m26.2.2.4"><minus id="S2.5.p2.26.m26.2.2.4.1.cmml" xref="S2.5.p2.26.m26.2.2.4.1"></minus><apply id="S2.5.p2.26.m26.2.2.4.2.cmml" xref="S2.5.p2.26.m26.2.2.4.2"><csymbol cd="ambiguous" id="S2.5.p2.26.m26.2.2.4.2.1.cmml" xref="S2.5.p2.26.m26.2.2.4.2">subscript</csymbol><ci id="S2.5.p2.26.m26.2.2.4.2.2.cmml" xref="S2.5.p2.26.m26.2.2.4.2.2">𝐺</ci><ci id="S2.5.p2.26.m26.2.2.4.2.3.cmml" xref="S2.5.p2.26.m26.2.2.4.2.3">𝑥</ci></apply><apply id="S2.5.p2.26.m26.2.2.4.3.cmml" xref="S2.5.p2.26.m26.2.2.4.3"><csymbol cd="ambiguous" id="S2.5.p2.26.m26.2.2.4.3.1.cmml" xref="S2.5.p2.26.m26.2.2.4.3">subscript</csymbol><partialdiff id="S2.5.p2.26.m26.2.2.4.3.2.cmml" xref="S2.5.p2.26.m26.2.2.4.3.2"></partialdiff><ci id="S2.5.p2.26.m26.2.2.4.3.3.cmml" xref="S2.5.p2.26.m26.2.2.4.3.3">𝑥</ci></apply></apply><apply id="S2.5.p2.26.m26.1.1.1.cmml" xref="S2.5.p2.26.m26.1.1.1"><times id="S2.5.p2.26.m26.1.1.1.2.cmml" xref="S2.5.p2.26.m26.1.1.1.2"></times><ci id="S2.5.p2.26.m26.1.1.1.3.cmml" xref="S2.5.p2.26.m26.1.1.1.3">𝐺</ci><apply id="S2.5.p2.26.m26.1.1.1.1.2.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1"><csymbol cd="latexml" id="S2.5.p2.26.m26.1.1.1.1.2.1.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.5.p2.26.m26.1.1.1.1.1.1.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1"><setdiff id="S2.5.p2.26.m26.1.1.1.1.1.1.2.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.2"></setdiff><apply id="S2.5.p2.26.m26.1.1.1.1.1.1.3.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.26.m26.1.1.1.1.1.1.3.1.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.5.p2.26.m26.1.1.1.1.1.1.3.2.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.3.2">𝐴</ci><ci id="S2.5.p2.26.m26.1.1.1.1.1.1.3.3.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.3.3">𝑦</ci></apply><apply id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1"><intersect id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.1"></intersect><apply id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.2">𝐴</ci><ci id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.2.3">𝑦</ci></apply><apply id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.2">𝐵</ci><ci id="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.5.p2.26.m26.1.1.1.1.1.1.1.1.1.3.3">𝑦</ci></apply></apply></apply></apply></apply></apply><apply id="S2.5.p2.26.m26.2.2c.cmml" xref="S2.5.p2.26.m26.2.2"><eq id="S2.5.p2.26.m26.2.2.6.cmml" xref="S2.5.p2.26.m26.2.2.6"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.5.p2.26.m26.1.1.1.cmml" id="S2.5.p2.26.m26.2.2d.cmml" xref="S2.5.p2.26.m26.2.2"></share><apply id="S2.5.p2.26.m26.2.2.2.cmml" xref="S2.5.p2.26.m26.2.2.2"><times id="S2.5.p2.26.m26.2.2.2.2.cmml" xref="S2.5.p2.26.m26.2.2.2.2"></times><ci id="S2.5.p2.26.m26.2.2.2.3.cmml" xref="S2.5.p2.26.m26.2.2.2.3">𝐺</ci><apply id="S2.5.p2.26.m26.2.2.2.1.2.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1"><csymbol cd="latexml" id="S2.5.p2.26.m26.2.2.2.1.2.1.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.2">delimited-[]</csymbol><apply id="S2.5.p2.26.m26.2.2.2.1.1.1.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1"><setdiff id="S2.5.p2.26.m26.2.2.2.1.1.1.1.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.1"></setdiff><apply id="S2.5.p2.26.m26.2.2.2.1.1.1.2.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.5.p2.26.m26.2.2.2.1.1.1.2.1.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.2">superscript</csymbol><ci id="S2.5.p2.26.m26.2.2.2.1.1.1.2.2.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.2.2">𝐴</ci><ci id="S2.5.p2.26.m26.2.2.2.1.1.1.2.3.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.2.3">𝑦</ci></apply><apply id="S2.5.p2.26.m26.2.2.2.1.1.1.3.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.26.m26.2.2.2.1.1.1.3.1.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.3">subscript</csymbol><ci id="S2.5.p2.26.m26.2.2.2.1.1.1.3.2.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.3.2">𝐵</ci><ci id="S2.5.p2.26.m26.2.2.2.1.1.1.3.3.cmml" xref="S2.5.p2.26.m26.2.2.2.1.1.1.3.3">𝑦</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.26.m26.2c">G_{x}-\partial_{x}=G[A^{y}\setminus(A^{y}\cap B_{y})]=G[A^{y}\setminus B_{y}]</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.26.m26.2d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_G [ italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∖ ( italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) ] = italic_G [ italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∖ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ]</annotation></semantics></math>. In particular, <math alttext="A^{x}\cup B^{x}" class="ltx_Math" display="inline" id="S2.5.p2.27.m27.1"><semantics id="S2.5.p2.27.m27.1a"><mrow id="S2.5.p2.27.m27.1.1" xref="S2.5.p2.27.m27.1.1.cmml"><msup id="S2.5.p2.27.m27.1.1.2" xref="S2.5.p2.27.m27.1.1.2.cmml"><mi id="S2.5.p2.27.m27.1.1.2.2" xref="S2.5.p2.27.m27.1.1.2.2.cmml">A</mi><mi id="S2.5.p2.27.m27.1.1.2.3" xref="S2.5.p2.27.m27.1.1.2.3.cmml">x</mi></msup><mo id="S2.5.p2.27.m27.1.1.1" xref="S2.5.p2.27.m27.1.1.1.cmml">∪</mo><msup id="S2.5.p2.27.m27.1.1.3" xref="S2.5.p2.27.m27.1.1.3.cmml"><mi id="S2.5.p2.27.m27.1.1.3.2" xref="S2.5.p2.27.m27.1.1.3.2.cmml">B</mi><mi id="S2.5.p2.27.m27.1.1.3.3" xref="S2.5.p2.27.m27.1.1.3.3.cmml">x</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.27.m27.1b"><apply id="S2.5.p2.27.m27.1.1.cmml" xref="S2.5.p2.27.m27.1.1"><union id="S2.5.p2.27.m27.1.1.1.cmml" xref="S2.5.p2.27.m27.1.1.1"></union><apply id="S2.5.p2.27.m27.1.1.2.cmml" xref="S2.5.p2.27.m27.1.1.2"><csymbol cd="ambiguous" id="S2.5.p2.27.m27.1.1.2.1.cmml" xref="S2.5.p2.27.m27.1.1.2">superscript</csymbol><ci id="S2.5.p2.27.m27.1.1.2.2.cmml" xref="S2.5.p2.27.m27.1.1.2.2">𝐴</ci><ci id="S2.5.p2.27.m27.1.1.2.3.cmml" xref="S2.5.p2.27.m27.1.1.2.3">𝑥</ci></apply><apply id="S2.5.p2.27.m27.1.1.3.cmml" xref="S2.5.p2.27.m27.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.27.m27.1.1.3.1.cmml" xref="S2.5.p2.27.m27.1.1.3">superscript</csymbol><ci id="S2.5.p2.27.m27.1.1.3.2.cmml" xref="S2.5.p2.27.m27.1.1.3.2">𝐵</ci><ci id="S2.5.p2.27.m27.1.1.3.3.cmml" xref="S2.5.p2.27.m27.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.27.m27.1c">A^{x}\cup B^{x}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.27.m27.1d">italic_A start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ∪ italic_B start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT</annotation></semantics></math> contains no vertex of <math alttext="B_{y}" class="ltx_Math" display="inline" id="S2.5.p2.28.m28.1"><semantics id="S2.5.p2.28.m28.1a"><msub id="S2.5.p2.28.m28.1.1" xref="S2.5.p2.28.m28.1.1.cmml"><mi id="S2.5.p2.28.m28.1.1.2" xref="S2.5.p2.28.m28.1.1.2.cmml">B</mi><mi id="S2.5.p2.28.m28.1.1.3" xref="S2.5.p2.28.m28.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S2.5.p2.28.m28.1b"><apply id="S2.5.p2.28.m28.1.1.cmml" xref="S2.5.p2.28.m28.1.1"><csymbol cd="ambiguous" id="S2.5.p2.28.m28.1.1.1.cmml" xref="S2.5.p2.28.m28.1.1">subscript</csymbol><ci id="S2.5.p2.28.m28.1.1.2.cmml" xref="S2.5.p2.28.m28.1.1.2">𝐵</ci><ci id="S2.5.p2.28.m28.1.1.3.cmml" xref="S2.5.p2.28.m28.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.28.m28.1c">B_{y}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.28.m28.1d">italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>, so <math alttext="B_{x}\cap B_{y}=(\partial_{x}\cup(A^{x}\cap B^{x}))\cap B_{y}=\partial_{x}\cap B% _{y}=\partial_{x}" class="ltx_Math" display="inline" id="S2.5.p2.29.m29.1"><semantics id="S2.5.p2.29.m29.1a"><mrow id="S2.5.p2.29.m29.1.1" xref="S2.5.p2.29.m29.1.1.cmml"><mrow id="S2.5.p2.29.m29.1.1.3" xref="S2.5.p2.29.m29.1.1.3.cmml"><msub id="S2.5.p2.29.m29.1.1.3.2" xref="S2.5.p2.29.m29.1.1.3.2.cmml"><mi id="S2.5.p2.29.m29.1.1.3.2.2" xref="S2.5.p2.29.m29.1.1.3.2.2.cmml">B</mi><mi id="S2.5.p2.29.m29.1.1.3.2.3" xref="S2.5.p2.29.m29.1.1.3.2.3.cmml">x</mi></msub><mo id="S2.5.p2.29.m29.1.1.3.1" xref="S2.5.p2.29.m29.1.1.3.1.cmml">∩</mo><msub id="S2.5.p2.29.m29.1.1.3.3" xref="S2.5.p2.29.m29.1.1.3.3.cmml"><mi id="S2.5.p2.29.m29.1.1.3.3.2" xref="S2.5.p2.29.m29.1.1.3.3.2.cmml">B</mi><mi id="S2.5.p2.29.m29.1.1.3.3.3" xref="S2.5.p2.29.m29.1.1.3.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.29.m29.1.1.4" xref="S2.5.p2.29.m29.1.1.4.cmml">=</mo><mrow id="S2.5.p2.29.m29.1.1.1" xref="S2.5.p2.29.m29.1.1.1.cmml"><mrow id="S2.5.p2.29.m29.1.1.1.1.1" xref="S2.5.p2.29.m29.1.1.1.1.1.1.cmml"><mo id="S2.5.p2.29.m29.1.1.1.1.1.2" stretchy="false" xref="S2.5.p2.29.m29.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.5.p2.29.m29.1.1.1.1.1.1" xref="S2.5.p2.29.m29.1.1.1.1.1.1.cmml"><msub id="S2.5.p2.29.m29.1.1.1.1.1.1.3" xref="S2.5.p2.29.m29.1.1.1.1.1.1.3.cmml"><mo id="S2.5.p2.29.m29.1.1.1.1.1.1.3.2" lspace="0em" rspace="0em" xref="S2.5.p2.29.m29.1.1.1.1.1.1.3.2.cmml">∂</mo><mi id="S2.5.p2.29.m29.1.1.1.1.1.1.3.3" xref="S2.5.p2.29.m29.1.1.1.1.1.1.3.3.cmml">x</mi></msub><mo id="S2.5.p2.29.m29.1.1.1.1.1.1.2" xref="S2.5.p2.29.m29.1.1.1.1.1.1.2.cmml">∪</mo><mrow id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.cmml"><msup id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.2" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.3" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.3.cmml">x</mi></msup><mo id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.1" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.1.cmml">∩</mo><msup id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.2" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.2.cmml">B</mi><mi id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.3" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.3.cmml">x</mi></msup></mrow><mo id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.29.m29.1.1.1.1.1.3" stretchy="false" xref="S2.5.p2.29.m29.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.5.p2.29.m29.1.1.1.2" xref="S2.5.p2.29.m29.1.1.1.2.cmml">∩</mo><msub id="S2.5.p2.29.m29.1.1.1.3" xref="S2.5.p2.29.m29.1.1.1.3.cmml"><mi id="S2.5.p2.29.m29.1.1.1.3.2" xref="S2.5.p2.29.m29.1.1.1.3.2.cmml">B</mi><mi id="S2.5.p2.29.m29.1.1.1.3.3" xref="S2.5.p2.29.m29.1.1.1.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.29.m29.1.1.5" rspace="0.1389em" xref="S2.5.p2.29.m29.1.1.5.cmml">=</mo><mrow id="S2.5.p2.29.m29.1.1.6" xref="S2.5.p2.29.m29.1.1.6.cmml"><msub id="S2.5.p2.29.m29.1.1.6.2" xref="S2.5.p2.29.m29.1.1.6.2.cmml"><mo id="S2.5.p2.29.m29.1.1.6.2.2" lspace="0.1389em" rspace="0em" xref="S2.5.p2.29.m29.1.1.6.2.2.cmml">∂</mo><mi id="S2.5.p2.29.m29.1.1.6.2.3" xref="S2.5.p2.29.m29.1.1.6.2.3.cmml">x</mi></msub><mo id="S2.5.p2.29.m29.1.1.6.1" xref="S2.5.p2.29.m29.1.1.6.1.cmml">∩</mo><msub id="S2.5.p2.29.m29.1.1.6.3" xref="S2.5.p2.29.m29.1.1.6.3.cmml"><mi id="S2.5.p2.29.m29.1.1.6.3.2" xref="S2.5.p2.29.m29.1.1.6.3.2.cmml">B</mi><mi id="S2.5.p2.29.m29.1.1.6.3.3" xref="S2.5.p2.29.m29.1.1.6.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.29.m29.1.1.7" rspace="0.1389em" xref="S2.5.p2.29.m29.1.1.7.cmml">=</mo><msub id="S2.5.p2.29.m29.1.1.8" xref="S2.5.p2.29.m29.1.1.8.cmml"><mo id="S2.5.p2.29.m29.1.1.8.2" lspace="0.1389em" xref="S2.5.p2.29.m29.1.1.8.2.cmml">∂</mo><mi id="S2.5.p2.29.m29.1.1.8.3" xref="S2.5.p2.29.m29.1.1.8.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.29.m29.1b"><apply id="S2.5.p2.29.m29.1.1.cmml" xref="S2.5.p2.29.m29.1.1"><and id="S2.5.p2.29.m29.1.1a.cmml" xref="S2.5.p2.29.m29.1.1"></and><apply id="S2.5.p2.29.m29.1.1b.cmml" xref="S2.5.p2.29.m29.1.1"><eq id="S2.5.p2.29.m29.1.1.4.cmml" xref="S2.5.p2.29.m29.1.1.4"></eq><apply id="S2.5.p2.29.m29.1.1.3.cmml" xref="S2.5.p2.29.m29.1.1.3"><intersect id="S2.5.p2.29.m29.1.1.3.1.cmml" xref="S2.5.p2.29.m29.1.1.3.1"></intersect><apply id="S2.5.p2.29.m29.1.1.3.2.cmml" xref="S2.5.p2.29.m29.1.1.3.2"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.3.2.1.cmml" xref="S2.5.p2.29.m29.1.1.3.2">subscript</csymbol><ci id="S2.5.p2.29.m29.1.1.3.2.2.cmml" xref="S2.5.p2.29.m29.1.1.3.2.2">𝐵</ci><ci id="S2.5.p2.29.m29.1.1.3.2.3.cmml" xref="S2.5.p2.29.m29.1.1.3.2.3">𝑥</ci></apply><apply id="S2.5.p2.29.m29.1.1.3.3.cmml" xref="S2.5.p2.29.m29.1.1.3.3"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.3.3.1.cmml" xref="S2.5.p2.29.m29.1.1.3.3">subscript</csymbol><ci id="S2.5.p2.29.m29.1.1.3.3.2.cmml" xref="S2.5.p2.29.m29.1.1.3.3.2">𝐵</ci><ci id="S2.5.p2.29.m29.1.1.3.3.3.cmml" xref="S2.5.p2.29.m29.1.1.3.3.3">𝑦</ci></apply></apply><apply id="S2.5.p2.29.m29.1.1.1.cmml" xref="S2.5.p2.29.m29.1.1.1"><intersect id="S2.5.p2.29.m29.1.1.1.2.cmml" xref="S2.5.p2.29.m29.1.1.1.2"></intersect><apply id="S2.5.p2.29.m29.1.1.1.1.1.1.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1"><union id="S2.5.p2.29.m29.1.1.1.1.1.1.2.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.2"></union><apply id="S2.5.p2.29.m29.1.1.1.1.1.1.3.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.1.1.1.1.3.1.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.3">subscript</csymbol><partialdiff id="S2.5.p2.29.m29.1.1.1.1.1.1.3.2.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.3.2"></partialdiff><ci id="S2.5.p2.29.m29.1.1.1.1.1.1.3.3.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.3.3">𝑥</ci></apply><apply id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1"><intersect id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.1"></intersect><apply id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.2">𝐴</ci><ci id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.2.3">𝑥</ci></apply><apply id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.2">𝐵</ci><ci id="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.5.p2.29.m29.1.1.1.1.1.1.1.1.1.3.3">𝑥</ci></apply></apply></apply><apply id="S2.5.p2.29.m29.1.1.1.3.cmml" xref="S2.5.p2.29.m29.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.1.3.1.cmml" xref="S2.5.p2.29.m29.1.1.1.3">subscript</csymbol><ci id="S2.5.p2.29.m29.1.1.1.3.2.cmml" xref="S2.5.p2.29.m29.1.1.1.3.2">𝐵</ci><ci id="S2.5.p2.29.m29.1.1.1.3.3.cmml" xref="S2.5.p2.29.m29.1.1.1.3.3">𝑦</ci></apply></apply></apply><apply id="S2.5.p2.29.m29.1.1c.cmml" xref="S2.5.p2.29.m29.1.1"><eq id="S2.5.p2.29.m29.1.1.5.cmml" xref="S2.5.p2.29.m29.1.1.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.5.p2.29.m29.1.1.1.cmml" id="S2.5.p2.29.m29.1.1d.cmml" xref="S2.5.p2.29.m29.1.1"></share><apply id="S2.5.p2.29.m29.1.1.6.cmml" xref="S2.5.p2.29.m29.1.1.6"><intersect id="S2.5.p2.29.m29.1.1.6.1.cmml" xref="S2.5.p2.29.m29.1.1.6.1"></intersect><apply id="S2.5.p2.29.m29.1.1.6.2.cmml" xref="S2.5.p2.29.m29.1.1.6.2"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.6.2.1.cmml" xref="S2.5.p2.29.m29.1.1.6.2">subscript</csymbol><partialdiff id="S2.5.p2.29.m29.1.1.6.2.2.cmml" xref="S2.5.p2.29.m29.1.1.6.2.2"></partialdiff><ci id="S2.5.p2.29.m29.1.1.6.2.3.cmml" xref="S2.5.p2.29.m29.1.1.6.2.3">𝑥</ci></apply><apply id="S2.5.p2.29.m29.1.1.6.3.cmml" xref="S2.5.p2.29.m29.1.1.6.3"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.6.3.1.cmml" xref="S2.5.p2.29.m29.1.1.6.3">subscript</csymbol><ci id="S2.5.p2.29.m29.1.1.6.3.2.cmml" xref="S2.5.p2.29.m29.1.1.6.3.2">𝐵</ci><ci id="S2.5.p2.29.m29.1.1.6.3.3.cmml" xref="S2.5.p2.29.m29.1.1.6.3.3">𝑦</ci></apply></apply></apply><apply id="S2.5.p2.29.m29.1.1e.cmml" xref="S2.5.p2.29.m29.1.1"><eq id="S2.5.p2.29.m29.1.1.7.cmml" xref="S2.5.p2.29.m29.1.1.7"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.5.p2.29.m29.1.1.6.cmml" id="S2.5.p2.29.m29.1.1f.cmml" xref="S2.5.p2.29.m29.1.1"></share><apply id="S2.5.p2.29.m29.1.1.8.cmml" xref="S2.5.p2.29.m29.1.1.8"><csymbol cd="ambiguous" id="S2.5.p2.29.m29.1.1.8.1.cmml" xref="S2.5.p2.29.m29.1.1.8">subscript</csymbol><partialdiff id="S2.5.p2.29.m29.1.1.8.2.cmml" xref="S2.5.p2.29.m29.1.1.8.2"></partialdiff><ci id="S2.5.p2.29.m29.1.1.8.3.cmml" xref="S2.5.p2.29.m29.1.1.8.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.29.m29.1c">B_{x}\cap B_{y}=(\partial_{x}\cup(A^{x}\cap B^{x}))\cap B_{y}=\partial_{x}\cap B% _{y}=\partial_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.29.m29.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = ( ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∪ ( italic_A start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ) ) ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> since <math alttext="\partial_{x}=A^{y}\cap B_{y}\subseteq B_{y}" class="ltx_Math" display="inline" id="S2.5.p2.30.m30.1"><semantics id="S2.5.p2.30.m30.1a"><mrow id="S2.5.p2.30.m30.1.1" xref="S2.5.p2.30.m30.1.1.cmml"><msub id="S2.5.p2.30.m30.1.1.2" xref="S2.5.p2.30.m30.1.1.2.cmml"><mo id="S2.5.p2.30.m30.1.1.2.2" xref="S2.5.p2.30.m30.1.1.2.2.cmml">∂</mo><mi id="S2.5.p2.30.m30.1.1.2.3" xref="S2.5.p2.30.m30.1.1.2.3.cmml">x</mi></msub><mo id="S2.5.p2.30.m30.1.1.3" lspace="0.278em" xref="S2.5.p2.30.m30.1.1.3.cmml">=</mo><mrow id="S2.5.p2.30.m30.1.1.4" xref="S2.5.p2.30.m30.1.1.4.cmml"><msup id="S2.5.p2.30.m30.1.1.4.2" xref="S2.5.p2.30.m30.1.1.4.2.cmml"><mi id="S2.5.p2.30.m30.1.1.4.2.2" xref="S2.5.p2.30.m30.1.1.4.2.2.cmml">A</mi><mi id="S2.5.p2.30.m30.1.1.4.2.3" xref="S2.5.p2.30.m30.1.1.4.2.3.cmml">y</mi></msup><mo id="S2.5.p2.30.m30.1.1.4.1" xref="S2.5.p2.30.m30.1.1.4.1.cmml">∩</mo><msub id="S2.5.p2.30.m30.1.1.4.3" xref="S2.5.p2.30.m30.1.1.4.3.cmml"><mi id="S2.5.p2.30.m30.1.1.4.3.2" xref="S2.5.p2.30.m30.1.1.4.3.2.cmml">B</mi><mi id="S2.5.p2.30.m30.1.1.4.3.3" xref="S2.5.p2.30.m30.1.1.4.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.30.m30.1.1.5" xref="S2.5.p2.30.m30.1.1.5.cmml">⊆</mo><msub id="S2.5.p2.30.m30.1.1.6" xref="S2.5.p2.30.m30.1.1.6.cmml"><mi id="S2.5.p2.30.m30.1.1.6.2" xref="S2.5.p2.30.m30.1.1.6.2.cmml">B</mi><mi id="S2.5.p2.30.m30.1.1.6.3" xref="S2.5.p2.30.m30.1.1.6.3.cmml">y</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.30.m30.1b"><apply id="S2.5.p2.30.m30.1.1.cmml" xref="S2.5.p2.30.m30.1.1"><and id="S2.5.p2.30.m30.1.1a.cmml" xref="S2.5.p2.30.m30.1.1"></and><apply id="S2.5.p2.30.m30.1.1b.cmml" xref="S2.5.p2.30.m30.1.1"><eq id="S2.5.p2.30.m30.1.1.3.cmml" xref="S2.5.p2.30.m30.1.1.3"></eq><apply id="S2.5.p2.30.m30.1.1.2.cmml" xref="S2.5.p2.30.m30.1.1.2"><csymbol cd="ambiguous" id="S2.5.p2.30.m30.1.1.2.1.cmml" xref="S2.5.p2.30.m30.1.1.2">subscript</csymbol><partialdiff id="S2.5.p2.30.m30.1.1.2.2.cmml" xref="S2.5.p2.30.m30.1.1.2.2"></partialdiff><ci id="S2.5.p2.30.m30.1.1.2.3.cmml" xref="S2.5.p2.30.m30.1.1.2.3">𝑥</ci></apply><apply id="S2.5.p2.30.m30.1.1.4.cmml" xref="S2.5.p2.30.m30.1.1.4"><intersect id="S2.5.p2.30.m30.1.1.4.1.cmml" xref="S2.5.p2.30.m30.1.1.4.1"></intersect><apply id="S2.5.p2.30.m30.1.1.4.2.cmml" xref="S2.5.p2.30.m30.1.1.4.2"><csymbol cd="ambiguous" id="S2.5.p2.30.m30.1.1.4.2.1.cmml" xref="S2.5.p2.30.m30.1.1.4.2">superscript</csymbol><ci id="S2.5.p2.30.m30.1.1.4.2.2.cmml" xref="S2.5.p2.30.m30.1.1.4.2.2">𝐴</ci><ci id="S2.5.p2.30.m30.1.1.4.2.3.cmml" xref="S2.5.p2.30.m30.1.1.4.2.3">𝑦</ci></apply><apply id="S2.5.p2.30.m30.1.1.4.3.cmml" xref="S2.5.p2.30.m30.1.1.4.3"><csymbol cd="ambiguous" id="S2.5.p2.30.m30.1.1.4.3.1.cmml" xref="S2.5.p2.30.m30.1.1.4.3">subscript</csymbol><ci id="S2.5.p2.30.m30.1.1.4.3.2.cmml" xref="S2.5.p2.30.m30.1.1.4.3.2">𝐵</ci><ci id="S2.5.p2.30.m30.1.1.4.3.3.cmml" xref="S2.5.p2.30.m30.1.1.4.3.3">𝑦</ci></apply></apply></apply><apply id="S2.5.p2.30.m30.1.1c.cmml" xref="S2.5.p2.30.m30.1.1"><subset id="S2.5.p2.30.m30.1.1.5.cmml" xref="S2.5.p2.30.m30.1.1.5"></subset><share href="https://arxiv.org/html/2503.17112v1#S2.5.p2.30.m30.1.1.4.cmml" id="S2.5.p2.30.m30.1.1d.cmml" xref="S2.5.p2.30.m30.1.1"></share><apply id="S2.5.p2.30.m30.1.1.6.cmml" xref="S2.5.p2.30.m30.1.1.6"><csymbol cd="ambiguous" id="S2.5.p2.30.m30.1.1.6.1.cmml" xref="S2.5.p2.30.m30.1.1.6">subscript</csymbol><ci id="S2.5.p2.30.m30.1.1.6.2.cmml" xref="S2.5.p2.30.m30.1.1.6.2">𝐵</ci><ci id="S2.5.p2.30.m30.1.1.6.3.cmml" xref="S2.5.p2.30.m30.1.1.6.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.30.m30.1c">\partial_{x}=A^{y}\cap B_{y}\subseteq B_{y}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.30.m30.1d">∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⊆ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. In either case <math alttext="\operatorname{\partial}_{\mathcal{T}}(x)=B_{x}\cap B_{y}=\partial_{x}" class="ltx_Math" display="inline" id="S2.5.p2.31.m31.2"><semantics id="S2.5.p2.31.m31.2a"><mrow id="S2.5.p2.31.m31.2.2" xref="S2.5.p2.31.m31.2.2.cmml"><mrow id="S2.5.p2.31.m31.2.2.1.1" xref="S2.5.p2.31.m31.2.2.1.2.cmml"><msub id="S2.5.p2.31.m31.2.2.1.1.1" xref="S2.5.p2.31.m31.2.2.1.1.1.cmml"><mi id="S2.5.p2.31.m31.2.2.1.1.1.2" mathvariant="normal" xref="S2.5.p2.31.m31.2.2.1.1.1.2.cmml">∂</mi><mi class="ltx_font_mathcaligraphic" id="S2.5.p2.31.m31.2.2.1.1.1.3" xref="S2.5.p2.31.m31.2.2.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.5.p2.31.m31.2.2.1.1a" xref="S2.5.p2.31.m31.2.2.1.2.cmml">⁡</mo><mrow id="S2.5.p2.31.m31.2.2.1.1.2" xref="S2.5.p2.31.m31.2.2.1.2.cmml"><mo id="S2.5.p2.31.m31.2.2.1.1.2.1" stretchy="false" xref="S2.5.p2.31.m31.2.2.1.2.cmml">(</mo><mi id="S2.5.p2.31.m31.1.1" xref="S2.5.p2.31.m31.1.1.cmml">x</mi><mo id="S2.5.p2.31.m31.2.2.1.1.2.2" stretchy="false" xref="S2.5.p2.31.m31.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.31.m31.2.2.3" xref="S2.5.p2.31.m31.2.2.3.cmml">=</mo><mrow id="S2.5.p2.31.m31.2.2.4" xref="S2.5.p2.31.m31.2.2.4.cmml"><msub id="S2.5.p2.31.m31.2.2.4.2" xref="S2.5.p2.31.m31.2.2.4.2.cmml"><mi id="S2.5.p2.31.m31.2.2.4.2.2" xref="S2.5.p2.31.m31.2.2.4.2.2.cmml">B</mi><mi id="S2.5.p2.31.m31.2.2.4.2.3" xref="S2.5.p2.31.m31.2.2.4.2.3.cmml">x</mi></msub><mo id="S2.5.p2.31.m31.2.2.4.1" xref="S2.5.p2.31.m31.2.2.4.1.cmml">∩</mo><msub id="S2.5.p2.31.m31.2.2.4.3" xref="S2.5.p2.31.m31.2.2.4.3.cmml"><mi id="S2.5.p2.31.m31.2.2.4.3.2" xref="S2.5.p2.31.m31.2.2.4.3.2.cmml">B</mi><mi id="S2.5.p2.31.m31.2.2.4.3.3" xref="S2.5.p2.31.m31.2.2.4.3.3.cmml">y</mi></msub></mrow><mo id="S2.5.p2.31.m31.2.2.5" rspace="0.1389em" xref="S2.5.p2.31.m31.2.2.5.cmml">=</mo><msub id="S2.5.p2.31.m31.2.2.6" xref="S2.5.p2.31.m31.2.2.6.cmml"><mo id="S2.5.p2.31.m31.2.2.6.2" lspace="0.1389em" xref="S2.5.p2.31.m31.2.2.6.2.cmml">∂</mo><mi id="S2.5.p2.31.m31.2.2.6.3" xref="S2.5.p2.31.m31.2.2.6.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.31.m31.2b"><apply id="S2.5.p2.31.m31.2.2.cmml" xref="S2.5.p2.31.m31.2.2"><and id="S2.5.p2.31.m31.2.2a.cmml" xref="S2.5.p2.31.m31.2.2"></and><apply id="S2.5.p2.31.m31.2.2b.cmml" xref="S2.5.p2.31.m31.2.2"><eq id="S2.5.p2.31.m31.2.2.3.cmml" xref="S2.5.p2.31.m31.2.2.3"></eq><apply id="S2.5.p2.31.m31.2.2.1.2.cmml" xref="S2.5.p2.31.m31.2.2.1.1"><apply id="S2.5.p2.31.m31.2.2.1.1.1.cmml" xref="S2.5.p2.31.m31.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.31.m31.2.2.1.1.1.1.cmml" xref="S2.5.p2.31.m31.2.2.1.1.1">subscript</csymbol><partialdiff id="S2.5.p2.31.m31.2.2.1.1.1.2.cmml" xref="S2.5.p2.31.m31.2.2.1.1.1.2"></partialdiff><ci id="S2.5.p2.31.m31.2.2.1.1.1.3.cmml" xref="S2.5.p2.31.m31.2.2.1.1.1.3">𝒯</ci></apply><ci id="S2.5.p2.31.m31.1.1.cmml" xref="S2.5.p2.31.m31.1.1">𝑥</ci></apply><apply id="S2.5.p2.31.m31.2.2.4.cmml" xref="S2.5.p2.31.m31.2.2.4"><intersect id="S2.5.p2.31.m31.2.2.4.1.cmml" xref="S2.5.p2.31.m31.2.2.4.1"></intersect><apply id="S2.5.p2.31.m31.2.2.4.2.cmml" xref="S2.5.p2.31.m31.2.2.4.2"><csymbol cd="ambiguous" id="S2.5.p2.31.m31.2.2.4.2.1.cmml" xref="S2.5.p2.31.m31.2.2.4.2">subscript</csymbol><ci id="S2.5.p2.31.m31.2.2.4.2.2.cmml" xref="S2.5.p2.31.m31.2.2.4.2.2">𝐵</ci><ci id="S2.5.p2.31.m31.2.2.4.2.3.cmml" xref="S2.5.p2.31.m31.2.2.4.2.3">𝑥</ci></apply><apply id="S2.5.p2.31.m31.2.2.4.3.cmml" xref="S2.5.p2.31.m31.2.2.4.3"><csymbol cd="ambiguous" id="S2.5.p2.31.m31.2.2.4.3.1.cmml" xref="S2.5.p2.31.m31.2.2.4.3">subscript</csymbol><ci id="S2.5.p2.31.m31.2.2.4.3.2.cmml" xref="S2.5.p2.31.m31.2.2.4.3.2">𝐵</ci><ci id="S2.5.p2.31.m31.2.2.4.3.3.cmml" xref="S2.5.p2.31.m31.2.2.4.3.3">𝑦</ci></apply></apply></apply><apply id="S2.5.p2.31.m31.2.2c.cmml" xref="S2.5.p2.31.m31.2.2"><eq id="S2.5.p2.31.m31.2.2.5.cmml" xref="S2.5.p2.31.m31.2.2.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.5.p2.31.m31.2.2.4.cmml" id="S2.5.p2.31.m31.2.2d.cmml" xref="S2.5.p2.31.m31.2.2"></share><apply id="S2.5.p2.31.m31.2.2.6.cmml" xref="S2.5.p2.31.m31.2.2.6"><csymbol cd="ambiguous" id="S2.5.p2.31.m31.2.2.6.1.cmml" xref="S2.5.p2.31.m31.2.2.6">subscript</csymbol><partialdiff id="S2.5.p2.31.m31.2.2.6.2.cmml" xref="S2.5.p2.31.m31.2.2.6.2"></partialdiff><ci id="S2.5.p2.31.m31.2.2.6.3.cmml" xref="S2.5.p2.31.m31.2.2.6.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.31.m31.2c">\operatorname{\partial}_{\mathcal{T}}(x)=B_{x}\cap B_{y}=\partial_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.31.m31.2d">∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) = italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.6.p3"> <p class="ltx_p" id="S2.6.p3.16">Now the bounds on <math alttext="|\operatorname{int}_{\mathcal{T}}(x)|" class="ltx_Math" display="inline" id="S2.6.p3.1.m1.2"><semantics id="S2.6.p3.1.m1.2a"><mrow id="S2.6.p3.1.m1.2.2.1" xref="S2.6.p3.1.m1.2.2.2.cmml"><mo id="S2.6.p3.1.m1.2.2.1.2" stretchy="false" xref="S2.6.p3.1.m1.2.2.2.1.cmml">|</mo><mrow id="S2.6.p3.1.m1.2.2.1.1.1" xref="S2.6.p3.1.m1.2.2.1.1.2.cmml"><msub id="S2.6.p3.1.m1.2.2.1.1.1.1" xref="S2.6.p3.1.m1.2.2.1.1.1.1.cmml"><mi id="S2.6.p3.1.m1.2.2.1.1.1.1.2" xref="S2.6.p3.1.m1.2.2.1.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.6.p3.1.m1.2.2.1.1.1.1.3" xref="S2.6.p3.1.m1.2.2.1.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.6.p3.1.m1.2.2.1.1.1a" xref="S2.6.p3.1.m1.2.2.1.1.2.cmml">⁡</mo><mrow id="S2.6.p3.1.m1.2.2.1.1.1.2" xref="S2.6.p3.1.m1.2.2.1.1.2.cmml"><mo id="S2.6.p3.1.m1.2.2.1.1.1.2.1" stretchy="false" xref="S2.6.p3.1.m1.2.2.1.1.2.cmml">(</mo><mi id="S2.6.p3.1.m1.1.1" xref="S2.6.p3.1.m1.1.1.cmml">x</mi><mo id="S2.6.p3.1.m1.2.2.1.1.1.2.2" stretchy="false" xref="S2.6.p3.1.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.6.p3.1.m1.2.2.1.3" stretchy="false" xref="S2.6.p3.1.m1.2.2.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.1.m1.2b"><apply id="S2.6.p3.1.m1.2.2.2.cmml" xref="S2.6.p3.1.m1.2.2.1"><abs id="S2.6.p3.1.m1.2.2.2.1.cmml" xref="S2.6.p3.1.m1.2.2.1.2"></abs><apply id="S2.6.p3.1.m1.2.2.1.1.2.cmml" xref="S2.6.p3.1.m1.2.2.1.1.1"><apply id="S2.6.p3.1.m1.2.2.1.1.1.1.cmml" xref="S2.6.p3.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.1.m1.2.2.1.1.1.1.1.cmml" xref="S2.6.p3.1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S2.6.p3.1.m1.2.2.1.1.1.1.2.cmml" xref="S2.6.p3.1.m1.2.2.1.1.1.1.2">int</ci><ci id="S2.6.p3.1.m1.2.2.1.1.1.1.3.cmml" xref="S2.6.p3.1.m1.2.2.1.1.1.1.3">𝒯</ci></apply><ci id="S2.6.p3.1.m1.1.1.cmml" xref="S2.6.p3.1.m1.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.1.m1.2c">|\operatorname{int}_{\mathcal{T}}(x)|</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.1.m1.2d">| roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) |</annotation></semantics></math> and <math alttext="|\operatorname{\partial}_{\mathcal{T}}(x)|" class="ltx_Math" display="inline" id="S2.6.p3.2.m2.2"><semantics id="S2.6.p3.2.m2.2a"><mrow id="S2.6.p3.2.m2.2.2.1" xref="S2.6.p3.2.m2.2.2.2.cmml"><mo id="S2.6.p3.2.m2.2.2.1.2" stretchy="false" xref="S2.6.p3.2.m2.2.2.2.1.cmml">|</mo><mrow id="S2.6.p3.2.m2.2.2.1.1.1" xref="S2.6.p3.2.m2.2.2.1.1.2.cmml"><msub id="S2.6.p3.2.m2.2.2.1.1.1.1" xref="S2.6.p3.2.m2.2.2.1.1.1.1.cmml"><mi id="S2.6.p3.2.m2.2.2.1.1.1.1.2" mathvariant="normal" xref="S2.6.p3.2.m2.2.2.1.1.1.1.2.cmml">∂</mi><mi class="ltx_font_mathcaligraphic" id="S2.6.p3.2.m2.2.2.1.1.1.1.3" xref="S2.6.p3.2.m2.2.2.1.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.6.p3.2.m2.2.2.1.1.1a" xref="S2.6.p3.2.m2.2.2.1.1.2.cmml">⁡</mo><mrow id="S2.6.p3.2.m2.2.2.1.1.1.2" xref="S2.6.p3.2.m2.2.2.1.1.2.cmml"><mo id="S2.6.p3.2.m2.2.2.1.1.1.2.1" stretchy="false" xref="S2.6.p3.2.m2.2.2.1.1.2.cmml">(</mo><mi id="S2.6.p3.2.m2.1.1" xref="S2.6.p3.2.m2.1.1.cmml">x</mi><mo id="S2.6.p3.2.m2.2.2.1.1.1.2.2" stretchy="false" xref="S2.6.p3.2.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.6.p3.2.m2.2.2.1.3" stretchy="false" xref="S2.6.p3.2.m2.2.2.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.2.m2.2b"><apply id="S2.6.p3.2.m2.2.2.2.cmml" xref="S2.6.p3.2.m2.2.2.1"><abs id="S2.6.p3.2.m2.2.2.2.1.cmml" xref="S2.6.p3.2.m2.2.2.1.2"></abs><apply id="S2.6.p3.2.m2.2.2.1.1.2.cmml" xref="S2.6.p3.2.m2.2.2.1.1.1"><apply id="S2.6.p3.2.m2.2.2.1.1.1.1.cmml" xref="S2.6.p3.2.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.2.m2.2.2.1.1.1.1.1.cmml" xref="S2.6.p3.2.m2.2.2.1.1.1.1">subscript</csymbol><partialdiff id="S2.6.p3.2.m2.2.2.1.1.1.1.2.cmml" xref="S2.6.p3.2.m2.2.2.1.1.1.1.2"></partialdiff><ci id="S2.6.p3.2.m2.2.2.1.1.1.1.3.cmml" xref="S2.6.p3.2.m2.2.2.1.1.1.1.3">𝒯</ci></apply><ci id="S2.6.p3.2.m2.1.1.cmml" xref="S2.6.p3.2.m2.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.2.m2.2c">|\operatorname{\partial}_{\mathcal{T}}(x)|</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.2.m2.2d">| ∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) |</annotation></semantics></math> are easily established by induction on <math alttext="d:=\operatorname{depth}_{T}(x)" class="ltx_Math" display="inline" id="S2.6.p3.3.m3.2"><semantics id="S2.6.p3.3.m3.2a"><mrow id="S2.6.p3.3.m3.2.2" xref="S2.6.p3.3.m3.2.2.cmml"><mi id="S2.6.p3.3.m3.2.2.3" xref="S2.6.p3.3.m3.2.2.3.cmml">d</mi><mo id="S2.6.p3.3.m3.2.2.2" lspace="0.278em" rspace="0.278em" xref="S2.6.p3.3.m3.2.2.2.cmml">:=</mo><mrow id="S2.6.p3.3.m3.2.2.1.1" xref="S2.6.p3.3.m3.2.2.1.2.cmml"><msub id="S2.6.p3.3.m3.2.2.1.1.1" xref="S2.6.p3.3.m3.2.2.1.1.1.cmml"><mi id="S2.6.p3.3.m3.2.2.1.1.1.2" xref="S2.6.p3.3.m3.2.2.1.1.1.2.cmml">depth</mi><mi id="S2.6.p3.3.m3.2.2.1.1.1.3" xref="S2.6.p3.3.m3.2.2.1.1.1.3.cmml">T</mi></msub><mo id="S2.6.p3.3.m3.2.2.1.1a" xref="S2.6.p3.3.m3.2.2.1.2.cmml">⁡</mo><mrow id="S2.6.p3.3.m3.2.2.1.1.2" xref="S2.6.p3.3.m3.2.2.1.2.cmml"><mo id="S2.6.p3.3.m3.2.2.1.1.2.1" stretchy="false" xref="S2.6.p3.3.m3.2.2.1.2.cmml">(</mo><mi id="S2.6.p3.3.m3.1.1" xref="S2.6.p3.3.m3.1.1.cmml">x</mi><mo id="S2.6.p3.3.m3.2.2.1.1.2.2" stretchy="false" xref="S2.6.p3.3.m3.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.3.m3.2b"><apply id="S2.6.p3.3.m3.2.2.cmml" xref="S2.6.p3.3.m3.2.2"><csymbol cd="latexml" id="S2.6.p3.3.m3.2.2.2.cmml" xref="S2.6.p3.3.m3.2.2.2">assign</csymbol><ci id="S2.6.p3.3.m3.2.2.3.cmml" xref="S2.6.p3.3.m3.2.2.3">𝑑</ci><apply id="S2.6.p3.3.m3.2.2.1.2.cmml" xref="S2.6.p3.3.m3.2.2.1.1"><apply id="S2.6.p3.3.m3.2.2.1.1.1.cmml" xref="S2.6.p3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" 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xref="S2.6.p3.6.m6.6.6.2.1.1.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.6.p3.6.m6.6.6.2.1.1.1.1a" xref="S2.6.p3.6.m6.6.6.2.1.1.1.2.cmml">⁡</mo><mrow id="S2.6.p3.6.m6.6.6.2.1.1.1.1.2" xref="S2.6.p3.6.m6.6.6.2.1.1.1.2.cmml"><mo id="S2.6.p3.6.m6.6.6.2.1.1.1.1.2.1" stretchy="false" xref="S2.6.p3.6.m6.6.6.2.1.1.1.2.cmml">(</mo><mi id="S2.6.p3.6.m6.3.3" xref="S2.6.p3.6.m6.3.3.cmml">x</mi><mo id="S2.6.p3.6.m6.6.6.2.1.1.1.1.2.2" stretchy="false" xref="S2.6.p3.6.m6.6.6.2.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.6.p3.6.m6.6.6.2.1.3" stretchy="false" xref="S2.6.p3.6.m6.6.6.2.2.1.cmml">|</mo></mrow><mo id="S2.6.p3.6.m6.6.6.5" xref="S2.6.p3.6.m6.6.6.5.cmml">=</mo><mi id="S2.6.p3.6.m6.6.6.6" xref="S2.6.p3.6.m6.6.6.6.cmml">n</mi><mo id="S2.6.p3.6.m6.6.6.7" xref="S2.6.p3.6.m6.6.6.7.cmml">=</mo><mrow id="S2.6.p3.6.m6.6.6.8" xref="S2.6.p3.6.m6.6.6.8.cmml"><mi id="S2.6.p3.6.m6.6.6.8.2" xref="S2.6.p3.6.m6.6.6.8.2.cmml">n</mi><mo id="S2.6.p3.6.m6.6.6.8.1" lspace="0.222em" rspace="0.222em" xref="S2.6.p3.6.m6.6.6.8.1.cmml">⋅</mo><msup id="S2.6.p3.6.m6.6.6.8.3" xref="S2.6.p3.6.m6.6.6.8.3.cmml"><mrow id="S2.6.p3.6.m6.6.6.8.3.2.2" xref="S2.6.p3.6.m6.4.4.cmml"><mo id="S2.6.p3.6.m6.6.6.8.3.2.2.1" stretchy="false" xref="S2.6.p3.6.m6.4.4.cmml">(</mo><mfrac id="S2.6.p3.6.m6.4.4" xref="S2.6.p3.6.m6.4.4.cmml"><mn id="S2.6.p3.6.m6.4.4.2" xref="S2.6.p3.6.m6.4.4.2.cmml">2</mn><mn id="S2.6.p3.6.m6.4.4.3" xref="S2.6.p3.6.m6.4.4.3.cmml">3</mn></mfrac><mo id="S2.6.p3.6.m6.6.6.8.3.2.2.2" stretchy="false" xref="S2.6.p3.6.m6.4.4.cmml">)</mo></mrow><mn id="S2.6.p3.6.m6.6.6.8.3.3" xref="S2.6.p3.6.m6.6.6.8.3.3.cmml">0</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.6.m6.6b"><apply id="S2.6.p3.6.m6.6.6.cmml" xref="S2.6.p3.6.m6.6.6"><and id="S2.6.p3.6.m6.6.6a.cmml" xref="S2.6.p3.6.m6.6.6"></and><apply id="S2.6.p3.6.m6.6.6b.cmml" xref="S2.6.p3.6.m6.6.6"><eq id="S2.6.p3.6.m6.6.6.4.cmml" xref="S2.6.p3.6.m6.6.6.4"></eq><apply id="S2.6.p3.6.m6.5.5.1.2.cmml" xref="S2.6.p3.6.m6.5.5.1.1"><abs id="S2.6.p3.6.m6.5.5.1.2.1.cmml" xref="S2.6.p3.6.m6.5.5.1.1.2"></abs><apply id="S2.6.p3.6.m6.5.5.1.1.1.2.cmml" xref="S2.6.p3.6.m6.5.5.1.1.1.1"><apply id="S2.6.p3.6.m6.5.5.1.1.1.1.1.cmml" xref="S2.6.p3.6.m6.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.6.m6.5.5.1.1.1.1.1.1.cmml" xref="S2.6.p3.6.m6.5.5.1.1.1.1.1">subscript</csymbol><ci id="S2.6.p3.6.m6.5.5.1.1.1.1.1.2.cmml" xref="S2.6.p3.6.m6.5.5.1.1.1.1.1.2">int</ci><ci id="S2.6.p3.6.m6.5.5.1.1.1.1.1.3.cmml" xref="S2.6.p3.6.m6.5.5.1.1.1.1.1.3">𝒯</ci></apply><ci id="S2.6.p3.6.m6.1.1.cmml" xref="S2.6.p3.6.m6.1.1">𝑥</ci></apply></apply><apply id="S2.6.p3.6.m6.6.6.2.2.cmml" xref="S2.6.p3.6.m6.6.6.2.1"><abs id="S2.6.p3.6.m6.6.6.2.2.1.cmml" xref="S2.6.p3.6.m6.6.6.2.1.2"></abs><apply id="S2.6.p3.6.m6.6.6.2.1.1.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1"><setdiff id="S2.6.p3.6.m6.6.6.2.1.1.2.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.2"></setdiff><apply id="S2.6.p3.6.m6.6.6.2.1.1.3.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.3"><times id="S2.6.p3.6.m6.6.6.2.1.1.3.1.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.3.1"></times><ci id="S2.6.p3.6.m6.6.6.2.1.1.3.2.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.3.2">𝑉</ci><ci id="S2.6.p3.6.m6.2.2.cmml" xref="S2.6.p3.6.m6.2.2">𝐺</ci></apply><apply id="S2.6.p3.6.m6.6.6.2.1.1.1.2.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.1.1"><apply id="S2.6.p3.6.m6.6.6.2.1.1.1.1.1.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.6.m6.6.6.2.1.1.1.1.1.1.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.1.1.1">subscript</csymbol><partialdiff id="S2.6.p3.6.m6.6.6.2.1.1.1.1.1.2.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.1.1.1.2"></partialdiff><ci id="S2.6.p3.6.m6.6.6.2.1.1.1.1.1.3.cmml" xref="S2.6.p3.6.m6.6.6.2.1.1.1.1.1.3">𝒯</ci></apply><ci id="S2.6.p3.6.m6.3.3.cmml" xref="S2.6.p3.6.m6.3.3">𝑥</ci></apply></apply></apply></apply><apply id="S2.6.p3.6.m6.6.6c.cmml" xref="S2.6.p3.6.m6.6.6"><eq id="S2.6.p3.6.m6.6.6.5.cmml" xref="S2.6.p3.6.m6.6.6.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.6.p3.6.m6.6.6.2.cmml" id="S2.6.p3.6.m6.6.6d.cmml" xref="S2.6.p3.6.m6.6.6"></share><ci id="S2.6.p3.6.m6.6.6.6.cmml" xref="S2.6.p3.6.m6.6.6.6">𝑛</ci></apply><apply id="S2.6.p3.6.m6.6.6e.cmml" xref="S2.6.p3.6.m6.6.6"><eq id="S2.6.p3.6.m6.6.6.7.cmml" xref="S2.6.p3.6.m6.6.6.7"></eq><share href="https://arxiv.org/html/2503.17112v1#S2.6.p3.6.m6.6.6.6.cmml" id="S2.6.p3.6.m6.6.6f.cmml" xref="S2.6.p3.6.m6.6.6"></share><apply id="S2.6.p3.6.m6.6.6.8.cmml" xref="S2.6.p3.6.m6.6.6.8"><ci id="S2.6.p3.6.m6.6.6.8.1.cmml" xref="S2.6.p3.6.m6.6.6.8.1">⋅</ci><ci id="S2.6.p3.6.m6.6.6.8.2.cmml" xref="S2.6.p3.6.m6.6.6.8.2">𝑛</ci><apply id="S2.6.p3.6.m6.6.6.8.3.cmml" xref="S2.6.p3.6.m6.6.6.8.3"><csymbol cd="ambiguous" id="S2.6.p3.6.m6.6.6.8.3.1.cmml" xref="S2.6.p3.6.m6.6.6.8.3">superscript</csymbol><apply id="S2.6.p3.6.m6.4.4.cmml" xref="S2.6.p3.6.m6.6.6.8.3.2.2"><divide id="S2.6.p3.6.m6.4.4.1.cmml" xref="S2.6.p3.6.m6.6.6.8.3.2.2"></divide><cn id="S2.6.p3.6.m6.4.4.2.cmml" type="integer" xref="S2.6.p3.6.m6.4.4.2">2</cn><cn id="S2.6.p3.6.m6.4.4.3.cmml" type="integer" xref="S2.6.p3.6.m6.4.4.3">3</cn></apply><cn id="S2.6.p3.6.m6.6.6.8.3.3.cmml" type="integer" xref="S2.6.p3.6.m6.6.6.8.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.6.m6.6c">|\operatorname{int}_{\mathcal{T}}(x)|=|V(G)\setminus\operatorname{\partial}_{% \mathcal{T}}(x)|=n=n\cdot(\tfrac{2}{3})^{0}</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.6.m6.6d">| roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) | = | italic_V ( italic_G ) ∖ ∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) | = italic_n = italic_n ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT</annotation></semantics></math>. Now consider a node <math alttext="x" class="ltx_Math" display="inline" id="S2.6.p3.7.m7.1"><semantics id="S2.6.p3.7.m7.1a"><mi id="S2.6.p3.7.m7.1.1" xref="S2.6.p3.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.6.p3.7.m7.1b"><ci id="S2.6.p3.7.m7.1.1.cmml" xref="S2.6.p3.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.7.m7.1d">italic_x</annotation></semantics></math> of depth <math alttext="d\geq 1" class="ltx_Math" display="inline" id="S2.6.p3.8.m8.1"><semantics id="S2.6.p3.8.m8.1a"><mrow id="S2.6.p3.8.m8.1.1" xref="S2.6.p3.8.m8.1.1.cmml"><mi id="S2.6.p3.8.m8.1.1.2" xref="S2.6.p3.8.m8.1.1.2.cmml">d</mi><mo id="S2.6.p3.8.m8.1.1.1" xref="S2.6.p3.8.m8.1.1.1.cmml">≥</mo><mn id="S2.6.p3.8.m8.1.1.3" xref="S2.6.p3.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.8.m8.1b"><apply id="S2.6.p3.8.m8.1.1.cmml" xref="S2.6.p3.8.m8.1.1"><geq id="S2.6.p3.8.m8.1.1.1.cmml" xref="S2.6.p3.8.m8.1.1.1"></geq><ci id="S2.6.p3.8.m8.1.1.2.cmml" xref="S2.6.p3.8.m8.1.1.2">𝑑</ci><cn id="S2.6.p3.8.m8.1.1.3.cmml" type="integer" xref="S2.6.p3.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.8.m8.1c">d\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.8.m8.1d">italic_d ≥ 1</annotation></semantics></math> with parent <math alttext="y" class="ltx_Math" display="inline" id="S2.6.p3.9.m9.1"><semantics id="S2.6.p3.9.m9.1a"><mi id="S2.6.p3.9.m9.1.1" xref="S2.6.p3.9.m9.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.6.p3.9.m9.1b"><ci id="S2.6.p3.9.m9.1.1.cmml" xref="S2.6.p3.9.m9.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.9.m9.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.9.m9.1d">italic_y</annotation></semantics></math> that was created by a recursive invocation on <math alttext="(G_{y},\partial_{y})=(G_{y},\operatorname{\partial}_{\mathcal{T}}(y))" class="ltx_Math" display="inline" id="S2.6.p3.10.m10.5"><semantics id="S2.6.p3.10.m10.5a"><mrow id="S2.6.p3.10.m10.5.5" xref="S2.6.p3.10.m10.5.5.cmml"><mrow id="S2.6.p3.10.m10.3.3.2.2" xref="S2.6.p3.10.m10.3.3.2.3.cmml"><mo id="S2.6.p3.10.m10.3.3.2.2.3" stretchy="false" xref="S2.6.p3.10.m10.3.3.2.3.cmml">(</mo><msub id="S2.6.p3.10.m10.2.2.1.1.1" xref="S2.6.p3.10.m10.2.2.1.1.1.cmml"><mi id="S2.6.p3.10.m10.2.2.1.1.1.2" xref="S2.6.p3.10.m10.2.2.1.1.1.2.cmml">G</mi><mi id="S2.6.p3.10.m10.2.2.1.1.1.3" xref="S2.6.p3.10.m10.2.2.1.1.1.3.cmml">y</mi></msub><mo id="S2.6.p3.10.m10.3.3.2.2.4" xref="S2.6.p3.10.m10.3.3.2.3.cmml">,</mo><msub id="S2.6.p3.10.m10.3.3.2.2.2" xref="S2.6.p3.10.m10.3.3.2.2.2.cmml"><mo id="S2.6.p3.10.m10.3.3.2.2.2.2" lspace="0em" rspace="0em" xref="S2.6.p3.10.m10.3.3.2.2.2.2.cmml">∂</mo><mi id="S2.6.p3.10.m10.3.3.2.2.2.3" xref="S2.6.p3.10.m10.3.3.2.2.2.3.cmml">y</mi></msub><mo id="S2.6.p3.10.m10.3.3.2.2.5" stretchy="false" xref="S2.6.p3.10.m10.3.3.2.3.cmml">)</mo></mrow><mo id="S2.6.p3.10.m10.5.5.5" xref="S2.6.p3.10.m10.5.5.5.cmml">=</mo><mrow id="S2.6.p3.10.m10.5.5.4.2" xref="S2.6.p3.10.m10.5.5.4.3.cmml"><mo id="S2.6.p3.10.m10.5.5.4.2.3" stretchy="false" xref="S2.6.p3.10.m10.5.5.4.3.cmml">(</mo><msub id="S2.6.p3.10.m10.4.4.3.1.1" xref="S2.6.p3.10.m10.4.4.3.1.1.cmml"><mi id="S2.6.p3.10.m10.4.4.3.1.1.2" xref="S2.6.p3.10.m10.4.4.3.1.1.2.cmml">G</mi><mi id="S2.6.p3.10.m10.4.4.3.1.1.3" xref="S2.6.p3.10.m10.4.4.3.1.1.3.cmml">y</mi></msub><mo id="S2.6.p3.10.m10.5.5.4.2.4" xref="S2.6.p3.10.m10.5.5.4.3.cmml">,</mo><mrow id="S2.6.p3.10.m10.5.5.4.2.2.1" xref="S2.6.p3.10.m10.5.5.4.2.2.2.cmml"><msub id="S2.6.p3.10.m10.5.5.4.2.2.1.1" xref="S2.6.p3.10.m10.5.5.4.2.2.1.1.cmml"><mi id="S2.6.p3.10.m10.5.5.4.2.2.1.1.2" mathvariant="normal" xref="S2.6.p3.10.m10.5.5.4.2.2.1.1.2.cmml">∂</mi><mi class="ltx_font_mathcaligraphic" id="S2.6.p3.10.m10.5.5.4.2.2.1.1.3" xref="S2.6.p3.10.m10.5.5.4.2.2.1.1.3.cmml">𝒯</mi></msub><mo id="S2.6.p3.10.m10.5.5.4.2.2.1a" xref="S2.6.p3.10.m10.5.5.4.2.2.2.cmml">⁡</mo><mrow id="S2.6.p3.10.m10.5.5.4.2.2.1.2" xref="S2.6.p3.10.m10.5.5.4.2.2.2.cmml"><mo id="S2.6.p3.10.m10.5.5.4.2.2.1.2.1" stretchy="false" xref="S2.6.p3.10.m10.5.5.4.2.2.2.cmml">(</mo><mi id="S2.6.p3.10.m10.1.1" xref="S2.6.p3.10.m10.1.1.cmml">y</mi><mo id="S2.6.p3.10.m10.5.5.4.2.2.1.2.2" stretchy="false" xref="S2.6.p3.10.m10.5.5.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.6.p3.10.m10.5.5.4.2.5" stretchy="false" xref="S2.6.p3.10.m10.5.5.4.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.10.m10.5b"><apply id="S2.6.p3.10.m10.5.5.cmml" xref="S2.6.p3.10.m10.5.5"><eq id="S2.6.p3.10.m10.5.5.5.cmml" xref="S2.6.p3.10.m10.5.5.5"></eq><interval closure="open" id="S2.6.p3.10.m10.3.3.2.3.cmml" xref="S2.6.p3.10.m10.3.3.2.2"><apply id="S2.6.p3.10.m10.2.2.1.1.1.cmml" xref="S2.6.p3.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.10.m10.2.2.1.1.1.1.cmml" xref="S2.6.p3.10.m10.2.2.1.1.1">subscript</csymbol><ci id="S2.6.p3.10.m10.2.2.1.1.1.2.cmml" xref="S2.6.p3.10.m10.2.2.1.1.1.2">𝐺</ci><ci id="S2.6.p3.10.m10.2.2.1.1.1.3.cmml" xref="S2.6.p3.10.m10.2.2.1.1.1.3">𝑦</ci></apply><apply id="S2.6.p3.10.m10.3.3.2.2.2.cmml" xref="S2.6.p3.10.m10.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.6.p3.10.m10.3.3.2.2.2.1.cmml" xref="S2.6.p3.10.m10.3.3.2.2.2">subscript</csymbol><partialdiff id="S2.6.p3.10.m10.3.3.2.2.2.2.cmml" xref="S2.6.p3.10.m10.3.3.2.2.2.2"></partialdiff><ci id="S2.6.p3.10.m10.3.3.2.2.2.3.cmml" xref="S2.6.p3.10.m10.3.3.2.2.2.3">𝑦</ci></apply></interval><interval closure="open" id="S2.6.p3.10.m10.5.5.4.3.cmml" xref="S2.6.p3.10.m10.5.5.4.2"><apply id="S2.6.p3.10.m10.4.4.3.1.1.cmml" xref="S2.6.p3.10.m10.4.4.3.1.1"><csymbol cd="ambiguous" id="S2.6.p3.10.m10.4.4.3.1.1.1.cmml" xref="S2.6.p3.10.m10.4.4.3.1.1">subscript</csymbol><ci id="S2.6.p3.10.m10.4.4.3.1.1.2.cmml" xref="S2.6.p3.10.m10.4.4.3.1.1.2">𝐺</ci><ci id="S2.6.p3.10.m10.4.4.3.1.1.3.cmml" xref="S2.6.p3.10.m10.4.4.3.1.1.3">𝑦</ci></apply><apply id="S2.6.p3.10.m10.5.5.4.2.2.2.cmml" xref="S2.6.p3.10.m10.5.5.4.2.2.1"><apply id="S2.6.p3.10.m10.5.5.4.2.2.1.1.cmml" xref="S2.6.p3.10.m10.5.5.4.2.2.1.1"><csymbol cd="ambiguous" id="S2.6.p3.10.m10.5.5.4.2.2.1.1.1.cmml" xref="S2.6.p3.10.m10.5.5.4.2.2.1.1">subscript</csymbol><partialdiff id="S2.6.p3.10.m10.5.5.4.2.2.1.1.2.cmml" xref="S2.6.p3.10.m10.5.5.4.2.2.1.1.2"></partialdiff><ci id="S2.6.p3.10.m10.5.5.4.2.2.1.1.3.cmml" xref="S2.6.p3.10.m10.5.5.4.2.2.1.1.3">𝒯</ci></apply><ci id="S2.6.p3.10.m10.1.1.cmml" xref="S2.6.p3.10.m10.1.1">𝑦</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.10.m10.5c">(G_{y},\partial_{y})=(G_{y},\operatorname{\partial}_{\mathcal{T}}(y))</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.10.m10.5d">( italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , ∂ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) = ( italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , ∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_y ) )</annotation></semantics></math>. Without loss of generality <math alttext="T_{x}" class="ltx_Math" display="inline" id="S2.6.p3.11.m11.1"><semantics id="S2.6.p3.11.m11.1a"><msub id="S2.6.p3.11.m11.1.1" xref="S2.6.p3.11.m11.1.1.cmml"><mi id="S2.6.p3.11.m11.1.1.2" xref="S2.6.p3.11.m11.1.1.2.cmml">T</mi><mi id="S2.6.p3.11.m11.1.1.3" xref="S2.6.p3.11.m11.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S2.6.p3.11.m11.1b"><apply id="S2.6.p3.11.m11.1.1.cmml" xref="S2.6.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S2.6.p3.11.m11.1.1.1.cmml" xref="S2.6.p3.11.m11.1.1">subscript</csymbol><ci id="S2.6.p3.11.m11.1.1.2.cmml" xref="S2.6.p3.11.m11.1.1.2">𝑇</ci><ci id="S2.6.p3.11.m11.1.1.3.cmml" xref="S2.6.p3.11.m11.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.11.m11.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.11.m11.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> was created by a recursive invocation on <math alttext="(G[A^{y}],\partial_{x})" class="ltx_Math" display="inline" id="S2.6.p3.12.m12.2"><semantics id="S2.6.p3.12.m12.2a"><mrow id="S2.6.p3.12.m12.2.2.2" xref="S2.6.p3.12.m12.2.2.3.cmml"><mo id="S2.6.p3.12.m12.2.2.2.3" stretchy="false" xref="S2.6.p3.12.m12.2.2.3.cmml">(</mo><mrow id="S2.6.p3.12.m12.1.1.1.1" xref="S2.6.p3.12.m12.1.1.1.1.cmml"><mi id="S2.6.p3.12.m12.1.1.1.1.3" xref="S2.6.p3.12.m12.1.1.1.1.3.cmml">G</mi><mo id="S2.6.p3.12.m12.1.1.1.1.2" xref="S2.6.p3.12.m12.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.6.p3.12.m12.1.1.1.1.1.1" xref="S2.6.p3.12.m12.1.1.1.1.1.2.cmml"><mo id="S2.6.p3.12.m12.1.1.1.1.1.1.2" stretchy="false" xref="S2.6.p3.12.m12.1.1.1.1.1.2.1.cmml">[</mo><msup id="S2.6.p3.12.m12.1.1.1.1.1.1.1" xref="S2.6.p3.12.m12.1.1.1.1.1.1.1.cmml"><mi id="S2.6.p3.12.m12.1.1.1.1.1.1.1.2" xref="S2.6.p3.12.m12.1.1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.6.p3.12.m12.1.1.1.1.1.1.1.3" xref="S2.6.p3.12.m12.1.1.1.1.1.1.1.3.cmml">y</mi></msup><mo id="S2.6.p3.12.m12.1.1.1.1.1.1.3" stretchy="false" xref="S2.6.p3.12.m12.1.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.6.p3.12.m12.2.2.2.4" xref="S2.6.p3.12.m12.2.2.3.cmml">,</mo><msub id="S2.6.p3.12.m12.2.2.2.2" xref="S2.6.p3.12.m12.2.2.2.2.cmml"><mo id="S2.6.p3.12.m12.2.2.2.2.2" lspace="0em" rspace="0em" xref="S2.6.p3.12.m12.2.2.2.2.2.cmml">∂</mo><mi id="S2.6.p3.12.m12.2.2.2.2.3" xref="S2.6.p3.12.m12.2.2.2.2.3.cmml">x</mi></msub><mo id="S2.6.p3.12.m12.2.2.2.5" stretchy="false" xref="S2.6.p3.12.m12.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.12.m12.2b"><interval closure="open" id="S2.6.p3.12.m12.2.2.3.cmml" xref="S2.6.p3.12.m12.2.2.2"><apply id="S2.6.p3.12.m12.1.1.1.1.cmml" xref="S2.6.p3.12.m12.1.1.1.1"><times id="S2.6.p3.12.m12.1.1.1.1.2.cmml" xref="S2.6.p3.12.m12.1.1.1.1.2"></times><ci id="S2.6.p3.12.m12.1.1.1.1.3.cmml" xref="S2.6.p3.12.m12.1.1.1.1.3">𝐺</ci><apply id="S2.6.p3.12.m12.1.1.1.1.1.2.cmml" xref="S2.6.p3.12.m12.1.1.1.1.1.1"><csymbol cd="latexml" id="S2.6.p3.12.m12.1.1.1.1.1.2.1.cmml" xref="S2.6.p3.12.m12.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.6.p3.12.m12.1.1.1.1.1.1.1.cmml" xref="S2.6.p3.12.m12.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.12.m12.1.1.1.1.1.1.1.1.cmml" xref="S2.6.p3.12.m12.1.1.1.1.1.1.1">superscript</csymbol><ci id="S2.6.p3.12.m12.1.1.1.1.1.1.1.2.cmml" xref="S2.6.p3.12.m12.1.1.1.1.1.1.1.2">𝐴</ci><ci id="S2.6.p3.12.m12.1.1.1.1.1.1.1.3.cmml" xref="S2.6.p3.12.m12.1.1.1.1.1.1.1.3">𝑦</ci></apply></apply></apply><apply id="S2.6.p3.12.m12.2.2.2.2.cmml" xref="S2.6.p3.12.m12.2.2.2.2"><csymbol cd="ambiguous" id="S2.6.p3.12.m12.2.2.2.2.1.cmml" xref="S2.6.p3.12.m12.2.2.2.2">subscript</csymbol><partialdiff id="S2.6.p3.12.m12.2.2.2.2.2.cmml" xref="S2.6.p3.12.m12.2.2.2.2.2"></partialdiff><ci id="S2.6.p3.12.m12.2.2.2.2.3.cmml" xref="S2.6.p3.12.m12.2.2.2.2.3">𝑥</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.12.m12.2c">(G[A^{y}],\partial_{x})</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.12.m12.2d">( italic_G [ italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ] , ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> where <math alttext="(A^{y},B^{y})" class="ltx_Math" display="inline" id="S2.6.p3.13.m13.2"><semantics id="S2.6.p3.13.m13.2a"><mrow id="S2.6.p3.13.m13.2.2.2" xref="S2.6.p3.13.m13.2.2.3.cmml"><mo id="S2.6.p3.13.m13.2.2.2.3" stretchy="false" xref="S2.6.p3.13.m13.2.2.3.cmml">(</mo><msup id="S2.6.p3.13.m13.1.1.1.1" xref="S2.6.p3.13.m13.1.1.1.1.cmml"><mi id="S2.6.p3.13.m13.1.1.1.1.2" xref="S2.6.p3.13.m13.1.1.1.1.2.cmml">A</mi><mi id="S2.6.p3.13.m13.1.1.1.1.3" xref="S2.6.p3.13.m13.1.1.1.1.3.cmml">y</mi></msup><mo id="S2.6.p3.13.m13.2.2.2.4" xref="S2.6.p3.13.m13.2.2.3.cmml">,</mo><msup id="S2.6.p3.13.m13.2.2.2.2" xref="S2.6.p3.13.m13.2.2.2.2.cmml"><mi id="S2.6.p3.13.m13.2.2.2.2.2" xref="S2.6.p3.13.m13.2.2.2.2.2.cmml">B</mi><mi id="S2.6.p3.13.m13.2.2.2.2.3" xref="S2.6.p3.13.m13.2.2.2.2.3.cmml">y</mi></msup><mo id="S2.6.p3.13.m13.2.2.2.5" stretchy="false" xref="S2.6.p3.13.m13.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.13.m13.2b"><interval closure="open" id="S2.6.p3.13.m13.2.2.3.cmml" xref="S2.6.p3.13.m13.2.2.2"><apply id="S2.6.p3.13.m13.1.1.1.1.cmml" xref="S2.6.p3.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.13.m13.1.1.1.1.1.cmml" xref="S2.6.p3.13.m13.1.1.1.1">superscript</csymbol><ci id="S2.6.p3.13.m13.1.1.1.1.2.cmml" xref="S2.6.p3.13.m13.1.1.1.1.2">𝐴</ci><ci id="S2.6.p3.13.m13.1.1.1.1.3.cmml" xref="S2.6.p3.13.m13.1.1.1.1.3">𝑦</ci></apply><apply id="S2.6.p3.13.m13.2.2.2.2.cmml" xref="S2.6.p3.13.m13.2.2.2.2"><csymbol cd="ambiguous" id="S2.6.p3.13.m13.2.2.2.2.1.cmml" xref="S2.6.p3.13.m13.2.2.2.2">superscript</csymbol><ci id="S2.6.p3.13.m13.2.2.2.2.2.cmml" xref="S2.6.p3.13.m13.2.2.2.2.2">𝐵</ci><ci id="S2.6.p3.13.m13.2.2.2.2.3.cmml" xref="S2.6.p3.13.m13.2.2.2.2.3">𝑦</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.13.m13.2c">(A^{y},B^{y})</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.13.m13.2d">( italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT , italic_B start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT )</annotation></semantics></math> is a balanced separation of <math alttext="G_{y}-\operatorname{\partial}_{y}=\operatorname{int}_{\mathcal{T}}(y)" class="ltx_Math" display="inline" id="S2.6.p3.14.m14.2"><semantics id="S2.6.p3.14.m14.2a"><mrow id="S2.6.p3.14.m14.2.2" xref="S2.6.p3.14.m14.2.2.cmml"><mrow id="S2.6.p3.14.m14.2.2.3" xref="S2.6.p3.14.m14.2.2.3.cmml"><msub id="S2.6.p3.14.m14.2.2.3.2" xref="S2.6.p3.14.m14.2.2.3.2.cmml"><mi id="S2.6.p3.14.m14.2.2.3.2.2" xref="S2.6.p3.14.m14.2.2.3.2.2.cmml">G</mi><mi id="S2.6.p3.14.m14.2.2.3.2.3" xref="S2.6.p3.14.m14.2.2.3.2.3.cmml">y</mi></msub><mo id="S2.6.p3.14.m14.2.2.3.1" xref="S2.6.p3.14.m14.2.2.3.1.cmml">−</mo><msub id="S2.6.p3.14.m14.2.2.3.3" xref="S2.6.p3.14.m14.2.2.3.3.cmml"><mi id="S2.6.p3.14.m14.2.2.3.3.2" mathvariant="normal" xref="S2.6.p3.14.m14.2.2.3.3.2.cmml">∂</mi><mi id="S2.6.p3.14.m14.2.2.3.3.3" xref="S2.6.p3.14.m14.2.2.3.3.3.cmml">y</mi></msub></mrow><mo id="S2.6.p3.14.m14.2.2.2" xref="S2.6.p3.14.m14.2.2.2.cmml">=</mo><mrow id="S2.6.p3.14.m14.2.2.1.1" xref="S2.6.p3.14.m14.2.2.1.2.cmml"><msub id="S2.6.p3.14.m14.2.2.1.1.1" xref="S2.6.p3.14.m14.2.2.1.1.1.cmml"><mi id="S2.6.p3.14.m14.2.2.1.1.1.2" xref="S2.6.p3.14.m14.2.2.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.6.p3.14.m14.2.2.1.1.1.3" xref="S2.6.p3.14.m14.2.2.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.6.p3.14.m14.2.2.1.1a" xref="S2.6.p3.14.m14.2.2.1.2.cmml">⁡</mo><mrow id="S2.6.p3.14.m14.2.2.1.1.2" xref="S2.6.p3.14.m14.2.2.1.2.cmml"><mo id="S2.6.p3.14.m14.2.2.1.1.2.1" stretchy="false" xref="S2.6.p3.14.m14.2.2.1.2.cmml">(</mo><mi id="S2.6.p3.14.m14.1.1" xref="S2.6.p3.14.m14.1.1.cmml">y</mi><mo id="S2.6.p3.14.m14.2.2.1.1.2.2" stretchy="false" xref="S2.6.p3.14.m14.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.14.m14.2b"><apply id="S2.6.p3.14.m14.2.2.cmml" xref="S2.6.p3.14.m14.2.2"><eq id="S2.6.p3.14.m14.2.2.2.cmml" xref="S2.6.p3.14.m14.2.2.2"></eq><apply id="S2.6.p3.14.m14.2.2.3.cmml" xref="S2.6.p3.14.m14.2.2.3"><minus id="S2.6.p3.14.m14.2.2.3.1.cmml" xref="S2.6.p3.14.m14.2.2.3.1"></minus><apply id="S2.6.p3.14.m14.2.2.3.2.cmml" xref="S2.6.p3.14.m14.2.2.3.2"><csymbol cd="ambiguous" id="S2.6.p3.14.m14.2.2.3.2.1.cmml" xref="S2.6.p3.14.m14.2.2.3.2">subscript</csymbol><ci id="S2.6.p3.14.m14.2.2.3.2.2.cmml" xref="S2.6.p3.14.m14.2.2.3.2.2">𝐺</ci><ci id="S2.6.p3.14.m14.2.2.3.2.3.cmml" xref="S2.6.p3.14.m14.2.2.3.2.3">𝑦</ci></apply><apply id="S2.6.p3.14.m14.2.2.3.3.cmml" xref="S2.6.p3.14.m14.2.2.3.3"><csymbol cd="ambiguous" id="S2.6.p3.14.m14.2.2.3.3.1.cmml" xref="S2.6.p3.14.m14.2.2.3.3">subscript</csymbol><partialdiff id="S2.6.p3.14.m14.2.2.3.3.2.cmml" xref="S2.6.p3.14.m14.2.2.3.3.2"></partialdiff><ci id="S2.6.p3.14.m14.2.2.3.3.3.cmml" xref="S2.6.p3.14.m14.2.2.3.3.3">𝑦</ci></apply></apply><apply id="S2.6.p3.14.m14.2.2.1.2.cmml" xref="S2.6.p3.14.m14.2.2.1.1"><apply id="S2.6.p3.14.m14.2.2.1.1.1.cmml" xref="S2.6.p3.14.m14.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.14.m14.2.2.1.1.1.1.cmml" xref="S2.6.p3.14.m14.2.2.1.1.1">subscript</csymbol><ci id="S2.6.p3.14.m14.2.2.1.1.1.2.cmml" xref="S2.6.p3.14.m14.2.2.1.1.1.2">int</ci><ci id="S2.6.p3.14.m14.2.2.1.1.1.3.cmml" xref="S2.6.p3.14.m14.2.2.1.1.1.3">𝒯</ci></apply><ci id="S2.6.p3.14.m14.1.1.cmml" xref="S2.6.p3.14.m14.1.1">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.14.m14.2c">G_{y}-\operatorname{\partial}_{y}=\operatorname{int}_{\mathcal{T}}(y)</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.14.m14.2d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - ∂ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_y )</annotation></semantics></math>. Then <math alttext="|\operatorname{int}_{\mathcal{T}}(x)|\leq\tfrac{2}{3}\cdot|\operatorname{int}_% {\mathcal{T}}(y)|\leq\tfrac{2}{3}\cdot(\frac{2}{3})^{d-1}\cdot n=(\tfrac{2}{3}% )^{d}\cdot n" class="ltx_Math" display="inline" id="S2.6.p3.15.m15.6"><semantics id="S2.6.p3.15.m15.6a"><mrow id="S2.6.p3.15.m15.6.6" xref="S2.6.p3.15.m15.6.6.cmml"><mrow id="S2.6.p3.15.m15.5.5.1.1" xref="S2.6.p3.15.m15.5.5.1.2.cmml"><mo id="S2.6.p3.15.m15.5.5.1.1.2" stretchy="false" xref="S2.6.p3.15.m15.5.5.1.2.1.cmml">|</mo><mrow id="S2.6.p3.15.m15.5.5.1.1.1.1" xref="S2.6.p3.15.m15.5.5.1.1.1.2.cmml"><msub id="S2.6.p3.15.m15.5.5.1.1.1.1.1" xref="S2.6.p3.15.m15.5.5.1.1.1.1.1.cmml"><mi id="S2.6.p3.15.m15.5.5.1.1.1.1.1.2" xref="S2.6.p3.15.m15.5.5.1.1.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.6.p3.15.m15.5.5.1.1.1.1.1.3" xref="S2.6.p3.15.m15.5.5.1.1.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.6.p3.15.m15.5.5.1.1.1.1a" xref="S2.6.p3.15.m15.5.5.1.1.1.2.cmml">⁡</mo><mrow id="S2.6.p3.15.m15.5.5.1.1.1.1.2" xref="S2.6.p3.15.m15.5.5.1.1.1.2.cmml"><mo id="S2.6.p3.15.m15.5.5.1.1.1.1.2.1" stretchy="false" xref="S2.6.p3.15.m15.5.5.1.1.1.2.cmml">(</mo><mi id="S2.6.p3.15.m15.1.1" xref="S2.6.p3.15.m15.1.1.cmml">x</mi><mo id="S2.6.p3.15.m15.5.5.1.1.1.1.2.2" stretchy="false" xref="S2.6.p3.15.m15.5.5.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.6.p3.15.m15.5.5.1.1.3" stretchy="false" xref="S2.6.p3.15.m15.5.5.1.2.1.cmml">|</mo></mrow><mo id="S2.6.p3.15.m15.6.6.4" xref="S2.6.p3.15.m15.6.6.4.cmml">≤</mo><mrow id="S2.6.p3.15.m15.6.6.2" xref="S2.6.p3.15.m15.6.6.2.cmml"><mfrac id="S2.6.p3.15.m15.6.6.2.3" xref="S2.6.p3.15.m15.6.6.2.3.cmml"><mn id="S2.6.p3.15.m15.6.6.2.3.2" xref="S2.6.p3.15.m15.6.6.2.3.2.cmml">2</mn><mn id="S2.6.p3.15.m15.6.6.2.3.3" xref="S2.6.p3.15.m15.6.6.2.3.3.cmml">3</mn></mfrac><mo id="S2.6.p3.15.m15.6.6.2.2" lspace="0.222em" rspace="0.222em" xref="S2.6.p3.15.m15.6.6.2.2.cmml">⋅</mo><mrow id="S2.6.p3.15.m15.6.6.2.1.1" xref="S2.6.p3.15.m15.6.6.2.1.2.cmml"><mo id="S2.6.p3.15.m15.6.6.2.1.1.2" stretchy="false" xref="S2.6.p3.15.m15.6.6.2.1.2.1.cmml">|</mo><mrow id="S2.6.p3.15.m15.6.6.2.1.1.1.1" xref="S2.6.p3.15.m15.6.6.2.1.1.1.2.cmml"><msub id="S2.6.p3.15.m15.6.6.2.1.1.1.1.1" xref="S2.6.p3.15.m15.6.6.2.1.1.1.1.1.cmml"><mi id="S2.6.p3.15.m15.6.6.2.1.1.1.1.1.2" xref="S2.6.p3.15.m15.6.6.2.1.1.1.1.1.2.cmml">int</mi><mi class="ltx_font_mathcaligraphic" id="S2.6.p3.15.m15.6.6.2.1.1.1.1.1.3" xref="S2.6.p3.15.m15.6.6.2.1.1.1.1.1.3.cmml">𝒯</mi></msub><mo id="S2.6.p3.15.m15.6.6.2.1.1.1.1a" xref="S2.6.p3.15.m15.6.6.2.1.1.1.2.cmml">⁡</mo><mrow id="S2.6.p3.15.m15.6.6.2.1.1.1.1.2" xref="S2.6.p3.15.m15.6.6.2.1.1.1.2.cmml"><mo id="S2.6.p3.15.m15.6.6.2.1.1.1.1.2.1" stretchy="false" xref="S2.6.p3.15.m15.6.6.2.1.1.1.2.cmml">(</mo><mi id="S2.6.p3.15.m15.2.2" xref="S2.6.p3.15.m15.2.2.cmml">y</mi><mo id="S2.6.p3.15.m15.6.6.2.1.1.1.1.2.2" stretchy="false" xref="S2.6.p3.15.m15.6.6.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.6.p3.15.m15.6.6.2.1.1.3" stretchy="false" xref="S2.6.p3.15.m15.6.6.2.1.2.1.cmml">|</mo></mrow></mrow><mo id="S2.6.p3.15.m15.6.6.5" xref="S2.6.p3.15.m15.6.6.5.cmml">≤</mo><mrow id="S2.6.p3.15.m15.6.6.6" xref="S2.6.p3.15.m15.6.6.6.cmml"><mfrac id="S2.6.p3.15.m15.6.6.6.2" xref="S2.6.p3.15.m15.6.6.6.2.cmml"><mn id="S2.6.p3.15.m15.6.6.6.2.2" xref="S2.6.p3.15.m15.6.6.6.2.2.cmml">2</mn><mn id="S2.6.p3.15.m15.6.6.6.2.3" xref="S2.6.p3.15.m15.6.6.6.2.3.cmml">3</mn></mfrac><mo id="S2.6.p3.15.m15.6.6.6.1" lspace="0.222em" rspace="0.222em" xref="S2.6.p3.15.m15.6.6.6.1.cmml">⋅</mo><msup id="S2.6.p3.15.m15.6.6.6.3" xref="S2.6.p3.15.m15.6.6.6.3.cmml"><mrow id="S2.6.p3.15.m15.6.6.6.3.2.2" xref="S2.6.p3.15.m15.3.3.cmml"><mo id="S2.6.p3.15.m15.6.6.6.3.2.2.1" stretchy="false" xref="S2.6.p3.15.m15.3.3.cmml">(</mo><mfrac id="S2.6.p3.15.m15.3.3" xref="S2.6.p3.15.m15.3.3.cmml"><mn id="S2.6.p3.15.m15.3.3.2" xref="S2.6.p3.15.m15.3.3.2.cmml">2</mn><mn id="S2.6.p3.15.m15.3.3.3" xref="S2.6.p3.15.m15.3.3.3.cmml">3</mn></mfrac><mo id="S2.6.p3.15.m15.6.6.6.3.2.2.2" rspace="0.055em" stretchy="false" xref="S2.6.p3.15.m15.3.3.cmml">)</mo></mrow><mrow id="S2.6.p3.15.m15.6.6.6.3.3" xref="S2.6.p3.15.m15.6.6.6.3.3.cmml"><mi id="S2.6.p3.15.m15.6.6.6.3.3.2" xref="S2.6.p3.15.m15.6.6.6.3.3.2.cmml">d</mi><mo id="S2.6.p3.15.m15.6.6.6.3.3.1" xref="S2.6.p3.15.m15.6.6.6.3.3.1.cmml">−</mo><mn id="S2.6.p3.15.m15.6.6.6.3.3.3" xref="S2.6.p3.15.m15.6.6.6.3.3.3.cmml">1</mn></mrow></msup><mo id="S2.6.p3.15.m15.6.6.6.1a" rspace="0.222em" xref="S2.6.p3.15.m15.6.6.6.1.cmml">⋅</mo><mi id="S2.6.p3.15.m15.6.6.6.4" xref="S2.6.p3.15.m15.6.6.6.4.cmml">n</mi></mrow><mo id="S2.6.p3.15.m15.6.6.7" xref="S2.6.p3.15.m15.6.6.7.cmml">=</mo><mrow id="S2.6.p3.15.m15.6.6.8" xref="S2.6.p3.15.m15.6.6.8.cmml"><msup id="S2.6.p3.15.m15.6.6.8.2" xref="S2.6.p3.15.m15.6.6.8.2.cmml"><mrow id="S2.6.p3.15.m15.6.6.8.2.2.2" xref="S2.6.p3.15.m15.4.4.cmml"><mo id="S2.6.p3.15.m15.6.6.8.2.2.2.1" stretchy="false" xref="S2.6.p3.15.m15.4.4.cmml">(</mo><mfrac id="S2.6.p3.15.m15.4.4" xref="S2.6.p3.15.m15.4.4.cmml"><mn id="S2.6.p3.15.m15.4.4.2" xref="S2.6.p3.15.m15.4.4.2.cmml">2</mn><mn id="S2.6.p3.15.m15.4.4.3" xref="S2.6.p3.15.m15.4.4.3.cmml">3</mn></mfrac><mo id="S2.6.p3.15.m15.6.6.8.2.2.2.2" rspace="0.055em" stretchy="false" xref="S2.6.p3.15.m15.4.4.cmml">)</mo></mrow><mi id="S2.6.p3.15.m15.6.6.8.2.3" xref="S2.6.p3.15.m15.6.6.8.2.3.cmml">d</mi></msup><mo id="S2.6.p3.15.m15.6.6.8.1" rspace="0.222em" xref="S2.6.p3.15.m15.6.6.8.1.cmml">⋅</mo><mi id="S2.6.p3.15.m15.6.6.8.3" xref="S2.6.p3.15.m15.6.6.8.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p3.15.m15.6b"><apply id="S2.6.p3.15.m15.6.6.cmml" xref="S2.6.p3.15.m15.6.6"><and id="S2.6.p3.15.m15.6.6a.cmml" xref="S2.6.p3.15.m15.6.6"></and><apply id="S2.6.p3.15.m15.6.6b.cmml" xref="S2.6.p3.15.m15.6.6"><leq id="S2.6.p3.15.m15.6.6.4.cmml" xref="S2.6.p3.15.m15.6.6.4"></leq><apply id="S2.6.p3.15.m15.5.5.1.2.cmml" xref="S2.6.p3.15.m15.5.5.1.1"><abs id="S2.6.p3.15.m15.5.5.1.2.1.cmml" xref="S2.6.p3.15.m15.5.5.1.1.2"></abs><apply id="S2.6.p3.15.m15.5.5.1.1.1.2.cmml" xref="S2.6.p3.15.m15.5.5.1.1.1.1"><apply id="S2.6.p3.15.m15.5.5.1.1.1.1.1.cmml" xref="S2.6.p3.15.m15.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.6.p3.15.m15.5.5.1.1.1.1.1.1.cmml" xref="S2.6.p3.15.m15.5.5.1.1.1.1.1">subscript</csymbol><ci id="S2.6.p3.15.m15.5.5.1.1.1.1.1.2.cmml" xref="S2.6.p3.15.m15.5.5.1.1.1.1.1.2">int</ci><ci id="S2.6.p3.15.m15.5.5.1.1.1.1.1.3.cmml" xref="S2.6.p3.15.m15.5.5.1.1.1.1.1.3">𝒯</ci></apply><ci id="S2.6.p3.15.m15.1.1.cmml" xref="S2.6.p3.15.m15.1.1">𝑥</ci></apply></apply><apply id="S2.6.p3.15.m15.6.6.2.cmml" xref="S2.6.p3.15.m15.6.6.2"><ci id="S2.6.p3.15.m15.6.6.2.2.cmml" xref="S2.6.p3.15.m15.6.6.2.2">⋅</ci><apply id="S2.6.p3.15.m15.6.6.2.3.cmml" xref="S2.6.p3.15.m15.6.6.2.3"><divide id="S2.6.p3.15.m15.6.6.2.3.1.cmml" xref="S2.6.p3.15.m15.6.6.2.3"></divide><cn id="S2.6.p3.15.m15.6.6.2.3.2.cmml" 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xref="S2.6.p3.15.m15.6.6.8.1">⋅</ci><apply id="S2.6.p3.15.m15.6.6.8.2.cmml" xref="S2.6.p3.15.m15.6.6.8.2"><csymbol cd="ambiguous" id="S2.6.p3.15.m15.6.6.8.2.1.cmml" xref="S2.6.p3.15.m15.6.6.8.2">superscript</csymbol><apply id="S2.6.p3.15.m15.4.4.cmml" xref="S2.6.p3.15.m15.6.6.8.2.2.2"><divide id="S2.6.p3.15.m15.4.4.1.cmml" xref="S2.6.p3.15.m15.6.6.8.2.2.2"></divide><cn id="S2.6.p3.15.m15.4.4.2.cmml" type="integer" xref="S2.6.p3.15.m15.4.4.2">2</cn><cn id="S2.6.p3.15.m15.4.4.3.cmml" type="integer" xref="S2.6.p3.15.m15.4.4.3">3</cn></apply><ci id="S2.6.p3.15.m15.6.6.8.2.3.cmml" xref="S2.6.p3.15.m15.6.6.8.2.3">𝑑</ci></apply><ci id="S2.6.p3.15.m15.6.6.8.3.cmml" xref="S2.6.p3.15.m15.6.6.8.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.15.m15.6c">|\operatorname{int}_{\mathcal{T}}(x)|\leq\tfrac{2}{3}\cdot|\operatorname{int}_% {\mathcal{T}}(y)|\leq\tfrac{2}{3}\cdot(\frac{2}{3})^{d-1}\cdot n=(\tfrac{2}{3}% )^{d}\cdot n</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.15.m15.6d">| roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) | ≤ divide start_ARG 2 end_ARG start_ARG 3 end_ARG ⋅ | roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_y ) | ≤ divide start_ARG 2 end_ARG start_ARG 3 end_ARG ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ⋅ italic_n = ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ⋅ italic_n</annotation></semantics></math> and <math alttext="|\operatorname{\partial}_{\mathcal{T}}(x)|=|\partial_{x}|\leq|\partial_{y}|+|A% ^{y}\cap B^{y}|=|\operatorname{\partial}_{\mathcal{T}}(y)|+|A^{y}\cap B^{y}|% \leq(d-1)a+a=da" class="ltx_Math" display="inline" id="S2.6.p3.16.m16.9"><semantics id="S2.6.p3.16.m16.9a"><mrow id="S2.6.p3.16.m16.9.9" xref="S2.6.p3.16.m16.9.9.cmml"><mrow id="S2.6.p3.16.m16.3.3.1.1" 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href="https://arxiv.org/html/2503.17112v1#S2.6.p3.16.m16.9.9.7.cmml" id="S2.6.p3.16.m16.9.9j.cmml" xref="S2.6.p3.16.m16.9.9"></share><apply id="S2.6.p3.16.m16.9.9.14.cmml" xref="S2.6.p3.16.m16.9.9.14"><times id="S2.6.p3.16.m16.9.9.14.1.cmml" xref="S2.6.p3.16.m16.9.9.14.1"></times><ci id="S2.6.p3.16.m16.9.9.14.2.cmml" xref="S2.6.p3.16.m16.9.9.14.2">𝑑</ci><ci id="S2.6.p3.16.m16.9.9.14.3.cmml" xref="S2.6.p3.16.m16.9.9.14.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p3.16.m16.9c">|\operatorname{\partial}_{\mathcal{T}}(x)|=|\partial_{x}|\leq|\partial_{y}|+|A% ^{y}\cap B^{y}|=|\operatorname{\partial}_{\mathcal{T}}(y)|+|A^{y}\cap B^{y}|% \leq(d-1)a+a=da</annotation><annotation encoding="application/x-llamapun" id="S2.6.p3.16.m16.9d">| ∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) | = | ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | ≤ | ∂ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | + | italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT | = | ∂ start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_y ) | + | italic_A start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ∩ italic_B start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT | ≤ ( italic_d - 1 ) italic_a + italic_a = italic_d italic_a</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.7.p4"> <p class="ltx_p" id="S2.7.p4.4">The bound on <math alttext="\operatorname{height}(T)" class="ltx_Math" display="inline" id="S2.7.p4.1.m1.2"><semantics id="S2.7.p4.1.m1.2a"><mrow id="S2.7.p4.1.m1.2.3.2" xref="S2.7.p4.1.m1.2.3.1.cmml"><mi id="S2.7.p4.1.m1.1.1" xref="S2.7.p4.1.m1.1.1.cmml">height</mi><mo id="S2.7.p4.1.m1.2.3.2a" xref="S2.7.p4.1.m1.2.3.1.cmml">⁡</mo><mrow id="S2.7.p4.1.m1.2.3.2.1" xref="S2.7.p4.1.m1.2.3.1.cmml"><mo id="S2.7.p4.1.m1.2.3.2.1.1" stretchy="false" xref="S2.7.p4.1.m1.2.3.1.cmml">(</mo><mi id="S2.7.p4.1.m1.2.2" xref="S2.7.p4.1.m1.2.2.cmml">T</mi><mo id="S2.7.p4.1.m1.2.3.2.1.2" stretchy="false" xref="S2.7.p4.1.m1.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.7.p4.1.m1.2b"><apply id="S2.7.p4.1.m1.2.3.1.cmml" xref="S2.7.p4.1.m1.2.3.2"><ci id="S2.7.p4.1.m1.1.1.cmml" xref="S2.7.p4.1.m1.1.1">height</ci><ci id="S2.7.p4.1.m1.2.2.cmml" xref="S2.7.p4.1.m1.2.2">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.7.p4.1.m1.2c">\operatorname{height}(T)</annotation><annotation encoding="application/x-llamapun" id="S2.7.p4.1.m1.2d">roman_height ( italic_T )</annotation></semantics></math> follows from the bound <math alttext="|\operatorname{int}_{\mathcal{T}}(x)|\leq n\cdot(\tfrac{2}{3})^{\operatorname{% depth}_{T}(x)}" class="ltx_Math" display="inline" id="S2.7.p4.2.m2.5"><semantics id="S2.7.p4.2.m2.5a"><mrow id="S2.7.p4.2.m2.5.5" xref="S2.7.p4.2.m2.5.5.cmml"><mrow id="S2.7.p4.2.m2.5.5.1.1" xref="S2.7.p4.2.m2.5.5.1.2.cmml"><mo 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xref="S2.7.p4.2.m2.5.5.2.cmml">≤</mo><mrow id="S2.7.p4.2.m2.5.5.3" xref="S2.7.p4.2.m2.5.5.3.cmml"><mi id="S2.7.p4.2.m2.5.5.3.2" xref="S2.7.p4.2.m2.5.5.3.2.cmml">n</mi><mo id="S2.7.p4.2.m2.5.5.3.1" lspace="0.222em" rspace="0.222em" xref="S2.7.p4.2.m2.5.5.3.1.cmml">⋅</mo><msup id="S2.7.p4.2.m2.5.5.3.3" xref="S2.7.p4.2.m2.5.5.3.3.cmml"><mrow id="S2.7.p4.2.m2.5.5.3.3.2.2" xref="S2.7.p4.2.m2.4.4.cmml"><mo id="S2.7.p4.2.m2.5.5.3.3.2.2.1" stretchy="false" xref="S2.7.p4.2.m2.4.4.cmml">(</mo><mfrac id="S2.7.p4.2.m2.4.4" xref="S2.7.p4.2.m2.4.4.cmml"><mn id="S2.7.p4.2.m2.4.4.2" xref="S2.7.p4.2.m2.4.4.2.cmml">2</mn><mn id="S2.7.p4.2.m2.4.4.3" xref="S2.7.p4.2.m2.4.4.3.cmml">3</mn></mfrac><mo id="S2.7.p4.2.m2.5.5.3.3.2.2.2" stretchy="false" xref="S2.7.p4.2.m2.4.4.cmml">)</mo></mrow><mrow id="S2.7.p4.2.m2.2.2.2.2" xref="S2.7.p4.2.m2.2.2.2.3.cmml"><msub id="S2.7.p4.2.m2.2.2.2.2.1" xref="S2.7.p4.2.m2.2.2.2.2.1.cmml"><mi id="S2.7.p4.2.m2.2.2.2.2.1.2" xref="S2.7.p4.2.m2.2.2.2.2.1.2.cmml">depth</mi><mi 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type="integer" xref="S2.7.p4.2.m2.4.4.3">3</cn></apply><apply id="S2.7.p4.2.m2.2.2.2.3.cmml" xref="S2.7.p4.2.m2.2.2.2.2"><apply id="S2.7.p4.2.m2.2.2.2.2.1.cmml" xref="S2.7.p4.2.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S2.7.p4.2.m2.2.2.2.2.1.1.cmml" xref="S2.7.p4.2.m2.2.2.2.2.1">subscript</csymbol><ci id="S2.7.p4.2.m2.2.2.2.2.1.2.cmml" xref="S2.7.p4.2.m2.2.2.2.2.1.2">depth</ci><ci id="S2.7.p4.2.m2.2.2.2.2.1.3.cmml" xref="S2.7.p4.2.m2.2.2.2.2.1.3">𝑇</ci></apply><ci id="S2.7.p4.2.m2.1.1.1.1.cmml" xref="S2.7.p4.2.m2.1.1.1.1">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.7.p4.2.m2.5c">|\operatorname{int}_{\mathcal{T}}(x)|\leq n\cdot(\tfrac{2}{3})^{\operatorname{% depth}_{T}(x)}</annotation><annotation encoding="application/x-llamapun" id="S2.7.p4.2.m2.5d">| roman_int start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT ( italic_x ) | ≤ italic_n ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT roman_depth start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x ) end_POSTSUPERSCRIPT</annotation></semantics></math> and the fact that the algorithm returns a 1-node tree (of height <math alttext="0" class="ltx_Math" display="inline" id="S2.7.p4.3.m3.1"><semantics id="S2.7.p4.3.m3.1a"><mn id="S2.7.p4.3.m3.1.1" xref="S2.7.p4.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.7.p4.3.m3.1b"><cn id="S2.7.p4.3.m3.1.1.cmml" type="integer" xref="S2.7.p4.3.m3.1.1">0</cn></annotation-xml></semantics></math>) when <math alttext="|V(G^{\prime})\setminus\partial|\leq N=n\cdot(\tfrac{2}{3})^{h}" class="ltx_Math" display="inline" id="S2.7.p4.4.m4.2"><semantics id="S2.7.p4.4.m4.2a"><mrow id="S2.7.p4.4.m4.2.2" xref="S2.7.p4.4.m4.2.2.cmml"><mrow id="S2.7.p4.4.m4.2.2.1.1" xref="S2.7.p4.4.m4.2.2.1.2.cmml"><mo id="S2.7.p4.4.m4.2.2.1.1.2" stretchy="false" xref="S2.7.p4.4.m4.2.2.1.2.1.cmml">|</mo><mrow id="S2.7.p4.4.m4.2.2.1.1.1" xref="S2.7.p4.4.m4.2.2.1.1.1.cmml"><mrow 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xref="S2.7.p4.4.m4.2.2.6.1">⋅</ci><ci id="S2.7.p4.4.m4.2.2.6.2.cmml" xref="S2.7.p4.4.m4.2.2.6.2">𝑛</ci><apply id="S2.7.p4.4.m4.2.2.6.3.cmml" xref="S2.7.p4.4.m4.2.2.6.3"><csymbol cd="ambiguous" id="S2.7.p4.4.m4.2.2.6.3.1.cmml" xref="S2.7.p4.4.m4.2.2.6.3">superscript</csymbol><apply id="S2.7.p4.4.m4.1.1.cmml" xref="S2.7.p4.4.m4.2.2.6.3.2.2"><divide id="S2.7.p4.4.m4.1.1.1.cmml" xref="S2.7.p4.4.m4.2.2.6.3.2.2"></divide><cn id="S2.7.p4.4.m4.1.1.2.cmml" type="integer" xref="S2.7.p4.4.m4.1.1.2">2</cn><cn id="S2.7.p4.4.m4.1.1.3.cmml" type="integer" xref="S2.7.p4.4.m4.1.1.3">3</cn></apply><ci id="S2.7.p4.4.m4.2.2.6.3.3.cmml" xref="S2.7.p4.4.m4.2.2.6.3.3">ℎ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.7.p4.4.m4.2c">|V(G^{\prime})\setminus\partial|\leq N=n\cdot(\tfrac{2}{3})^{h}</annotation><annotation encoding="application/x-llamapun" id="S2.7.p4.4.m4.2d">| italic_V ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∖ ∂ | ≤ italic_N = italic_n ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>The Proof</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.13">The following definition replaces the notion of <math alttext="W" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_W</annotation></semantics></math>-clouds in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#bib.bib4" title="">4</a>]</cite>. Let <math alttext="W_{-1}:=\emptyset" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mrow id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><msub id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml"><mi id="S3.p1.2.m2.1.1.2.2" xref="S3.p1.2.m2.1.1.2.2.cmml">W</mi><mrow id="S3.p1.2.m2.1.1.2.3" xref="S3.p1.2.m2.1.1.2.3.cmml"><mo id="S3.p1.2.m2.1.1.2.3a" xref="S3.p1.2.m2.1.1.2.3.cmml">−</mo><mn id="S3.p1.2.m2.1.1.2.3.2" xref="S3.p1.2.m2.1.1.2.3.2.cmml">1</mn></mrow></msub><mo id="S3.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.p1.2.m2.1.1.1.cmml">:=</mo><mi id="S3.p1.2.m2.1.1.3" mathvariant="normal" xref="S3.p1.2.m2.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1.1">assign</csymbol><apply id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.2.1.cmml" xref="S3.p1.2.m2.1.1.2">subscript</csymbol><ci id="S3.p1.2.m2.1.1.2.2.cmml" xref="S3.p1.2.m2.1.1.2.2">𝑊</ci><apply id="S3.p1.2.m2.1.1.2.3.cmml" xref="S3.p1.2.m2.1.1.2.3"><minus id="S3.p1.2.m2.1.1.2.3.1.cmml" xref="S3.p1.2.m2.1.1.2.3"></minus><cn id="S3.p1.2.m2.1.1.2.3.2.cmml" type="integer" xref="S3.p1.2.m2.1.1.2.3.2">1</cn></apply></apply><emptyset id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">W_{-1}:=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_W start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT := ∅</annotation></semantics></math>, let <math alttext="G" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mi id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">italic_G</annotation></semantics></math> be a graph, let <math alttext="W\subseteq V(G)" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mrow id="S3.p1.4.m4.1.2" xref="S3.p1.4.m4.1.2.cmml"><mi id="S3.p1.4.m4.1.2.2" xref="S3.p1.4.m4.1.2.2.cmml">W</mi><mo id="S3.p1.4.m4.1.2.1" xref="S3.p1.4.m4.1.2.1.cmml">⊆</mo><mrow id="S3.p1.4.m4.1.2.3" xref="S3.p1.4.m4.1.2.3.cmml"><mi id="S3.p1.4.m4.1.2.3.2" xref="S3.p1.4.m4.1.2.3.2.cmml">V</mi><mo id="S3.p1.4.m4.1.2.3.1" xref="S3.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S3.p1.4.m4.1.2.3.3.2" xref="S3.p1.4.m4.1.2.3.cmml"><mo id="S3.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="S3.p1.4.m4.1.2.3.cmml">(</mo><mi id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml">G</mi><mo id="S3.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="S3.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><apply id="S3.p1.4.m4.1.2.cmml" xref="S3.p1.4.m4.1.2"><subset id="S3.p1.4.m4.1.2.1.cmml" xref="S3.p1.4.m4.1.2.1"></subset><ci id="S3.p1.4.m4.1.2.2.cmml" xref="S3.p1.4.m4.1.2.2">𝑊</ci><apply id="S3.p1.4.m4.1.2.3.cmml" xref="S3.p1.4.m4.1.2.3"><times id="S3.p1.4.m4.1.2.3.1.cmml" xref="S3.p1.4.m4.1.2.3.1"></times><ci id="S3.p1.4.m4.1.2.3.2.cmml" xref="S3.p1.4.m4.1.2.3.2">𝑉</ci><ci id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">W\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">italic_W ⊆ italic_V ( italic_G )</annotation></semantics></math>, let <math alttext="W_{0}\subseteq W_{1}\subseteq\cdots\subseteq W_{\ell}\subseteq W_{\ell+1}% \subseteq V(G)" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><mrow id="S3.p1.5.m5.1.2" xref="S3.p1.5.m5.1.2.cmml"><msub id="S3.p1.5.m5.1.2.2" xref="S3.p1.5.m5.1.2.2.cmml"><mi id="S3.p1.5.m5.1.2.2.2" xref="S3.p1.5.m5.1.2.2.2.cmml">W</mi><mn id="S3.p1.5.m5.1.2.2.3" xref="S3.p1.5.m5.1.2.2.3.cmml">0</mn></msub><mo id="S3.p1.5.m5.1.2.3" xref="S3.p1.5.m5.1.2.3.cmml">⊆</mo><msub id="S3.p1.5.m5.1.2.4" xref="S3.p1.5.m5.1.2.4.cmml"><mi id="S3.p1.5.m5.1.2.4.2" xref="S3.p1.5.m5.1.2.4.2.cmml">W</mi><mn id="S3.p1.5.m5.1.2.4.3" xref="S3.p1.5.m5.1.2.4.3.cmml">1</mn></msub><mo id="S3.p1.5.m5.1.2.5" xref="S3.p1.5.m5.1.2.5.cmml">⊆</mo><mi id="S3.p1.5.m5.1.2.6" mathvariant="normal" xref="S3.p1.5.m5.1.2.6.cmml">⋯</mi><mo id="S3.p1.5.m5.1.2.7" xref="S3.p1.5.m5.1.2.7.cmml">⊆</mo><msub id="S3.p1.5.m5.1.2.8" xref="S3.p1.5.m5.1.2.8.cmml"><mi id="S3.p1.5.m5.1.2.8.2" xref="S3.p1.5.m5.1.2.8.2.cmml">W</mi><mi id="S3.p1.5.m5.1.2.8.3" mathvariant="normal" xref="S3.p1.5.m5.1.2.8.3.cmml">ℓ</mi></msub><mo id="S3.p1.5.m5.1.2.9" xref="S3.p1.5.m5.1.2.9.cmml">⊆</mo><msub id="S3.p1.5.m5.1.2.10" xref="S3.p1.5.m5.1.2.10.cmml"><mi id="S3.p1.5.m5.1.2.10.2" xref="S3.p1.5.m5.1.2.10.2.cmml">W</mi><mrow id="S3.p1.5.m5.1.2.10.3" xref="S3.p1.5.m5.1.2.10.3.cmml"><mi id="S3.p1.5.m5.1.2.10.3.2" mathvariant="normal" xref="S3.p1.5.m5.1.2.10.3.2.cmml">ℓ</mi><mo id="S3.p1.5.m5.1.2.10.3.1" xref="S3.p1.5.m5.1.2.10.3.1.cmml">+</mo><mn id="S3.p1.5.m5.1.2.10.3.3" xref="S3.p1.5.m5.1.2.10.3.3.cmml">1</mn></mrow></msub><mo id="S3.p1.5.m5.1.2.11" xref="S3.p1.5.m5.1.2.11.cmml">⊆</mo><mrow id="S3.p1.5.m5.1.2.12" xref="S3.p1.5.m5.1.2.12.cmml"><mi id="S3.p1.5.m5.1.2.12.2" xref="S3.p1.5.m5.1.2.12.2.cmml">V</mi><mo id="S3.p1.5.m5.1.2.12.1" xref="S3.p1.5.m5.1.2.12.1.cmml">⁢</mo><mrow id="S3.p1.5.m5.1.2.12.3.2" xref="S3.p1.5.m5.1.2.12.cmml"><mo id="S3.p1.5.m5.1.2.12.3.2.1" stretchy="false" xref="S3.p1.5.m5.1.2.12.cmml">(</mo><mi id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml">G</mi><mo id="S3.p1.5.m5.1.2.12.3.2.2" stretchy="false" xref="S3.p1.5.m5.1.2.12.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><apply id="S3.p1.5.m5.1.2.cmml" xref="S3.p1.5.m5.1.2"><and id="S3.p1.5.m5.1.2a.cmml" xref="S3.p1.5.m5.1.2"></and><apply id="S3.p1.5.m5.1.2b.cmml" xref="S3.p1.5.m5.1.2"><subset id="S3.p1.5.m5.1.2.3.cmml" xref="S3.p1.5.m5.1.2.3"></subset><apply id="S3.p1.5.m5.1.2.2.cmml" xref="S3.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.2.2.1.cmml" xref="S3.p1.5.m5.1.2.2">subscript</csymbol><ci id="S3.p1.5.m5.1.2.2.2.cmml" xref="S3.p1.5.m5.1.2.2.2">𝑊</ci><cn id="S3.p1.5.m5.1.2.2.3.cmml" type="integer" xref="S3.p1.5.m5.1.2.2.3">0</cn></apply><apply id="S3.p1.5.m5.1.2.4.cmml" xref="S3.p1.5.m5.1.2.4"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.2.4.1.cmml" xref="S3.p1.5.m5.1.2.4">subscript</csymbol><ci id="S3.p1.5.m5.1.2.4.2.cmml" xref="S3.p1.5.m5.1.2.4.2">𝑊</ci><cn id="S3.p1.5.m5.1.2.4.3.cmml" type="integer" xref="S3.p1.5.m5.1.2.4.3">1</cn></apply></apply><apply id="S3.p1.5.m5.1.2c.cmml" xref="S3.p1.5.m5.1.2"><subset id="S3.p1.5.m5.1.2.5.cmml" xref="S3.p1.5.m5.1.2.5"></subset><share href="https://arxiv.org/html/2503.17112v1#S3.p1.5.m5.1.2.4.cmml" id="S3.p1.5.m5.1.2d.cmml" xref="S3.p1.5.m5.1.2"></share><ci id="S3.p1.5.m5.1.2.6.cmml" xref="S3.p1.5.m5.1.2.6">⋯</ci></apply><apply id="S3.p1.5.m5.1.2e.cmml" xref="S3.p1.5.m5.1.2"><subset id="S3.p1.5.m5.1.2.7.cmml" xref="S3.p1.5.m5.1.2.7"></subset><share href="https://arxiv.org/html/2503.17112v1#S3.p1.5.m5.1.2.6.cmml" id="S3.p1.5.m5.1.2f.cmml" xref="S3.p1.5.m5.1.2"></share><apply id="S3.p1.5.m5.1.2.8.cmml" xref="S3.p1.5.m5.1.2.8"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.2.8.1.cmml" xref="S3.p1.5.m5.1.2.8">subscript</csymbol><ci id="S3.p1.5.m5.1.2.8.2.cmml" xref="S3.p1.5.m5.1.2.8.2">𝑊</ci><ci id="S3.p1.5.m5.1.2.8.3.cmml" xref="S3.p1.5.m5.1.2.8.3">ℓ</ci></apply></apply><apply id="S3.p1.5.m5.1.2g.cmml" xref="S3.p1.5.m5.1.2"><subset id="S3.p1.5.m5.1.2.9.cmml" xref="S3.p1.5.m5.1.2.9"></subset><share href="https://arxiv.org/html/2503.17112v1#S3.p1.5.m5.1.2.8.cmml" id="S3.p1.5.m5.1.2h.cmml" xref="S3.p1.5.m5.1.2"></share><apply id="S3.p1.5.m5.1.2.10.cmml" xref="S3.p1.5.m5.1.2.10"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.2.10.1.cmml" xref="S3.p1.5.m5.1.2.10">subscript</csymbol><ci id="S3.p1.5.m5.1.2.10.2.cmml" xref="S3.p1.5.m5.1.2.10.2">𝑊</ci><apply id="S3.p1.5.m5.1.2.10.3.cmml" xref="S3.p1.5.m5.1.2.10.3"><plus id="S3.p1.5.m5.1.2.10.3.1.cmml" xref="S3.p1.5.m5.1.2.10.3.1"></plus><ci id="S3.p1.5.m5.1.2.10.3.2.cmml" xref="S3.p1.5.m5.1.2.10.3.2">ℓ</ci><cn id="S3.p1.5.m5.1.2.10.3.3.cmml" type="integer" xref="S3.p1.5.m5.1.2.10.3.3">1</cn></apply></apply></apply><apply id="S3.p1.5.m5.1.2i.cmml" xref="S3.p1.5.m5.1.2"><subset id="S3.p1.5.m5.1.2.11.cmml" xref="S3.p1.5.m5.1.2.11"></subset><share href="https://arxiv.org/html/2503.17112v1#S3.p1.5.m5.1.2.10.cmml" id="S3.p1.5.m5.1.2j.cmml" xref="S3.p1.5.m5.1.2"></share><apply id="S3.p1.5.m5.1.2.12.cmml" xref="S3.p1.5.m5.1.2.12"><times id="S3.p1.5.m5.1.2.12.1.cmml" xref="S3.p1.5.m5.1.2.12.1"></times><ci id="S3.p1.5.m5.1.2.12.2.cmml" xref="S3.p1.5.m5.1.2.12.2">𝑉</ci><ci id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">W_{0}\subseteq W_{1}\subseteq\cdots\subseteq W_{\ell}\subseteq W_{\ell+1}% \subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊆ italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊆ ⋯ ⊆ italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ⊆ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ⊆ italic_V ( italic_G )</annotation></semantics></math> be a nested sequence of vertex subsets of <math alttext="G" class="ltx_Math" display="inline" id="S3.p1.6.m6.1"><semantics id="S3.p1.6.m6.1a"><mi id="S3.p1.6.m6.1.1" xref="S3.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><ci id="S3.p1.6.m6.1.1.cmml" xref="S3.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.1d">italic_G</annotation></semantics></math>, and let <math alttext="\Delta_{i}:=W_{i}\setminus W_{i-1}" class="ltx_Math" display="inline" id="S3.p1.7.m7.1"><semantics id="S3.p1.7.m7.1a"><mrow id="S3.p1.7.m7.1.1" xref="S3.p1.7.m7.1.1.cmml"><msub id="S3.p1.7.m7.1.1.2" xref="S3.p1.7.m7.1.1.2.cmml"><mi id="S3.p1.7.m7.1.1.2.2" mathvariant="normal" xref="S3.p1.7.m7.1.1.2.2.cmml">Δ</mi><mi id="S3.p1.7.m7.1.1.2.3" xref="S3.p1.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="S3.p1.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.p1.7.m7.1.1.1.cmml">:=</mo><mrow id="S3.p1.7.m7.1.1.3" xref="S3.p1.7.m7.1.1.3.cmml"><msub id="S3.p1.7.m7.1.1.3.2" xref="S3.p1.7.m7.1.1.3.2.cmml"><mi id="S3.p1.7.m7.1.1.3.2.2" xref="S3.p1.7.m7.1.1.3.2.2.cmml">W</mi><mi id="S3.p1.7.m7.1.1.3.2.3" xref="S3.p1.7.m7.1.1.3.2.3.cmml">i</mi></msub><mo id="S3.p1.7.m7.1.1.3.1" xref="S3.p1.7.m7.1.1.3.1.cmml">∖</mo><msub id="S3.p1.7.m7.1.1.3.3" xref="S3.p1.7.m7.1.1.3.3.cmml"><mi id="S3.p1.7.m7.1.1.3.3.2" xref="S3.p1.7.m7.1.1.3.3.2.cmml">W</mi><mrow id="S3.p1.7.m7.1.1.3.3.3" xref="S3.p1.7.m7.1.1.3.3.3.cmml"><mi id="S3.p1.7.m7.1.1.3.3.3.2" xref="S3.p1.7.m7.1.1.3.3.3.2.cmml">i</mi><mo id="S3.p1.7.m7.1.1.3.3.3.1" xref="S3.p1.7.m7.1.1.3.3.3.1.cmml">−</mo><mn id="S3.p1.7.m7.1.1.3.3.3.3" xref="S3.p1.7.m7.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.1b"><apply id="S3.p1.7.m7.1.1.cmml" xref="S3.p1.7.m7.1.1"><csymbol cd="latexml" id="S3.p1.7.m7.1.1.1.cmml" xref="S3.p1.7.m7.1.1.1">assign</csymbol><apply id="S3.p1.7.m7.1.1.2.cmml" xref="S3.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.p1.7.m7.1.1.2.1.cmml" xref="S3.p1.7.m7.1.1.2">subscript</csymbol><ci id="S3.p1.7.m7.1.1.2.2.cmml" xref="S3.p1.7.m7.1.1.2.2">Δ</ci><ci id="S3.p1.7.m7.1.1.2.3.cmml" xref="S3.p1.7.m7.1.1.2.3">𝑖</ci></apply><apply id="S3.p1.7.m7.1.1.3.cmml" xref="S3.p1.7.m7.1.1.3"><setdiff id="S3.p1.7.m7.1.1.3.1.cmml" xref="S3.p1.7.m7.1.1.3.1"></setdiff><apply id="S3.p1.7.m7.1.1.3.2.cmml" xref="S3.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.p1.7.m7.1.1.3.2.1.cmml" xref="S3.p1.7.m7.1.1.3.2">subscript</csymbol><ci id="S3.p1.7.m7.1.1.3.2.2.cmml" xref="S3.p1.7.m7.1.1.3.2.2">𝑊</ci><ci id="S3.p1.7.m7.1.1.3.2.3.cmml" xref="S3.p1.7.m7.1.1.3.2.3">𝑖</ci></apply><apply id="S3.p1.7.m7.1.1.3.3.cmml" xref="S3.p1.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.p1.7.m7.1.1.3.3.1.cmml" xref="S3.p1.7.m7.1.1.3.3">subscript</csymbol><ci id="S3.p1.7.m7.1.1.3.3.2.cmml" xref="S3.p1.7.m7.1.1.3.3.2">𝑊</ci><apply id="S3.p1.7.m7.1.1.3.3.3.cmml" xref="S3.p1.7.m7.1.1.3.3.3"><minus id="S3.p1.7.m7.1.1.3.3.3.1.cmml" xref="S3.p1.7.m7.1.1.3.3.3.1"></minus><ci id="S3.p1.7.m7.1.1.3.3.3.2.cmml" xref="S3.p1.7.m7.1.1.3.3.3.2">𝑖</ci><cn id="S3.p1.7.m7.1.1.3.3.3.3.cmml" type="integer" xref="S3.p1.7.m7.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.1c">\Delta_{i}:=W_{i}\setminus W_{i-1}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m7.1d">roman_Δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ italic_W start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="s_{i}:=|\Delta_{i}|" class="ltx_Math" display="inline" id="S3.p1.8.m8.1"><semantics id="S3.p1.8.m8.1a"><mrow id="S3.p1.8.m8.1.1" xref="S3.p1.8.m8.1.1.cmml"><msub id="S3.p1.8.m8.1.1.3" xref="S3.p1.8.m8.1.1.3.cmml"><mi id="S3.p1.8.m8.1.1.3.2" xref="S3.p1.8.m8.1.1.3.2.cmml">s</mi><mi id="S3.p1.8.m8.1.1.3.3" xref="S3.p1.8.m8.1.1.3.3.cmml">i</mi></msub><mo id="S3.p1.8.m8.1.1.2" lspace="0.278em" rspace="0.278em" xref="S3.p1.8.m8.1.1.2.cmml">:=</mo><mrow id="S3.p1.8.m8.1.1.1.1" xref="S3.p1.8.m8.1.1.1.2.cmml"><mo id="S3.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S3.p1.8.m8.1.1.1.2.1.cmml">|</mo><msub id="S3.p1.8.m8.1.1.1.1.1" xref="S3.p1.8.m8.1.1.1.1.1.cmml"><mi id="S3.p1.8.m8.1.1.1.1.1.2" mathvariant="normal" xref="S3.p1.8.m8.1.1.1.1.1.2.cmml">Δ</mi><mi id="S3.p1.8.m8.1.1.1.1.1.3" xref="S3.p1.8.m8.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S3.p1.8.m8.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.8.m8.1b"><apply id="S3.p1.8.m8.1.1.cmml" xref="S3.p1.8.m8.1.1"><csymbol cd="latexml" id="S3.p1.8.m8.1.1.2.cmml" xref="S3.p1.8.m8.1.1.2">assign</csymbol><apply id="S3.p1.8.m8.1.1.3.cmml" xref="S3.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.p1.8.m8.1.1.3.1.cmml" xref="S3.p1.8.m8.1.1.3">subscript</csymbol><ci id="S3.p1.8.m8.1.1.3.2.cmml" xref="S3.p1.8.m8.1.1.3.2">𝑠</ci><ci id="S3.p1.8.m8.1.1.3.3.cmml" xref="S3.p1.8.m8.1.1.3.3">𝑖</ci></apply><apply id="S3.p1.8.m8.1.1.1.2.cmml" xref="S3.p1.8.m8.1.1.1.1"><abs id="S3.p1.8.m8.1.1.1.2.1.cmml" xref="S3.p1.8.m8.1.1.1.1.2"></abs><apply id="S3.p1.8.m8.1.1.1.1.1.cmml" xref="S3.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p1.8.m8.1.1.1.1.1.1.cmml" xref="S3.p1.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S3.p1.8.m8.1.1.1.1.1.2.cmml" xref="S3.p1.8.m8.1.1.1.1.1.2">Δ</ci><ci id="S3.p1.8.m8.1.1.1.1.1.3.cmml" xref="S3.p1.8.m8.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m8.1c">s_{i}:=|\Delta_{i}|</annotation><annotation encoding="application/x-llamapun" id="S3.p1.8.m8.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := | roman_Δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |</annotation></semantics></math>, for each <math alttext="i\in\{0,\ldots,\ell+1\}" class="ltx_Math" display="inline" id="S3.p1.9.m9.3"><semantics id="S3.p1.9.m9.3a"><mrow id="S3.p1.9.m9.3.3" xref="S3.p1.9.m9.3.3.cmml"><mi id="S3.p1.9.m9.3.3.3" xref="S3.p1.9.m9.3.3.3.cmml">i</mi><mo id="S3.p1.9.m9.3.3.2" xref="S3.p1.9.m9.3.3.2.cmml">∈</mo><mrow id="S3.p1.9.m9.3.3.1.1" xref="S3.p1.9.m9.3.3.1.2.cmml"><mo id="S3.p1.9.m9.3.3.1.1.2" stretchy="false" xref="S3.p1.9.m9.3.3.1.2.cmml">{</mo><mn id="S3.p1.9.m9.1.1" xref="S3.p1.9.m9.1.1.cmml">0</mn><mo id="S3.p1.9.m9.3.3.1.1.3" xref="S3.p1.9.m9.3.3.1.2.cmml">,</mo><mi id="S3.p1.9.m9.2.2" mathvariant="normal" xref="S3.p1.9.m9.2.2.cmml">…</mi><mo id="S3.p1.9.m9.3.3.1.1.4" xref="S3.p1.9.m9.3.3.1.2.cmml">,</mo><mrow id="S3.p1.9.m9.3.3.1.1.1" xref="S3.p1.9.m9.3.3.1.1.1.cmml"><mi id="S3.p1.9.m9.3.3.1.1.1.2" mathvariant="normal" xref="S3.p1.9.m9.3.3.1.1.1.2.cmml">ℓ</mi><mo id="S3.p1.9.m9.3.3.1.1.1.1" xref="S3.p1.9.m9.3.3.1.1.1.1.cmml">+</mo><mn id="S3.p1.9.m9.3.3.1.1.1.3" xref="S3.p1.9.m9.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p1.9.m9.3.3.1.1.5" stretchy="false" xref="S3.p1.9.m9.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.9.m9.3b"><apply id="S3.p1.9.m9.3.3.cmml" xref="S3.p1.9.m9.3.3"><in id="S3.p1.9.m9.3.3.2.cmml" xref="S3.p1.9.m9.3.3.2"></in><ci id="S3.p1.9.m9.3.3.3.cmml" xref="S3.p1.9.m9.3.3.3">𝑖</ci><set id="S3.p1.9.m9.3.3.1.2.cmml" xref="S3.p1.9.m9.3.3.1.1"><cn id="S3.p1.9.m9.1.1.cmml" type="integer" xref="S3.p1.9.m9.1.1">0</cn><ci id="S3.p1.9.m9.2.2.cmml" xref="S3.p1.9.m9.2.2">…</ci><apply id="S3.p1.9.m9.3.3.1.1.1.cmml" xref="S3.p1.9.m9.3.3.1.1.1"><plus id="S3.p1.9.m9.3.3.1.1.1.1.cmml" xref="S3.p1.9.m9.3.3.1.1.1.1"></plus><ci id="S3.p1.9.m9.3.3.1.1.1.2.cmml" xref="S3.p1.9.m9.3.3.1.1.1.2">ℓ</ci><cn id="S3.p1.9.m9.3.3.1.1.1.3.cmml" type="integer" xref="S3.p1.9.m9.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m9.3c">i\in\{0,\ldots,\ell+1\}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.9.m9.3d">italic_i ∈ { 0 , … , roman_ℓ + 1 }</annotation></semantics></math>. Then <math alttext="W_{0},\ldots,W_{\ell+1}" class="ltx_Math" display="inline" id="S3.p1.10.m10.3"><semantics id="S3.p1.10.m10.3a"><mrow id="S3.p1.10.m10.3.3.2" xref="S3.p1.10.m10.3.3.3.cmml"><msub id="S3.p1.10.m10.2.2.1.1" xref="S3.p1.10.m10.2.2.1.1.cmml"><mi id="S3.p1.10.m10.2.2.1.1.2" xref="S3.p1.10.m10.2.2.1.1.2.cmml">W</mi><mn id="S3.p1.10.m10.2.2.1.1.3" xref="S3.p1.10.m10.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.p1.10.m10.3.3.2.3" xref="S3.p1.10.m10.3.3.3.cmml">,</mo><mi id="S3.p1.10.m10.1.1" mathvariant="normal" xref="S3.p1.10.m10.1.1.cmml">…</mi><mo id="S3.p1.10.m10.3.3.2.4" xref="S3.p1.10.m10.3.3.3.cmml">,</mo><msub id="S3.p1.10.m10.3.3.2.2" xref="S3.p1.10.m10.3.3.2.2.cmml"><mi id="S3.p1.10.m10.3.3.2.2.2" xref="S3.p1.10.m10.3.3.2.2.2.cmml">W</mi><mrow id="S3.p1.10.m10.3.3.2.2.3" xref="S3.p1.10.m10.3.3.2.2.3.cmml"><mi id="S3.p1.10.m10.3.3.2.2.3.2" mathvariant="normal" xref="S3.p1.10.m10.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.p1.10.m10.3.3.2.2.3.1" xref="S3.p1.10.m10.3.3.2.2.3.1.cmml">+</mo><mn id="S3.p1.10.m10.3.3.2.2.3.3" xref="S3.p1.10.m10.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.10.m10.3b"><list id="S3.p1.10.m10.3.3.3.cmml" xref="S3.p1.10.m10.3.3.2"><apply id="S3.p1.10.m10.2.2.1.1.cmml" xref="S3.p1.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="S3.p1.10.m10.2.2.1.1.1.cmml" xref="S3.p1.10.m10.2.2.1.1">subscript</csymbol><ci id="S3.p1.10.m10.2.2.1.1.2.cmml" xref="S3.p1.10.m10.2.2.1.1.2">𝑊</ci><cn id="S3.p1.10.m10.2.2.1.1.3.cmml" type="integer" xref="S3.p1.10.m10.2.2.1.1.3">0</cn></apply><ci id="S3.p1.10.m10.1.1.cmml" xref="S3.p1.10.m10.1.1">…</ci><apply id="S3.p1.10.m10.3.3.2.2.cmml" xref="S3.p1.10.m10.3.3.2.2"><csymbol cd="ambiguous" id="S3.p1.10.m10.3.3.2.2.1.cmml" xref="S3.p1.10.m10.3.3.2.2">subscript</csymbol><ci id="S3.p1.10.m10.3.3.2.2.2.cmml" xref="S3.p1.10.m10.3.3.2.2.2">𝑊</ci><apply id="S3.p1.10.m10.3.3.2.2.3.cmml" xref="S3.p1.10.m10.3.3.2.2.3"><plus id="S3.p1.10.m10.3.3.2.2.3.1.cmml" xref="S3.p1.10.m10.3.3.2.2.3.1"></plus><ci id="S3.p1.10.m10.3.3.2.2.3.2.cmml" xref="S3.p1.10.m10.3.3.2.2.3.2">ℓ</ci><cn id="S3.p1.10.m10.3.3.2.2.3.3.cmml" type="integer" xref="S3.p1.10.m10.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.10.m10.3c">W_{0},\ldots,W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.10.m10.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> is a <em class="ltx_emph ltx_font_italic" id="S3.p1.12.2"><math alttext="W" class="ltx_Math" display="inline" id="S3.p1.11.1.m1.1"><semantics id="S3.p1.11.1.m1.1a"><mi id="S3.p1.11.1.m1.1.1" mathcolor="#C22147" xref="S3.p1.11.1.m1.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.p1.11.1.m1.1b"><ci id="S3.p1.11.1.m1.1.1.cmml" xref="S3.p1.11.1.m1.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.11.1.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.p1.11.1.m1.1d">italic_W</annotation></semantics></math><span class="ltx_text" id="S3.p1.12.2.1" style="color:#C22147;">-sequence of width <math alttext="w" class="ltx_Math" display="inline" id="S3.p1.12.2.1.m1.1"><semantics id="S3.p1.12.2.1.m1.1a"><mi id="S3.p1.12.2.1.m1.1.1" mathcolor="#C22147" xref="S3.p1.12.2.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.p1.12.2.1.m1.1b"><ci id="S3.p1.12.2.1.m1.1.1.cmml" xref="S3.p1.12.2.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.12.2.1.m1.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.p1.12.2.1.m1.1d">italic_w</annotation></semantics></math></span></em> in <math alttext="G" class="ltx_Math" display="inline" id="S3.p1.13.m11.1"><semantics id="S3.p1.13.m11.1a"><mi id="S3.p1.13.m11.1.1" xref="S3.p1.13.m11.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.p1.13.m11.1b"><ci id="S3.p1.13.m11.1.1.cmml" xref="S3.p1.13.m11.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.13.m11.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.p1.13.m11.1d">italic_G</annotation></semantics></math> if it satisfies the following conditions:</p> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.1"><math alttext="W_{0}=W" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><mrow id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><msub id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.2.2" xref="S3.I1.i1.p1.1.m1.1.1.2.2.cmml">W</mi><mn id="S3.I1.i1.p1.1.m1.1.1.2.3" xref="S3.I1.i1.p1.1.m1.1.1.2.3.cmml">0</mn></msub><mo id="S3.I1.i1.p1.1.m1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.cmml">=</mo><mi id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1"><eq id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1"></eq><apply id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2.2">𝑊</ci><cn id="S3.I1.i1.p1.1.m1.1.1.2.3.cmml" type="integer" xref="S3.I1.i1.p1.1.m1.1.1.2.3">0</cn></apply><ci id="S3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">W_{0}=W</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_W</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.2"><math alttext="s_{i}=w" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><mrow id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><msub id="S3.I1.i2.p1.1.m1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.2.2" xref="S3.I1.i2.p1.1.m1.1.1.2.2.cmml">s</mi><mi id="S3.I1.i2.p1.1.m1.1.1.2.3" xref="S3.I1.i2.p1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S3.I1.i2.p1.1.m1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.cmml">=</mo><mi id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><eq id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1"></eq><apply id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2.2">𝑠</ci><ci id="S3.I1.i2.p1.1.m1.1.1.2.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2.3">𝑖</ci></apply><ci id="S3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">s_{i}=w</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_w</annotation></semantics></math>, for each <math alttext="i\in\{0,\ldots,\ell\}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.3"><semantics id="S3.I1.i2.p1.2.m2.3a"><mrow id="S3.I1.i2.p1.2.m2.3.4" xref="S3.I1.i2.p1.2.m2.3.4.cmml"><mi id="S3.I1.i2.p1.2.m2.3.4.2" xref="S3.I1.i2.p1.2.m2.3.4.2.cmml">i</mi><mo id="S3.I1.i2.p1.2.m2.3.4.1" xref="S3.I1.i2.p1.2.m2.3.4.1.cmml">∈</mo><mrow id="S3.I1.i2.p1.2.m2.3.4.3.2" xref="S3.I1.i2.p1.2.m2.3.4.3.1.cmml"><mo id="S3.I1.i2.p1.2.m2.3.4.3.2.1" stretchy="false" xref="S3.I1.i2.p1.2.m2.3.4.3.1.cmml">{</mo><mn id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml">0</mn><mo id="S3.I1.i2.p1.2.m2.3.4.3.2.2" xref="S3.I1.i2.p1.2.m2.3.4.3.1.cmml">,</mo><mi id="S3.I1.i2.p1.2.m2.2.2" mathvariant="normal" xref="S3.I1.i2.p1.2.m2.2.2.cmml">…</mi><mo id="S3.I1.i2.p1.2.m2.3.4.3.2.3" xref="S3.I1.i2.p1.2.m2.3.4.3.1.cmml">,</mo><mi id="S3.I1.i2.p1.2.m2.3.3" mathvariant="normal" xref="S3.I1.i2.p1.2.m2.3.3.cmml">ℓ</mi><mo id="S3.I1.i2.p1.2.m2.3.4.3.2.4" stretchy="false" xref="S3.I1.i2.p1.2.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.3b"><apply id="S3.I1.i2.p1.2.m2.3.4.cmml" xref="S3.I1.i2.p1.2.m2.3.4"><in id="S3.I1.i2.p1.2.m2.3.4.1.cmml" xref="S3.I1.i2.p1.2.m2.3.4.1"></in><ci id="S3.I1.i2.p1.2.m2.3.4.2.cmml" xref="S3.I1.i2.p1.2.m2.3.4.2">𝑖</ci><set id="S3.I1.i2.p1.2.m2.3.4.3.1.cmml" xref="S3.I1.i2.p1.2.m2.3.4.3.2"><cn id="S3.I1.i2.p1.2.m2.1.1.cmml" type="integer" xref="S3.I1.i2.p1.2.m2.1.1">0</cn><ci id="S3.I1.i2.p1.2.m2.2.2.cmml" xref="S3.I1.i2.p1.2.m2.2.2">…</ci><ci id="S3.I1.i2.p1.2.m2.3.3.cmml" xref="S3.I1.i2.p1.2.m2.3.3">ℓ</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.3c">i\in\{0,\ldots,\ell\}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.3d">italic_i ∈ { 0 , … , roman_ℓ }</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(c)</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.1"><math alttext="s_{\ell+1}\in\{0,\ldots,w-1\}" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.m1.3"><semantics id="S3.I1.i3.p1.1.m1.3a"><mrow id="S3.I1.i3.p1.1.m1.3.3" xref="S3.I1.i3.p1.1.m1.3.3.cmml"><msub id="S3.I1.i3.p1.1.m1.3.3.3" xref="S3.I1.i3.p1.1.m1.3.3.3.cmml"><mi id="S3.I1.i3.p1.1.m1.3.3.3.2" xref="S3.I1.i3.p1.1.m1.3.3.3.2.cmml">s</mi><mrow id="S3.I1.i3.p1.1.m1.3.3.3.3" xref="S3.I1.i3.p1.1.m1.3.3.3.3.cmml"><mi id="S3.I1.i3.p1.1.m1.3.3.3.3.2" mathvariant="normal" xref="S3.I1.i3.p1.1.m1.3.3.3.3.2.cmml">ℓ</mi><mo id="S3.I1.i3.p1.1.m1.3.3.3.3.1" xref="S3.I1.i3.p1.1.m1.3.3.3.3.1.cmml">+</mo><mn id="S3.I1.i3.p1.1.m1.3.3.3.3.3" xref="S3.I1.i3.p1.1.m1.3.3.3.3.3.cmml">1</mn></mrow></msub><mo id="S3.I1.i3.p1.1.m1.3.3.2" xref="S3.I1.i3.p1.1.m1.3.3.2.cmml">∈</mo><mrow id="S3.I1.i3.p1.1.m1.3.3.1.1" xref="S3.I1.i3.p1.1.m1.3.3.1.2.cmml"><mo id="S3.I1.i3.p1.1.m1.3.3.1.1.2" stretchy="false" xref="S3.I1.i3.p1.1.m1.3.3.1.2.cmml">{</mo><mn id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml">0</mn><mo id="S3.I1.i3.p1.1.m1.3.3.1.1.3" xref="S3.I1.i3.p1.1.m1.3.3.1.2.cmml">,</mo><mi id="S3.I1.i3.p1.1.m1.2.2" mathvariant="normal" xref="S3.I1.i3.p1.1.m1.2.2.cmml">…</mi><mo id="S3.I1.i3.p1.1.m1.3.3.1.1.4" xref="S3.I1.i3.p1.1.m1.3.3.1.2.cmml">,</mo><mrow id="S3.I1.i3.p1.1.m1.3.3.1.1.1" xref="S3.I1.i3.p1.1.m1.3.3.1.1.1.cmml"><mi id="S3.I1.i3.p1.1.m1.3.3.1.1.1.2" xref="S3.I1.i3.p1.1.m1.3.3.1.1.1.2.cmml">w</mi><mo id="S3.I1.i3.p1.1.m1.3.3.1.1.1.1" xref="S3.I1.i3.p1.1.m1.3.3.1.1.1.1.cmml">−</mo><mn id="S3.I1.i3.p1.1.m1.3.3.1.1.1.3" xref="S3.I1.i3.p1.1.m1.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I1.i3.p1.1.m1.3.3.1.1.5" stretchy="false" xref="S3.I1.i3.p1.1.m1.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.m1.3b"><apply id="S3.I1.i3.p1.1.m1.3.3.cmml" xref="S3.I1.i3.p1.1.m1.3.3"><in id="S3.I1.i3.p1.1.m1.3.3.2.cmml" xref="S3.I1.i3.p1.1.m1.3.3.2"></in><apply id="S3.I1.i3.p1.1.m1.3.3.3.cmml" xref="S3.I1.i3.p1.1.m1.3.3.3"><csymbol cd="ambiguous" id="S3.I1.i3.p1.1.m1.3.3.3.1.cmml" xref="S3.I1.i3.p1.1.m1.3.3.3">subscript</csymbol><ci id="S3.I1.i3.p1.1.m1.3.3.3.2.cmml" xref="S3.I1.i3.p1.1.m1.3.3.3.2">𝑠</ci><apply id="S3.I1.i3.p1.1.m1.3.3.3.3.cmml" xref="S3.I1.i3.p1.1.m1.3.3.3.3"><plus id="S3.I1.i3.p1.1.m1.3.3.3.3.1.cmml" xref="S3.I1.i3.p1.1.m1.3.3.3.3.1"></plus><ci id="S3.I1.i3.p1.1.m1.3.3.3.3.2.cmml" xref="S3.I1.i3.p1.1.m1.3.3.3.3.2">ℓ</ci><cn id="S3.I1.i3.p1.1.m1.3.3.3.3.3.cmml" type="integer" xref="S3.I1.i3.p1.1.m1.3.3.3.3.3">1</cn></apply></apply><set id="S3.I1.i3.p1.1.m1.3.3.1.2.cmml" xref="S3.I1.i3.p1.1.m1.3.3.1.1"><cn id="S3.I1.i3.p1.1.m1.1.1.cmml" type="integer" xref="S3.I1.i3.p1.1.m1.1.1">0</cn><ci id="S3.I1.i3.p1.1.m1.2.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2">…</ci><apply id="S3.I1.i3.p1.1.m1.3.3.1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.3.3.1.1.1"><minus id="S3.I1.i3.p1.1.m1.3.3.1.1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.3.3.1.1.1.1"></minus><ci id="S3.I1.i3.p1.1.m1.3.3.1.1.1.2.cmml" xref="S3.I1.i3.p1.1.m1.3.3.1.1.1.2">𝑤</ci><cn id="S3.I1.i3.p1.1.m1.3.3.1.1.1.3.cmml" type="integer" xref="S3.I1.i3.p1.1.m1.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.m1.3c">s_{\ell+1}\in\{0,\ldots,w-1\}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.1.m1.3d">italic_s start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∈ { 0 , … , italic_w - 1 }</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(d)</span> <div class="ltx_para" id="S3.I1.i4.p1"> <p class="ltx_p" id="S3.I1.i4.p1.5"><math alttext="G[W_{i}]" class="ltx_Math" display="inline" id="S3.I1.i4.p1.1.m1.1"><semantics id="S3.I1.i4.p1.1.m1.1a"><mrow id="S3.I1.i4.p1.1.m1.1.1" xref="S3.I1.i4.p1.1.m1.1.1.cmml"><mi id="S3.I1.i4.p1.1.m1.1.1.3" xref="S3.I1.i4.p1.1.m1.1.1.3.cmml">G</mi><mo id="S3.I1.i4.p1.1.m1.1.1.2" xref="S3.I1.i4.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.I1.i4.p1.1.m1.1.1.1.1" xref="S3.I1.i4.p1.1.m1.1.1.1.2.cmml"><mo id="S3.I1.i4.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S3.I1.i4.p1.1.m1.1.1.1.2.1.cmml">[</mo><msub id="S3.I1.i4.p1.1.m1.1.1.1.1.1" xref="S3.I1.i4.p1.1.m1.1.1.1.1.1.cmml"><mi id="S3.I1.i4.p1.1.m1.1.1.1.1.1.2" xref="S3.I1.i4.p1.1.m1.1.1.1.1.1.2.cmml">W</mi><mi id="S3.I1.i4.p1.1.m1.1.1.1.1.1.3" xref="S3.I1.i4.p1.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.I1.i4.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S3.I1.i4.p1.1.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.1.m1.1b"><apply id="S3.I1.i4.p1.1.m1.1.1.cmml" xref="S3.I1.i4.p1.1.m1.1.1"><times id="S3.I1.i4.p1.1.m1.1.1.2.cmml" xref="S3.I1.i4.p1.1.m1.1.1.2"></times><ci id="S3.I1.i4.p1.1.m1.1.1.3.cmml" xref="S3.I1.i4.p1.1.m1.1.1.3">𝐺</ci><apply id="S3.I1.i4.p1.1.m1.1.1.1.2.cmml" xref="S3.I1.i4.p1.1.m1.1.1.1.1"><csymbol cd="latexml" id="S3.I1.i4.p1.1.m1.1.1.1.2.1.cmml" xref="S3.I1.i4.p1.1.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.I1.i4.p1.1.m1.1.1.1.1.1.cmml" xref="S3.I1.i4.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i4.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.I1.i4.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.I1.i4.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.I1.i4.p1.1.m1.1.1.1.1.1.2">𝑊</ci><ci id="S3.I1.i4.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.I1.i4.p1.1.m1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.1.m1.1c">G[W_{i}]</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.1.m1.1d">italic_G [ italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ]</annotation></semantics></math> contains <math alttext="s_{i}" class="ltx_Math" display="inline" id="S3.I1.i4.p1.2.m2.1"><semantics id="S3.I1.i4.p1.2.m2.1a"><msub id="S3.I1.i4.p1.2.m2.1.1" xref="S3.I1.i4.p1.2.m2.1.1.cmml"><mi id="S3.I1.i4.p1.2.m2.1.1.2" xref="S3.I1.i4.p1.2.m2.1.1.2.cmml">s</mi><mi id="S3.I1.i4.p1.2.m2.1.1.3" xref="S3.I1.i4.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.2.m2.1b"><apply id="S3.I1.i4.p1.2.m2.1.1.cmml" xref="S3.I1.i4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i4.p1.2.m2.1.1.1.cmml" xref="S3.I1.i4.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I1.i4.p1.2.m2.1.1.2.cmml" xref="S3.I1.i4.p1.2.m2.1.1.2">𝑠</ci><ci id="S3.I1.i4.p1.2.m2.1.1.3.cmml" xref="S3.I1.i4.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.2.m2.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.2.m2.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> pairwise vertex-disjoint <math alttext="\Delta_{i}" class="ltx_Math" display="inline" id="S3.I1.i4.p1.3.m3.1"><semantics id="S3.I1.i4.p1.3.m3.1a"><msub id="S3.I1.i4.p1.3.m3.1.1" xref="S3.I1.i4.p1.3.m3.1.1.cmml"><mi id="S3.I1.i4.p1.3.m3.1.1.2" mathvariant="normal" xref="S3.I1.i4.p1.3.m3.1.1.2.cmml">Δ</mi><mi id="S3.I1.i4.p1.3.m3.1.1.3" xref="S3.I1.i4.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.3.m3.1b"><apply id="S3.I1.i4.p1.3.m3.1.1.cmml" xref="S3.I1.i4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i4.p1.3.m3.1.1.1.cmml" xref="S3.I1.i4.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I1.i4.p1.3.m3.1.1.2.cmml" xref="S3.I1.i4.p1.3.m3.1.1.2">Δ</ci><ci id="S3.I1.i4.p1.3.m3.1.1.3.cmml" xref="S3.I1.i4.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.3.m3.1c">\Delta_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.3.m3.1d">roman_Δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>-<math alttext="W" class="ltx_Math" display="inline" id="S3.I1.i4.p1.4.m4.1"><semantics id="S3.I1.i4.p1.4.m4.1a"><mi id="S3.I1.i4.p1.4.m4.1.1" xref="S3.I1.i4.p1.4.m4.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.4.m4.1b"><ci id="S3.I1.i4.p1.4.m4.1.1.cmml" xref="S3.I1.i4.p1.4.m4.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.4.m4.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.4.m4.1d">italic_W</annotation></semantics></math> paths, for each <math alttext="i\in\{0,\ldots,\ell+1\}" class="ltx_Math" display="inline" id="S3.I1.i4.p1.5.m5.3"><semantics id="S3.I1.i4.p1.5.m5.3a"><mrow id="S3.I1.i4.p1.5.m5.3.3" xref="S3.I1.i4.p1.5.m5.3.3.cmml"><mi id="S3.I1.i4.p1.5.m5.3.3.3" xref="S3.I1.i4.p1.5.m5.3.3.3.cmml">i</mi><mo id="S3.I1.i4.p1.5.m5.3.3.2" xref="S3.I1.i4.p1.5.m5.3.3.2.cmml">∈</mo><mrow id="S3.I1.i4.p1.5.m5.3.3.1.1" xref="S3.I1.i4.p1.5.m5.3.3.1.2.cmml"><mo id="S3.I1.i4.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S3.I1.i4.p1.5.m5.3.3.1.2.cmml">{</mo><mn id="S3.I1.i4.p1.5.m5.1.1" xref="S3.I1.i4.p1.5.m5.1.1.cmml">0</mn><mo id="S3.I1.i4.p1.5.m5.3.3.1.1.3" xref="S3.I1.i4.p1.5.m5.3.3.1.2.cmml">,</mo><mi id="S3.I1.i4.p1.5.m5.2.2" mathvariant="normal" xref="S3.I1.i4.p1.5.m5.2.2.cmml">…</mi><mo id="S3.I1.i4.p1.5.m5.3.3.1.1.4" xref="S3.I1.i4.p1.5.m5.3.3.1.2.cmml">,</mo><mrow id="S3.I1.i4.p1.5.m5.3.3.1.1.1" xref="S3.I1.i4.p1.5.m5.3.3.1.1.1.cmml"><mi id="S3.I1.i4.p1.5.m5.3.3.1.1.1.2" mathvariant="normal" xref="S3.I1.i4.p1.5.m5.3.3.1.1.1.2.cmml">ℓ</mi><mo id="S3.I1.i4.p1.5.m5.3.3.1.1.1.1" xref="S3.I1.i4.p1.5.m5.3.3.1.1.1.1.cmml">+</mo><mn id="S3.I1.i4.p1.5.m5.3.3.1.1.1.3" xref="S3.I1.i4.p1.5.m5.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I1.i4.p1.5.m5.3.3.1.1.5" stretchy="false" xref="S3.I1.i4.p1.5.m5.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.5.m5.3b"><apply id="S3.I1.i4.p1.5.m5.3.3.cmml" xref="S3.I1.i4.p1.5.m5.3.3"><in id="S3.I1.i4.p1.5.m5.3.3.2.cmml" xref="S3.I1.i4.p1.5.m5.3.3.2"></in><ci id="S3.I1.i4.p1.5.m5.3.3.3.cmml" xref="S3.I1.i4.p1.5.m5.3.3.3">𝑖</ci><set id="S3.I1.i4.p1.5.m5.3.3.1.2.cmml" xref="S3.I1.i4.p1.5.m5.3.3.1.1"><cn id="S3.I1.i4.p1.5.m5.1.1.cmml" type="integer" xref="S3.I1.i4.p1.5.m5.1.1">0</cn><ci id="S3.I1.i4.p1.5.m5.2.2.cmml" xref="S3.I1.i4.p1.5.m5.2.2">…</ci><apply id="S3.I1.i4.p1.5.m5.3.3.1.1.1.cmml" xref="S3.I1.i4.p1.5.m5.3.3.1.1.1"><plus id="S3.I1.i4.p1.5.m5.3.3.1.1.1.1.cmml" xref="S3.I1.i4.p1.5.m5.3.3.1.1.1.1"></plus><ci id="S3.I1.i4.p1.5.m5.3.3.1.1.1.2.cmml" xref="S3.I1.i4.p1.5.m5.3.3.1.1.1.2">ℓ</ci><cn id="S3.I1.i4.p1.5.m5.3.3.1.1.1.3.cmml" type="integer" xref="S3.I1.i4.p1.5.m5.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.5.m5.3c">i\in\{0,\ldots,\ell+1\}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.5.m5.3d">italic_i ∈ { 0 , … , roman_ℓ + 1 }</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(e)</span> <div class="ltx_para" id="S3.I1.i5.p1"> <p class="ltx_p" id="S3.I1.i5.p1.4">There exists <math alttext="Z\subseteq W_{\ell+1}" class="ltx_Math" display="inline" id="S3.I1.i5.p1.1.m1.1"><semantics id="S3.I1.i5.p1.1.m1.1a"><mrow id="S3.I1.i5.p1.1.m1.1.1" xref="S3.I1.i5.p1.1.m1.1.1.cmml"><mi id="S3.I1.i5.p1.1.m1.1.1.2" xref="S3.I1.i5.p1.1.m1.1.1.2.cmml">Z</mi><mo id="S3.I1.i5.p1.1.m1.1.1.1" xref="S3.I1.i5.p1.1.m1.1.1.1.cmml">⊆</mo><msub id="S3.I1.i5.p1.1.m1.1.1.3" xref="S3.I1.i5.p1.1.m1.1.1.3.cmml"><mi id="S3.I1.i5.p1.1.m1.1.1.3.2" xref="S3.I1.i5.p1.1.m1.1.1.3.2.cmml">W</mi><mrow id="S3.I1.i5.p1.1.m1.1.1.3.3" xref="S3.I1.i5.p1.1.m1.1.1.3.3.cmml"><mi id="S3.I1.i5.p1.1.m1.1.1.3.3.2" mathvariant="normal" xref="S3.I1.i5.p1.1.m1.1.1.3.3.2.cmml">ℓ</mi><mo id="S3.I1.i5.p1.1.m1.1.1.3.3.1" xref="S3.I1.i5.p1.1.m1.1.1.3.3.1.cmml">+</mo><mn id="S3.I1.i5.p1.1.m1.1.1.3.3.3" xref="S3.I1.i5.p1.1.m1.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i5.p1.1.m1.1b"><apply id="S3.I1.i5.p1.1.m1.1.1.cmml" xref="S3.I1.i5.p1.1.m1.1.1"><subset id="S3.I1.i5.p1.1.m1.1.1.1.cmml" xref="S3.I1.i5.p1.1.m1.1.1.1"></subset><ci id="S3.I1.i5.p1.1.m1.1.1.2.cmml" xref="S3.I1.i5.p1.1.m1.1.1.2">𝑍</ci><apply id="S3.I1.i5.p1.1.m1.1.1.3.cmml" xref="S3.I1.i5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i5.p1.1.m1.1.1.3.1.cmml" xref="S3.I1.i5.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.I1.i5.p1.1.m1.1.1.3.2.cmml" xref="S3.I1.i5.p1.1.m1.1.1.3.2">𝑊</ci><apply id="S3.I1.i5.p1.1.m1.1.1.3.3.cmml" xref="S3.I1.i5.p1.1.m1.1.1.3.3"><plus id="S3.I1.i5.p1.1.m1.1.1.3.3.1.cmml" xref="S3.I1.i5.p1.1.m1.1.1.3.3.1"></plus><ci id="S3.I1.i5.p1.1.m1.1.1.3.3.2.cmml" xref="S3.I1.i5.p1.1.m1.1.1.3.3.2">ℓ</ci><cn id="S3.I1.i5.p1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S3.I1.i5.p1.1.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i5.p1.1.m1.1c">Z\subseteq W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i5.p1.1.m1.1d">italic_Z ⊆ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="|Z|=s_{\ell+1}" class="ltx_Math" display="inline" id="S3.I1.i5.p1.2.m2.1"><semantics id="S3.I1.i5.p1.2.m2.1a"><mrow id="S3.I1.i5.p1.2.m2.1.2" xref="S3.I1.i5.p1.2.m2.1.2.cmml"><mrow id="S3.I1.i5.p1.2.m2.1.2.2.2" xref="S3.I1.i5.p1.2.m2.1.2.2.1.cmml"><mo id="S3.I1.i5.p1.2.m2.1.2.2.2.1" stretchy="false" xref="S3.I1.i5.p1.2.m2.1.2.2.1.1.cmml">|</mo><mi id="S3.I1.i5.p1.2.m2.1.1" xref="S3.I1.i5.p1.2.m2.1.1.cmml">Z</mi><mo id="S3.I1.i5.p1.2.m2.1.2.2.2.2" stretchy="false" xref="S3.I1.i5.p1.2.m2.1.2.2.1.1.cmml">|</mo></mrow><mo id="S3.I1.i5.p1.2.m2.1.2.1" xref="S3.I1.i5.p1.2.m2.1.2.1.cmml">=</mo><msub id="S3.I1.i5.p1.2.m2.1.2.3" xref="S3.I1.i5.p1.2.m2.1.2.3.cmml"><mi id="S3.I1.i5.p1.2.m2.1.2.3.2" xref="S3.I1.i5.p1.2.m2.1.2.3.2.cmml">s</mi><mrow id="S3.I1.i5.p1.2.m2.1.2.3.3" xref="S3.I1.i5.p1.2.m2.1.2.3.3.cmml"><mi id="S3.I1.i5.p1.2.m2.1.2.3.3.2" mathvariant="normal" xref="S3.I1.i5.p1.2.m2.1.2.3.3.2.cmml">ℓ</mi><mo id="S3.I1.i5.p1.2.m2.1.2.3.3.1" xref="S3.I1.i5.p1.2.m2.1.2.3.3.1.cmml">+</mo><mn id="S3.I1.i5.p1.2.m2.1.2.3.3.3" xref="S3.I1.i5.p1.2.m2.1.2.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i5.p1.2.m2.1b"><apply id="S3.I1.i5.p1.2.m2.1.2.cmml" xref="S3.I1.i5.p1.2.m2.1.2"><eq id="S3.I1.i5.p1.2.m2.1.2.1.cmml" xref="S3.I1.i5.p1.2.m2.1.2.1"></eq><apply id="S3.I1.i5.p1.2.m2.1.2.2.1.cmml" xref="S3.I1.i5.p1.2.m2.1.2.2.2"><abs id="S3.I1.i5.p1.2.m2.1.2.2.1.1.cmml" xref="S3.I1.i5.p1.2.m2.1.2.2.2.1"></abs><ci id="S3.I1.i5.p1.2.m2.1.1.cmml" xref="S3.I1.i5.p1.2.m2.1.1">𝑍</ci></apply><apply id="S3.I1.i5.p1.2.m2.1.2.3.cmml" xref="S3.I1.i5.p1.2.m2.1.2.3"><csymbol cd="ambiguous" id="S3.I1.i5.p1.2.m2.1.2.3.1.cmml" xref="S3.I1.i5.p1.2.m2.1.2.3">subscript</csymbol><ci id="S3.I1.i5.p1.2.m2.1.2.3.2.cmml" xref="S3.I1.i5.p1.2.m2.1.2.3.2">𝑠</ci><apply id="S3.I1.i5.p1.2.m2.1.2.3.3.cmml" xref="S3.I1.i5.p1.2.m2.1.2.3.3"><plus id="S3.I1.i5.p1.2.m2.1.2.3.3.1.cmml" xref="S3.I1.i5.p1.2.m2.1.2.3.3.1"></plus><ci id="S3.I1.i5.p1.2.m2.1.2.3.3.2.cmml" xref="S3.I1.i5.p1.2.m2.1.2.3.3.2">ℓ</ci><cn id="S3.I1.i5.p1.2.m2.1.2.3.3.3.cmml" type="integer" xref="S3.I1.i5.p1.2.m2.1.2.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i5.p1.2.m2.1c">|Z|=s_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i5.p1.2.m2.1d">| italic_Z | = italic_s start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> that separates <math alttext="V(G)\setminus W_{\ell}" class="ltx_Math" display="inline" id="S3.I1.i5.p1.3.m3.1"><semantics id="S3.I1.i5.p1.3.m3.1a"><mrow id="S3.I1.i5.p1.3.m3.1.2" xref="S3.I1.i5.p1.3.m3.1.2.cmml"><mrow id="S3.I1.i5.p1.3.m3.1.2.2" xref="S3.I1.i5.p1.3.m3.1.2.2.cmml"><mi id="S3.I1.i5.p1.3.m3.1.2.2.2" xref="S3.I1.i5.p1.3.m3.1.2.2.2.cmml">V</mi><mo id="S3.I1.i5.p1.3.m3.1.2.2.1" xref="S3.I1.i5.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.I1.i5.p1.3.m3.1.2.2.3.2" xref="S3.I1.i5.p1.3.m3.1.2.2.cmml"><mo id="S3.I1.i5.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.I1.i5.p1.3.m3.1.2.2.cmml">(</mo><mi id="S3.I1.i5.p1.3.m3.1.1" xref="S3.I1.i5.p1.3.m3.1.1.cmml">G</mi><mo id="S3.I1.i5.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="S3.I1.i5.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I1.i5.p1.3.m3.1.2.1" xref="S3.I1.i5.p1.3.m3.1.2.1.cmml">∖</mo><msub id="S3.I1.i5.p1.3.m3.1.2.3" xref="S3.I1.i5.p1.3.m3.1.2.3.cmml"><mi id="S3.I1.i5.p1.3.m3.1.2.3.2" xref="S3.I1.i5.p1.3.m3.1.2.3.2.cmml">W</mi><mi id="S3.I1.i5.p1.3.m3.1.2.3.3" mathvariant="normal" xref="S3.I1.i5.p1.3.m3.1.2.3.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i5.p1.3.m3.1b"><apply id="S3.I1.i5.p1.3.m3.1.2.cmml" xref="S3.I1.i5.p1.3.m3.1.2"><setdiff id="S3.I1.i5.p1.3.m3.1.2.1.cmml" xref="S3.I1.i5.p1.3.m3.1.2.1"></setdiff><apply id="S3.I1.i5.p1.3.m3.1.2.2.cmml" xref="S3.I1.i5.p1.3.m3.1.2.2"><times id="S3.I1.i5.p1.3.m3.1.2.2.1.cmml" xref="S3.I1.i5.p1.3.m3.1.2.2.1"></times><ci id="S3.I1.i5.p1.3.m3.1.2.2.2.cmml" xref="S3.I1.i5.p1.3.m3.1.2.2.2">𝑉</ci><ci id="S3.I1.i5.p1.3.m3.1.1.cmml" xref="S3.I1.i5.p1.3.m3.1.1">𝐺</ci></apply><apply id="S3.I1.i5.p1.3.m3.1.2.3.cmml" xref="S3.I1.i5.p1.3.m3.1.2.3"><csymbol cd="ambiguous" id="S3.I1.i5.p1.3.m3.1.2.3.1.cmml" xref="S3.I1.i5.p1.3.m3.1.2.3">subscript</csymbol><ci id="S3.I1.i5.p1.3.m3.1.2.3.2.cmml" xref="S3.I1.i5.p1.3.m3.1.2.3.2">𝑊</ci><ci id="S3.I1.i5.p1.3.m3.1.2.3.3.cmml" xref="S3.I1.i5.p1.3.m3.1.2.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i5.p1.3.m3.1c">V(G)\setminus W_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i5.p1.3.m3.1d">italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="W" class="ltx_Math" display="inline" id="S3.I1.i5.p1.4.m4.1"><semantics id="S3.I1.i5.p1.4.m4.1a"><mi id="S3.I1.i5.p1.4.m4.1.1" xref="S3.I1.i5.p1.4.m4.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i5.p1.4.m4.1b"><ci id="S3.I1.i5.p1.4.m4.1.1.cmml" xref="S3.I1.i5.p1.4.m4.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i5.p1.4.m4.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i5.p1.4.m4.1d">italic_W</annotation></semantics></math>.</p> </div> </li> </ol> </div> <div class="ltx_theorem ltx_theorem_lem" id="Thmthm6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm6.1.1.1">Lemma 6</span></span><span class="ltx_text ltx_font_bold" id="Thmthm6.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm6.p1"> <p class="ltx_p" id="Thmthm6.p1.6"><span class="ltx_text ltx_font_italic" id="Thmthm6.p1.6.6">For every graph <math alttext="G" class="ltx_Math" display="inline" id="Thmthm6.p1.1.1.m1.1"><semantics id="Thmthm6.p1.1.1.m1.1a"><mi id="Thmthm6.p1.1.1.m1.1.1" xref="Thmthm6.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.1.1.m1.1b"><ci id="Thmthm6.p1.1.1.m1.1.1.cmml" xref="Thmthm6.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.1.1.m1.1d">italic_G</annotation></semantics></math>, every <math alttext="W\subseteq V(G)" class="ltx_Math" display="inline" id="Thmthm6.p1.2.2.m2.1"><semantics id="Thmthm6.p1.2.2.m2.1a"><mrow id="Thmthm6.p1.2.2.m2.1.2" xref="Thmthm6.p1.2.2.m2.1.2.cmml"><mi id="Thmthm6.p1.2.2.m2.1.2.2" xref="Thmthm6.p1.2.2.m2.1.2.2.cmml">W</mi><mo id="Thmthm6.p1.2.2.m2.1.2.1" xref="Thmthm6.p1.2.2.m2.1.2.1.cmml">⊆</mo><mrow id="Thmthm6.p1.2.2.m2.1.2.3" xref="Thmthm6.p1.2.2.m2.1.2.3.cmml"><mi id="Thmthm6.p1.2.2.m2.1.2.3.2" xref="Thmthm6.p1.2.2.m2.1.2.3.2.cmml">V</mi><mo id="Thmthm6.p1.2.2.m2.1.2.3.1" xref="Thmthm6.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmthm6.p1.2.2.m2.1.2.3.3.2" xref="Thmthm6.p1.2.2.m2.1.2.3.cmml"><mo id="Thmthm6.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmthm6.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="Thmthm6.p1.2.2.m2.1.1" xref="Thmthm6.p1.2.2.m2.1.1.cmml">G</mi><mo id="Thmthm6.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmthm6.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.2.2.m2.1b"><apply id="Thmthm6.p1.2.2.m2.1.2.cmml" xref="Thmthm6.p1.2.2.m2.1.2"><subset id="Thmthm6.p1.2.2.m2.1.2.1.cmml" xref="Thmthm6.p1.2.2.m2.1.2.1"></subset><ci id="Thmthm6.p1.2.2.m2.1.2.2.cmml" xref="Thmthm6.p1.2.2.m2.1.2.2">𝑊</ci><apply id="Thmthm6.p1.2.2.m2.1.2.3.cmml" xref="Thmthm6.p1.2.2.m2.1.2.3"><times id="Thmthm6.p1.2.2.m2.1.2.3.1.cmml" xref="Thmthm6.p1.2.2.m2.1.2.3.1"></times><ci id="Thmthm6.p1.2.2.m2.1.2.3.2.cmml" xref="Thmthm6.p1.2.2.m2.1.2.3.2">𝑉</ci><ci id="Thmthm6.p1.2.2.m2.1.1.cmml" xref="Thmthm6.p1.2.2.m2.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.2.2.m2.1c">W\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.2.2.m2.1d">italic_W ⊆ italic_V ( italic_G )</annotation></semantics></math> and every non-negative integer <math alttext="w\leq|W|" class="ltx_Math" display="inline" id="Thmthm6.p1.3.3.m3.1"><semantics id="Thmthm6.p1.3.3.m3.1a"><mrow id="Thmthm6.p1.3.3.m3.1.2" xref="Thmthm6.p1.3.3.m3.1.2.cmml"><mi id="Thmthm6.p1.3.3.m3.1.2.2" xref="Thmthm6.p1.3.3.m3.1.2.2.cmml">w</mi><mo id="Thmthm6.p1.3.3.m3.1.2.1" xref="Thmthm6.p1.3.3.m3.1.2.1.cmml">≤</mo><mrow id="Thmthm6.p1.3.3.m3.1.2.3.2" xref="Thmthm6.p1.3.3.m3.1.2.3.1.cmml"><mo id="Thmthm6.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="Thmthm6.p1.3.3.m3.1.2.3.1.1.cmml">|</mo><mi id="Thmthm6.p1.3.3.m3.1.1" xref="Thmthm6.p1.3.3.m3.1.1.cmml">W</mi><mo id="Thmthm6.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="Thmthm6.p1.3.3.m3.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.3.3.m3.1b"><apply id="Thmthm6.p1.3.3.m3.1.2.cmml" xref="Thmthm6.p1.3.3.m3.1.2"><leq id="Thmthm6.p1.3.3.m3.1.2.1.cmml" xref="Thmthm6.p1.3.3.m3.1.2.1"></leq><ci id="Thmthm6.p1.3.3.m3.1.2.2.cmml" xref="Thmthm6.p1.3.3.m3.1.2.2">𝑤</ci><apply id="Thmthm6.p1.3.3.m3.1.2.3.1.cmml" xref="Thmthm6.p1.3.3.m3.1.2.3.2"><abs id="Thmthm6.p1.3.3.m3.1.2.3.1.1.cmml" xref="Thmthm6.p1.3.3.m3.1.2.3.2.1"></abs><ci id="Thmthm6.p1.3.3.m3.1.1.cmml" xref="Thmthm6.p1.3.3.m3.1.1">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.3.3.m3.1c">w\leq|W|</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.3.3.m3.1d">italic_w ≤ | italic_W |</annotation></semantics></math>, there exists a <math alttext="W" class="ltx_Math" display="inline" id="Thmthm6.p1.4.4.m4.1"><semantics id="Thmthm6.p1.4.4.m4.1a"><mi id="Thmthm6.p1.4.4.m4.1.1" xref="Thmthm6.p1.4.4.m4.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.4.4.m4.1b"><ci id="Thmthm6.p1.4.4.m4.1.1.cmml" xref="Thmthm6.p1.4.4.m4.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.4.4.m4.1c">W</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.4.4.m4.1d">italic_W</annotation></semantics></math>-sequence of width <math alttext="w" class="ltx_Math" display="inline" id="Thmthm6.p1.5.5.m5.1"><semantics id="Thmthm6.p1.5.5.m5.1a"><mi id="Thmthm6.p1.5.5.m5.1.1" xref="Thmthm6.p1.5.5.m5.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.5.5.m5.1b"><ci id="Thmthm6.p1.5.5.m5.1.1.cmml" xref="Thmthm6.p1.5.5.m5.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.5.5.m5.1c">w</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.5.5.m5.1d">italic_w</annotation></semantics></math> in <math alttext="G" class="ltx_Math" display="inline" id="Thmthm6.p1.6.6.m6.1"><semantics id="Thmthm6.p1.6.6.m6.1a"><mi id="Thmthm6.p1.6.6.m6.1.1" xref="Thmthm6.p1.6.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.6.6.m6.1b"><ci id="Thmthm6.p1.6.6.m6.1.1.cmml" xref="Thmthm6.p1.6.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.6.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.6.6.m6.1d">italic_G</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.1.p1"> <p class="ltx_p" id="S3.1.p1.5">Let <math alttext="W_{0}:=W" class="ltx_Math" display="inline" id="S3.1.p1.1.m1.1"><semantics id="S3.1.p1.1.m1.1a"><mrow id="S3.1.p1.1.m1.1.1" xref="S3.1.p1.1.m1.1.1.cmml"><msub id="S3.1.p1.1.m1.1.1.2" xref="S3.1.p1.1.m1.1.1.2.cmml"><mi id="S3.1.p1.1.m1.1.1.2.2" xref="S3.1.p1.1.m1.1.1.2.2.cmml">W</mi><mn id="S3.1.p1.1.m1.1.1.2.3" xref="S3.1.p1.1.m1.1.1.2.3.cmml">0</mn></msub><mo id="S3.1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.1.p1.1.m1.1.1.1.cmml">:=</mo><mi id="S3.1.p1.1.m1.1.1.3" xref="S3.1.p1.1.m1.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.1.m1.1b"><apply id="S3.1.p1.1.m1.1.1.cmml" xref="S3.1.p1.1.m1.1.1"><csymbol cd="latexml" id="S3.1.p1.1.m1.1.1.1.cmml" xref="S3.1.p1.1.m1.1.1.1">assign</csymbol><apply id="S3.1.p1.1.m1.1.1.2.cmml" xref="S3.1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.1.p1.1.m1.1.1.2.1.cmml" xref="S3.1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.1.p1.1.m1.1.1.2.2.cmml" xref="S3.1.p1.1.m1.1.1.2.2">𝑊</ci><cn id="S3.1.p1.1.m1.1.1.2.3.cmml" type="integer" xref="S3.1.p1.1.m1.1.1.2.3">0</cn></apply><ci id="S3.1.p1.1.m1.1.1.3.cmml" xref="S3.1.p1.1.m1.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.1.m1.1c">W_{0}:=W</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.1.m1.1d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := italic_W</annotation></semantics></math> and suppose that sets <math alttext="W_{0},\ldots,W_{i}" class="ltx_Math" display="inline" id="S3.1.p1.2.m2.3"><semantics id="S3.1.p1.2.m2.3a"><mrow id="S3.1.p1.2.m2.3.3.2" xref="S3.1.p1.2.m2.3.3.3.cmml"><msub id="S3.1.p1.2.m2.2.2.1.1" xref="S3.1.p1.2.m2.2.2.1.1.cmml"><mi id="S3.1.p1.2.m2.2.2.1.1.2" xref="S3.1.p1.2.m2.2.2.1.1.2.cmml">W</mi><mn id="S3.1.p1.2.m2.2.2.1.1.3" xref="S3.1.p1.2.m2.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.1.p1.2.m2.3.3.2.3" xref="S3.1.p1.2.m2.3.3.3.cmml">,</mo><mi id="S3.1.p1.2.m2.1.1" mathvariant="normal" xref="S3.1.p1.2.m2.1.1.cmml">…</mi><mo id="S3.1.p1.2.m2.3.3.2.4" xref="S3.1.p1.2.m2.3.3.3.cmml">,</mo><msub id="S3.1.p1.2.m2.3.3.2.2" xref="S3.1.p1.2.m2.3.3.2.2.cmml"><mi id="S3.1.p1.2.m2.3.3.2.2.2" xref="S3.1.p1.2.m2.3.3.2.2.2.cmml">W</mi><mi id="S3.1.p1.2.m2.3.3.2.2.3" xref="S3.1.p1.2.m2.3.3.2.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.2.m2.3b"><list id="S3.1.p1.2.m2.3.3.3.cmml" xref="S3.1.p1.2.m2.3.3.2"><apply id="S3.1.p1.2.m2.2.2.1.1.cmml" xref="S3.1.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3.1.p1.2.m2.2.2.1.1.1.cmml" xref="S3.1.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S3.1.p1.2.m2.2.2.1.1.2.cmml" xref="S3.1.p1.2.m2.2.2.1.1.2">𝑊</ci><cn id="S3.1.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S3.1.p1.2.m2.2.2.1.1.3">0</cn></apply><ci id="S3.1.p1.2.m2.1.1.cmml" xref="S3.1.p1.2.m2.1.1">…</ci><apply id="S3.1.p1.2.m2.3.3.2.2.cmml" xref="S3.1.p1.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S3.1.p1.2.m2.3.3.2.2.1.cmml" xref="S3.1.p1.2.m2.3.3.2.2">subscript</csymbol><ci id="S3.1.p1.2.m2.3.3.2.2.2.cmml" xref="S3.1.p1.2.m2.3.3.2.2.2">𝑊</ci><ci id="S3.1.p1.2.m2.3.3.2.2.3.cmml" xref="S3.1.p1.2.m2.3.3.2.2.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.2.m2.3c">W_{0},\ldots,W_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.2.m2.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> have been defined that satisfy <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i1" title="Item (a) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(a)</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i2" title="Item (b) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(b)</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i4" title="Item (d) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(d)</span></a> for some <math alttext="i\geq 0" class="ltx_Math" display="inline" id="S3.1.p1.3.m3.1"><semantics id="S3.1.p1.3.m3.1a"><mrow id="S3.1.p1.3.m3.1.1" xref="S3.1.p1.3.m3.1.1.cmml"><mi id="S3.1.p1.3.m3.1.1.2" xref="S3.1.p1.3.m3.1.1.2.cmml">i</mi><mo id="S3.1.p1.3.m3.1.1.1" xref="S3.1.p1.3.m3.1.1.1.cmml">≥</mo><mn id="S3.1.p1.3.m3.1.1.3" xref="S3.1.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.3.m3.1b"><apply id="S3.1.p1.3.m3.1.1.cmml" xref="S3.1.p1.3.m3.1.1"><geq id="S3.1.p1.3.m3.1.1.1.cmml" xref="S3.1.p1.3.m3.1.1.1"></geq><ci id="S3.1.p1.3.m3.1.1.2.cmml" xref="S3.1.p1.3.m3.1.1.2">𝑖</ci><cn id="S3.1.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.3.m3.1c">i\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.3.m3.1d">italic_i ≥ 0</annotation></semantics></math>. (These conditions are trivially satisfied for <math alttext="i=0" class="ltx_Math" display="inline" id="S3.1.p1.4.m4.1"><semantics id="S3.1.p1.4.m4.1a"><mrow id="S3.1.p1.4.m4.1.1" xref="S3.1.p1.4.m4.1.1.cmml"><mi id="S3.1.p1.4.m4.1.1.2" xref="S3.1.p1.4.m4.1.1.2.cmml">i</mi><mo id="S3.1.p1.4.m4.1.1.1" xref="S3.1.p1.4.m4.1.1.1.cmml">=</mo><mn id="S3.1.p1.4.m4.1.1.3" xref="S3.1.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.4.m4.1b"><apply id="S3.1.p1.4.m4.1.1.cmml" xref="S3.1.p1.4.m4.1.1"><eq id="S3.1.p1.4.m4.1.1.1.cmml" xref="S3.1.p1.4.m4.1.1.1"></eq><ci id="S3.1.p1.4.m4.1.1.2.cmml" xref="S3.1.p1.4.m4.1.1.2">𝑖</ci><cn id="S3.1.p1.4.m4.1.1.3.cmml" type="integer" xref="S3.1.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.4.m4.1c">i=0</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.4.m4.1d">italic_i = 0</annotation></semantics></math>.) We now show how to construct <math alttext="W_{i+1}" class="ltx_Math" display="inline" id="S3.1.p1.5.m5.1"><semantics id="S3.1.p1.5.m5.1a"><msub id="S3.1.p1.5.m5.1.1" xref="S3.1.p1.5.m5.1.1.cmml"><mi id="S3.1.p1.5.m5.1.1.2" xref="S3.1.p1.5.m5.1.1.2.cmml">W</mi><mrow id="S3.1.p1.5.m5.1.1.3" xref="S3.1.p1.5.m5.1.1.3.cmml"><mi id="S3.1.p1.5.m5.1.1.3.2" xref="S3.1.p1.5.m5.1.1.3.2.cmml">i</mi><mo id="S3.1.p1.5.m5.1.1.3.1" xref="S3.1.p1.5.m5.1.1.3.1.cmml">+</mo><mn id="S3.1.p1.5.m5.1.1.3.3" xref="S3.1.p1.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.1.p1.5.m5.1b"><apply id="S3.1.p1.5.m5.1.1.cmml" xref="S3.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.1.p1.5.m5.1.1.1.cmml" xref="S3.1.p1.5.m5.1.1">subscript</csymbol><ci id="S3.1.p1.5.m5.1.1.2.cmml" xref="S3.1.p1.5.m5.1.1.2">𝑊</ci><apply id="S3.1.p1.5.m5.1.1.3.cmml" xref="S3.1.p1.5.m5.1.1.3"><plus id="S3.1.p1.5.m5.1.1.3.1.cmml" xref="S3.1.p1.5.m5.1.1.3.1"></plus><ci id="S3.1.p1.5.m5.1.1.3.2.cmml" xref="S3.1.p1.5.m5.1.1.3.2">𝑖</ci><cn id="S3.1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S3.1.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.5.m5.1c">W_{i+1}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.5.m5.1d">italic_W start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.2.p2"> <p class="ltx_p" id="S3.2.p2.22">Let <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S3.2.p2.1.m1.1"><semantics id="S3.2.p2.1.m1.1a"><msup id="S3.2.p2.1.m1.1.1" xref="S3.2.p2.1.m1.1.1.cmml"><mi id="S3.2.p2.1.m1.1.1.2" xref="S3.2.p2.1.m1.1.1.2.cmml">w</mi><mo id="S3.2.p2.1.m1.1.1.3" xref="S3.2.p2.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.2.p2.1.m1.1b"><apply id="S3.2.p2.1.m1.1.1.cmml" xref="S3.2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.2.p2.1.m1.1.1.1.cmml" xref="S3.2.p2.1.m1.1.1">superscript</csymbol><ci id="S3.2.p2.1.m1.1.1.2.cmml" xref="S3.2.p2.1.m1.1.1.2">𝑤</ci><ci id="S3.2.p2.1.m1.1.1.3.cmml" xref="S3.2.p2.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.1.m1.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the maximum number of pairwise vertex-disjoint <math alttext="(V(G)\setminus W_{i})" class="ltx_Math" display="inline" id="S3.2.p2.2.m2.2"><semantics id="S3.2.p2.2.m2.2a"><mrow id="S3.2.p2.2.m2.2.2.1" xref="S3.2.p2.2.m2.2.2.1.1.cmml"><mo id="S3.2.p2.2.m2.2.2.1.2" stretchy="false" xref="S3.2.p2.2.m2.2.2.1.1.cmml">(</mo><mrow id="S3.2.p2.2.m2.2.2.1.1" xref="S3.2.p2.2.m2.2.2.1.1.cmml"><mrow id="S3.2.p2.2.m2.2.2.1.1.2" xref="S3.2.p2.2.m2.2.2.1.1.2.cmml"><mi id="S3.2.p2.2.m2.2.2.1.1.2.2" xref="S3.2.p2.2.m2.2.2.1.1.2.2.cmml">V</mi><mo id="S3.2.p2.2.m2.2.2.1.1.2.1" xref="S3.2.p2.2.m2.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S3.2.p2.2.m2.2.2.1.1.2.3.2" xref="S3.2.p2.2.m2.2.2.1.1.2.cmml"><mo id="S3.2.p2.2.m2.2.2.1.1.2.3.2.1" stretchy="false" xref="S3.2.p2.2.m2.2.2.1.1.2.cmml">(</mo><mi id="S3.2.p2.2.m2.1.1" xref="S3.2.p2.2.m2.1.1.cmml">G</mi><mo id="S3.2.p2.2.m2.2.2.1.1.2.3.2.2" stretchy="false" xref="S3.2.p2.2.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.2.p2.2.m2.2.2.1.1.1" xref="S3.2.p2.2.m2.2.2.1.1.1.cmml">∖</mo><msub id="S3.2.p2.2.m2.2.2.1.1.3" xref="S3.2.p2.2.m2.2.2.1.1.3.cmml"><mi id="S3.2.p2.2.m2.2.2.1.1.3.2" xref="S3.2.p2.2.m2.2.2.1.1.3.2.cmml">W</mi><mi id="S3.2.p2.2.m2.2.2.1.1.3.3" xref="S3.2.p2.2.m2.2.2.1.1.3.3.cmml">i</mi></msub></mrow><mo id="S3.2.p2.2.m2.2.2.1.3" stretchy="false" xref="S3.2.p2.2.m2.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.2.m2.2b"><apply id="S3.2.p2.2.m2.2.2.1.1.cmml" xref="S3.2.p2.2.m2.2.2.1"><setdiff id="S3.2.p2.2.m2.2.2.1.1.1.cmml" xref="S3.2.p2.2.m2.2.2.1.1.1"></setdiff><apply id="S3.2.p2.2.m2.2.2.1.1.2.cmml" xref="S3.2.p2.2.m2.2.2.1.1.2"><times id="S3.2.p2.2.m2.2.2.1.1.2.1.cmml" xref="S3.2.p2.2.m2.2.2.1.1.2.1"></times><ci id="S3.2.p2.2.m2.2.2.1.1.2.2.cmml" xref="S3.2.p2.2.m2.2.2.1.1.2.2">𝑉</ci><ci id="S3.2.p2.2.m2.1.1.cmml" xref="S3.2.p2.2.m2.1.1">𝐺</ci></apply><apply id="S3.2.p2.2.m2.2.2.1.1.3.cmml" xref="S3.2.p2.2.m2.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.2.p2.2.m2.2.2.1.1.3.1.cmml" xref="S3.2.p2.2.m2.2.2.1.1.3">subscript</csymbol><ci id="S3.2.p2.2.m2.2.2.1.1.3.2.cmml" xref="S3.2.p2.2.m2.2.2.1.1.3.2">𝑊</ci><ci id="S3.2.p2.2.m2.2.2.1.1.3.3.cmml" xref="S3.2.p2.2.m2.2.2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.2.m2.2c">(V(G)\setminus W_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.2.m2.2d">( italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>-<math alttext="W" class="ltx_Math" display="inline" id="S3.2.p2.3.m3.1"><semantics id="S3.2.p2.3.m3.1a"><mi id="S3.2.p2.3.m3.1.1" xref="S3.2.p2.3.m3.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.3.m3.1b"><ci id="S3.2.p2.3.m3.1.1.cmml" xref="S3.2.p2.3.m3.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.3.m3.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.3.m3.1d">italic_W</annotation></semantics></math> paths in <math alttext="G" class="ltx_Math" display="inline" id="S3.2.p2.4.m4.1"><semantics id="S3.2.p2.4.m4.1a"><mi id="S3.2.p2.4.m4.1.1" xref="S3.2.p2.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.4.m4.1b"><ci id="S3.2.p2.4.m4.1.1.cmml" xref="S3.2.p2.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.4.m4.1d">italic_G</annotation></semantics></math>, let <math alttext="r:=\min\{w,w^{\prime}\}" class="ltx_Math" display="inline" id="S3.2.p2.5.m5.3"><semantics id="S3.2.p2.5.m5.3a"><mrow id="S3.2.p2.5.m5.3.3" xref="S3.2.p2.5.m5.3.3.cmml"><mi id="S3.2.p2.5.m5.3.3.3" xref="S3.2.p2.5.m5.3.3.3.cmml">r</mi><mo id="S3.2.p2.5.m5.3.3.2" lspace="0.278em" rspace="0.278em" xref="S3.2.p2.5.m5.3.3.2.cmml">:=</mo><mrow id="S3.2.p2.5.m5.3.3.1.1" xref="S3.2.p2.5.m5.3.3.1.2.cmml"><mi id="S3.2.p2.5.m5.1.1" xref="S3.2.p2.5.m5.1.1.cmml">min</mi><mo id="S3.2.p2.5.m5.3.3.1.1a" xref="S3.2.p2.5.m5.3.3.1.2.cmml">⁡</mo><mrow id="S3.2.p2.5.m5.3.3.1.1.1" xref="S3.2.p2.5.m5.3.3.1.2.cmml"><mo id="S3.2.p2.5.m5.3.3.1.1.1.2" stretchy="false" xref="S3.2.p2.5.m5.3.3.1.2.cmml">{</mo><mi id="S3.2.p2.5.m5.2.2" xref="S3.2.p2.5.m5.2.2.cmml">w</mi><mo id="S3.2.p2.5.m5.3.3.1.1.1.3" xref="S3.2.p2.5.m5.3.3.1.2.cmml">,</mo><msup id="S3.2.p2.5.m5.3.3.1.1.1.1" xref="S3.2.p2.5.m5.3.3.1.1.1.1.cmml"><mi id="S3.2.p2.5.m5.3.3.1.1.1.1.2" xref="S3.2.p2.5.m5.3.3.1.1.1.1.2.cmml">w</mi><mo id="S3.2.p2.5.m5.3.3.1.1.1.1.3" xref="S3.2.p2.5.m5.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.2.p2.5.m5.3.3.1.1.1.4" stretchy="false" xref="S3.2.p2.5.m5.3.3.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.5.m5.3b"><apply id="S3.2.p2.5.m5.3.3.cmml" xref="S3.2.p2.5.m5.3.3"><csymbol cd="latexml" id="S3.2.p2.5.m5.3.3.2.cmml" xref="S3.2.p2.5.m5.3.3.2">assign</csymbol><ci id="S3.2.p2.5.m5.3.3.3.cmml" xref="S3.2.p2.5.m5.3.3.3">𝑟</ci><apply id="S3.2.p2.5.m5.3.3.1.2.cmml" xref="S3.2.p2.5.m5.3.3.1.1"><min id="S3.2.p2.5.m5.1.1.cmml" xref="S3.2.p2.5.m5.1.1"></min><ci id="S3.2.p2.5.m5.2.2.cmml" xref="S3.2.p2.5.m5.2.2">𝑤</ci><apply id="S3.2.p2.5.m5.3.3.1.1.1.1.cmml" xref="S3.2.p2.5.m5.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.2.p2.5.m5.3.3.1.1.1.1.1.cmml" xref="S3.2.p2.5.m5.3.3.1.1.1.1">superscript</csymbol><ci id="S3.2.p2.5.m5.3.3.1.1.1.1.2.cmml" xref="S3.2.p2.5.m5.3.3.1.1.1.1.2">𝑤</ci><ci id="S3.2.p2.5.m5.3.3.1.1.1.1.3.cmml" xref="S3.2.p2.5.m5.3.3.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.5.m5.3c">r:=\min\{w,w^{\prime}\}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.5.m5.3d">italic_r := roman_min { italic_w , italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }</annotation></semantics></math>, and let <math alttext="P_{1},\ldots,P_{r}" class="ltx_Math" display="inline" id="S3.2.p2.6.m6.3"><semantics id="S3.2.p2.6.m6.3a"><mrow id="S3.2.p2.6.m6.3.3.2" xref="S3.2.p2.6.m6.3.3.3.cmml"><msub id="S3.2.p2.6.m6.2.2.1.1" xref="S3.2.p2.6.m6.2.2.1.1.cmml"><mi id="S3.2.p2.6.m6.2.2.1.1.2" xref="S3.2.p2.6.m6.2.2.1.1.2.cmml">P</mi><mn id="S3.2.p2.6.m6.2.2.1.1.3" xref="S3.2.p2.6.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.2.p2.6.m6.3.3.2.3" xref="S3.2.p2.6.m6.3.3.3.cmml">,</mo><mi id="S3.2.p2.6.m6.1.1" mathvariant="normal" xref="S3.2.p2.6.m6.1.1.cmml">…</mi><mo id="S3.2.p2.6.m6.3.3.2.4" xref="S3.2.p2.6.m6.3.3.3.cmml">,</mo><msub id="S3.2.p2.6.m6.3.3.2.2" xref="S3.2.p2.6.m6.3.3.2.2.cmml"><mi id="S3.2.p2.6.m6.3.3.2.2.2" xref="S3.2.p2.6.m6.3.3.2.2.2.cmml">P</mi><mi id="S3.2.p2.6.m6.3.3.2.2.3" xref="S3.2.p2.6.m6.3.3.2.2.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.6.m6.3b"><list id="S3.2.p2.6.m6.3.3.3.cmml" xref="S3.2.p2.6.m6.3.3.2"><apply id="S3.2.p2.6.m6.2.2.1.1.cmml" xref="S3.2.p2.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S3.2.p2.6.m6.2.2.1.1.1.cmml" xref="S3.2.p2.6.m6.2.2.1.1">subscript</csymbol><ci id="S3.2.p2.6.m6.2.2.1.1.2.cmml" xref="S3.2.p2.6.m6.2.2.1.1.2">𝑃</ci><cn id="S3.2.p2.6.m6.2.2.1.1.3.cmml" type="integer" xref="S3.2.p2.6.m6.2.2.1.1.3">1</cn></apply><ci id="S3.2.p2.6.m6.1.1.cmml" xref="S3.2.p2.6.m6.1.1">…</ci><apply id="S3.2.p2.6.m6.3.3.2.2.cmml" xref="S3.2.p2.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S3.2.p2.6.m6.3.3.2.2.1.cmml" xref="S3.2.p2.6.m6.3.3.2.2">subscript</csymbol><ci id="S3.2.p2.6.m6.3.3.2.2.2.cmml" xref="S3.2.p2.6.m6.3.3.2.2.2">𝑃</ci><ci id="S3.2.p2.6.m6.3.3.2.2.3.cmml" xref="S3.2.p2.6.m6.3.3.2.2.3">𝑟</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.6.m6.3c">P_{1},\ldots,P_{r}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.6.m6.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> be pairwise vertex-disjoint <math alttext="(V(G)\setminus W_{i})" class="ltx_Math" display="inline" id="S3.2.p2.7.m7.2"><semantics id="S3.2.p2.7.m7.2a"><mrow id="S3.2.p2.7.m7.2.2.1" xref="S3.2.p2.7.m7.2.2.1.1.cmml"><mo id="S3.2.p2.7.m7.2.2.1.2" stretchy="false" xref="S3.2.p2.7.m7.2.2.1.1.cmml">(</mo><mrow id="S3.2.p2.7.m7.2.2.1.1" xref="S3.2.p2.7.m7.2.2.1.1.cmml"><mrow id="S3.2.p2.7.m7.2.2.1.1.2" xref="S3.2.p2.7.m7.2.2.1.1.2.cmml"><mi id="S3.2.p2.7.m7.2.2.1.1.2.2" xref="S3.2.p2.7.m7.2.2.1.1.2.2.cmml">V</mi><mo id="S3.2.p2.7.m7.2.2.1.1.2.1" xref="S3.2.p2.7.m7.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S3.2.p2.7.m7.2.2.1.1.2.3.2" xref="S3.2.p2.7.m7.2.2.1.1.2.cmml"><mo id="S3.2.p2.7.m7.2.2.1.1.2.3.2.1" stretchy="false" xref="S3.2.p2.7.m7.2.2.1.1.2.cmml">(</mo><mi id="S3.2.p2.7.m7.1.1" xref="S3.2.p2.7.m7.1.1.cmml">G</mi><mo id="S3.2.p2.7.m7.2.2.1.1.2.3.2.2" stretchy="false" xref="S3.2.p2.7.m7.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.2.p2.7.m7.2.2.1.1.1" xref="S3.2.p2.7.m7.2.2.1.1.1.cmml">∖</mo><msub id="S3.2.p2.7.m7.2.2.1.1.3" xref="S3.2.p2.7.m7.2.2.1.1.3.cmml"><mi id="S3.2.p2.7.m7.2.2.1.1.3.2" xref="S3.2.p2.7.m7.2.2.1.1.3.2.cmml">W</mi><mi id="S3.2.p2.7.m7.2.2.1.1.3.3" xref="S3.2.p2.7.m7.2.2.1.1.3.3.cmml">i</mi></msub></mrow><mo id="S3.2.p2.7.m7.2.2.1.3" stretchy="false" xref="S3.2.p2.7.m7.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.7.m7.2b"><apply id="S3.2.p2.7.m7.2.2.1.1.cmml" xref="S3.2.p2.7.m7.2.2.1"><setdiff id="S3.2.p2.7.m7.2.2.1.1.1.cmml" xref="S3.2.p2.7.m7.2.2.1.1.1"></setdiff><apply id="S3.2.p2.7.m7.2.2.1.1.2.cmml" xref="S3.2.p2.7.m7.2.2.1.1.2"><times id="S3.2.p2.7.m7.2.2.1.1.2.1.cmml" xref="S3.2.p2.7.m7.2.2.1.1.2.1"></times><ci id="S3.2.p2.7.m7.2.2.1.1.2.2.cmml" xref="S3.2.p2.7.m7.2.2.1.1.2.2">𝑉</ci><ci id="S3.2.p2.7.m7.1.1.cmml" xref="S3.2.p2.7.m7.1.1">𝐺</ci></apply><apply id="S3.2.p2.7.m7.2.2.1.1.3.cmml" xref="S3.2.p2.7.m7.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.2.p2.7.m7.2.2.1.1.3.1.cmml" xref="S3.2.p2.7.m7.2.2.1.1.3">subscript</csymbol><ci id="S3.2.p2.7.m7.2.2.1.1.3.2.cmml" xref="S3.2.p2.7.m7.2.2.1.1.3.2">𝑊</ci><ci id="S3.2.p2.7.m7.2.2.1.1.3.3.cmml" xref="S3.2.p2.7.m7.2.2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.7.m7.2c">(V(G)\setminus W_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.7.m7.2d">( italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>-<math alttext="W" class="ltx_Math" display="inline" id="S3.2.p2.8.m8.1"><semantics id="S3.2.p2.8.m8.1a"><mi id="S3.2.p2.8.m8.1.1" xref="S3.2.p2.8.m8.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.8.m8.1b"><ci id="S3.2.p2.8.m8.1.1.cmml" xref="S3.2.p2.8.m8.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.8.m8.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.8.m8.1d">italic_W</annotation></semantics></math> paths in <math alttext="G" class="ltx_Math" display="inline" id="S3.2.p2.9.m9.1"><semantics id="S3.2.p2.9.m9.1a"><mi id="S3.2.p2.9.m9.1.1" xref="S3.2.p2.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.9.m9.1b"><ci id="S3.2.p2.9.m9.1.1.cmml" xref="S3.2.p2.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.9.m9.1d">italic_G</annotation></semantics></math>. For each <math alttext="j\in\{1,\ldots,r\}" class="ltx_Math" display="inline" id="S3.2.p2.10.m10.3"><semantics id="S3.2.p2.10.m10.3a"><mrow id="S3.2.p2.10.m10.3.4" xref="S3.2.p2.10.m10.3.4.cmml"><mi id="S3.2.p2.10.m10.3.4.2" xref="S3.2.p2.10.m10.3.4.2.cmml">j</mi><mo id="S3.2.p2.10.m10.3.4.1" xref="S3.2.p2.10.m10.3.4.1.cmml">∈</mo><mrow id="S3.2.p2.10.m10.3.4.3.2" xref="S3.2.p2.10.m10.3.4.3.1.cmml"><mo id="S3.2.p2.10.m10.3.4.3.2.1" stretchy="false" xref="S3.2.p2.10.m10.3.4.3.1.cmml">{</mo><mn id="S3.2.p2.10.m10.1.1" xref="S3.2.p2.10.m10.1.1.cmml">1</mn><mo id="S3.2.p2.10.m10.3.4.3.2.2" xref="S3.2.p2.10.m10.3.4.3.1.cmml">,</mo><mi id="S3.2.p2.10.m10.2.2" mathvariant="normal" xref="S3.2.p2.10.m10.2.2.cmml">…</mi><mo id="S3.2.p2.10.m10.3.4.3.2.3" xref="S3.2.p2.10.m10.3.4.3.1.cmml">,</mo><mi id="S3.2.p2.10.m10.3.3" xref="S3.2.p2.10.m10.3.3.cmml">r</mi><mo id="S3.2.p2.10.m10.3.4.3.2.4" stretchy="false" xref="S3.2.p2.10.m10.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.10.m10.3b"><apply id="S3.2.p2.10.m10.3.4.cmml" xref="S3.2.p2.10.m10.3.4"><in id="S3.2.p2.10.m10.3.4.1.cmml" xref="S3.2.p2.10.m10.3.4.1"></in><ci id="S3.2.p2.10.m10.3.4.2.cmml" xref="S3.2.p2.10.m10.3.4.2">𝑗</ci><set id="S3.2.p2.10.m10.3.4.3.1.cmml" xref="S3.2.p2.10.m10.3.4.3.2"><cn id="S3.2.p2.10.m10.1.1.cmml" type="integer" xref="S3.2.p2.10.m10.1.1">1</cn><ci id="S3.2.p2.10.m10.2.2.cmml" xref="S3.2.p2.10.m10.2.2">…</ci><ci id="S3.2.p2.10.m10.3.3.cmml" xref="S3.2.p2.10.m10.3.3">𝑟</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.10.m10.3c">j\in\{1,\ldots,r\}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.10.m10.3d">italic_j ∈ { 1 , … , italic_r }</annotation></semantics></math> let <math alttext="v_{j}" class="ltx_Math" display="inline" id="S3.2.p2.11.m11.1"><semantics id="S3.2.p2.11.m11.1a"><msub id="S3.2.p2.11.m11.1.1" xref="S3.2.p2.11.m11.1.1.cmml"><mi id="S3.2.p2.11.m11.1.1.2" xref="S3.2.p2.11.m11.1.1.2.cmml">v</mi><mi id="S3.2.p2.11.m11.1.1.3" xref="S3.2.p2.11.m11.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.2.p2.11.m11.1b"><apply id="S3.2.p2.11.m11.1.1.cmml" xref="S3.2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S3.2.p2.11.m11.1.1.1.cmml" xref="S3.2.p2.11.m11.1.1">subscript</csymbol><ci id="S3.2.p2.11.m11.1.1.2.cmml" xref="S3.2.p2.11.m11.1.1.2">𝑣</ci><ci id="S3.2.p2.11.m11.1.1.3.cmml" xref="S3.2.p2.11.m11.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.11.m11.1c">v_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.11.m11.1d">italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> be the last vertex of <math alttext="P_{j}" class="ltx_Math" display="inline" id="S3.2.p2.12.m12.1"><semantics id="S3.2.p2.12.m12.1a"><msub id="S3.2.p2.12.m12.1.1" xref="S3.2.p2.12.m12.1.1.cmml"><mi id="S3.2.p2.12.m12.1.1.2" xref="S3.2.p2.12.m12.1.1.2.cmml">P</mi><mi id="S3.2.p2.12.m12.1.1.3" xref="S3.2.p2.12.m12.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.2.p2.12.m12.1b"><apply id="S3.2.p2.12.m12.1.1.cmml" xref="S3.2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S3.2.p2.12.m12.1.1.1.cmml" xref="S3.2.p2.12.m12.1.1">subscript</csymbol><ci id="S3.2.p2.12.m12.1.1.2.cmml" xref="S3.2.p2.12.m12.1.1.2">𝑃</ci><ci id="S3.2.p2.12.m12.1.1.3.cmml" xref="S3.2.p2.12.m12.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.12.m12.1c">P_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.12.m12.1d">italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> contained in <math alttext="V(G)\setminus W_{i}" class="ltx_Math" display="inline" id="S3.2.p2.13.m13.1"><semantics id="S3.2.p2.13.m13.1a"><mrow id="S3.2.p2.13.m13.1.2" xref="S3.2.p2.13.m13.1.2.cmml"><mrow id="S3.2.p2.13.m13.1.2.2" xref="S3.2.p2.13.m13.1.2.2.cmml"><mi id="S3.2.p2.13.m13.1.2.2.2" xref="S3.2.p2.13.m13.1.2.2.2.cmml">V</mi><mo id="S3.2.p2.13.m13.1.2.2.1" xref="S3.2.p2.13.m13.1.2.2.1.cmml">⁢</mo><mrow id="S3.2.p2.13.m13.1.2.2.3.2" xref="S3.2.p2.13.m13.1.2.2.cmml"><mo id="S3.2.p2.13.m13.1.2.2.3.2.1" stretchy="false" xref="S3.2.p2.13.m13.1.2.2.cmml">(</mo><mi id="S3.2.p2.13.m13.1.1" xref="S3.2.p2.13.m13.1.1.cmml">G</mi><mo id="S3.2.p2.13.m13.1.2.2.3.2.2" stretchy="false" xref="S3.2.p2.13.m13.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.2.p2.13.m13.1.2.1" xref="S3.2.p2.13.m13.1.2.1.cmml">∖</mo><msub id="S3.2.p2.13.m13.1.2.3" xref="S3.2.p2.13.m13.1.2.3.cmml"><mi id="S3.2.p2.13.m13.1.2.3.2" xref="S3.2.p2.13.m13.1.2.3.2.cmml">W</mi><mi id="S3.2.p2.13.m13.1.2.3.3" xref="S3.2.p2.13.m13.1.2.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.13.m13.1b"><apply id="S3.2.p2.13.m13.1.2.cmml" xref="S3.2.p2.13.m13.1.2"><setdiff id="S3.2.p2.13.m13.1.2.1.cmml" xref="S3.2.p2.13.m13.1.2.1"></setdiff><apply id="S3.2.p2.13.m13.1.2.2.cmml" xref="S3.2.p2.13.m13.1.2.2"><times id="S3.2.p2.13.m13.1.2.2.1.cmml" xref="S3.2.p2.13.m13.1.2.2.1"></times><ci id="S3.2.p2.13.m13.1.2.2.2.cmml" xref="S3.2.p2.13.m13.1.2.2.2">𝑉</ci><ci id="S3.2.p2.13.m13.1.1.cmml" xref="S3.2.p2.13.m13.1.1">𝐺</ci></apply><apply id="S3.2.p2.13.m13.1.2.3.cmml" xref="S3.2.p2.13.m13.1.2.3"><csymbol cd="ambiguous" id="S3.2.p2.13.m13.1.2.3.1.cmml" xref="S3.2.p2.13.m13.1.2.3">subscript</csymbol><ci id="S3.2.p2.13.m13.1.2.3.2.cmml" xref="S3.2.p2.13.m13.1.2.3.2">𝑊</ci><ci id="S3.2.p2.13.m13.1.2.3.3.cmml" xref="S3.2.p2.13.m13.1.2.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.13.m13.1c">V(G)\setminus W_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.13.m13.1d">italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and let <math alttext="P_{j}^{\prime}" class="ltx_Math" display="inline" id="S3.2.p2.14.m14.1"><semantics id="S3.2.p2.14.m14.1a"><msubsup id="S3.2.p2.14.m14.1.1" xref="S3.2.p2.14.m14.1.1.cmml"><mi id="S3.2.p2.14.m14.1.1.2.2" xref="S3.2.p2.14.m14.1.1.2.2.cmml">P</mi><mi id="S3.2.p2.14.m14.1.1.2.3" xref="S3.2.p2.14.m14.1.1.2.3.cmml">j</mi><mo id="S3.2.p2.14.m14.1.1.3" xref="S3.2.p2.14.m14.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.2.p2.14.m14.1b"><apply id="S3.2.p2.14.m14.1.1.cmml" xref="S3.2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S3.2.p2.14.m14.1.1.1.cmml" xref="S3.2.p2.14.m14.1.1">superscript</csymbol><apply id="S3.2.p2.14.m14.1.1.2.cmml" xref="S3.2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S3.2.p2.14.m14.1.1.2.1.cmml" xref="S3.2.p2.14.m14.1.1">subscript</csymbol><ci id="S3.2.p2.14.m14.1.1.2.2.cmml" xref="S3.2.p2.14.m14.1.1.2.2">𝑃</ci><ci id="S3.2.p2.14.m14.1.1.2.3.cmml" xref="S3.2.p2.14.m14.1.1.2.3">𝑗</ci></apply><ci id="S3.2.p2.14.m14.1.1.3.cmml" xref="S3.2.p2.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.14.m14.1c">P_{j}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.14.m14.1d">italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the subpath of <math alttext="P_{j}" class="ltx_Math" display="inline" id="S3.2.p2.15.m15.1"><semantics id="S3.2.p2.15.m15.1a"><msub id="S3.2.p2.15.m15.1.1" xref="S3.2.p2.15.m15.1.1.cmml"><mi id="S3.2.p2.15.m15.1.1.2" xref="S3.2.p2.15.m15.1.1.2.cmml">P</mi><mi id="S3.2.p2.15.m15.1.1.3" xref="S3.2.p2.15.m15.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.2.p2.15.m15.1b"><apply id="S3.2.p2.15.m15.1.1.cmml" xref="S3.2.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S3.2.p2.15.m15.1.1.1.cmml" xref="S3.2.p2.15.m15.1.1">subscript</csymbol><ci id="S3.2.p2.15.m15.1.1.2.cmml" xref="S3.2.p2.15.m15.1.1.2">𝑃</ci><ci id="S3.2.p2.15.m15.1.1.3.cmml" xref="S3.2.p2.15.m15.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.15.m15.1c">P_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.15.m15.1d">italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> that begins at <math alttext="v_{j}" class="ltx_Math" display="inline" id="S3.2.p2.16.m16.1"><semantics id="S3.2.p2.16.m16.1a"><msub id="S3.2.p2.16.m16.1.1" xref="S3.2.p2.16.m16.1.1.cmml"><mi id="S3.2.p2.16.m16.1.1.2" xref="S3.2.p2.16.m16.1.1.2.cmml">v</mi><mi id="S3.2.p2.16.m16.1.1.3" xref="S3.2.p2.16.m16.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.2.p2.16.m16.1b"><apply id="S3.2.p2.16.m16.1.1.cmml" xref="S3.2.p2.16.m16.1.1"><csymbol cd="ambiguous" id="S3.2.p2.16.m16.1.1.1.cmml" xref="S3.2.p2.16.m16.1.1">subscript</csymbol><ci id="S3.2.p2.16.m16.1.1.2.cmml" xref="S3.2.p2.16.m16.1.1.2">𝑣</ci><ci id="S3.2.p2.16.m16.1.1.3.cmml" xref="S3.2.p2.16.m16.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.16.m16.1c">v_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.16.m16.1d">italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and ends at the first vertex of <math alttext="P_{j}" class="ltx_Math" display="inline" id="S3.2.p2.17.m17.1"><semantics id="S3.2.p2.17.m17.1a"><msub id="S3.2.p2.17.m17.1.1" xref="S3.2.p2.17.m17.1.1.cmml"><mi id="S3.2.p2.17.m17.1.1.2" xref="S3.2.p2.17.m17.1.1.2.cmml">P</mi><mi id="S3.2.p2.17.m17.1.1.3" xref="S3.2.p2.17.m17.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.2.p2.17.m17.1b"><apply id="S3.2.p2.17.m17.1.1.cmml" xref="S3.2.p2.17.m17.1.1"><csymbol cd="ambiguous" id="S3.2.p2.17.m17.1.1.1.cmml" xref="S3.2.p2.17.m17.1.1">subscript</csymbol><ci id="S3.2.p2.17.m17.1.1.2.cmml" xref="S3.2.p2.17.m17.1.1.2">𝑃</ci><ci id="S3.2.p2.17.m17.1.1.3.cmml" xref="S3.2.p2.17.m17.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.17.m17.1c">P_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.17.m17.1d">italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> contained in <math alttext="W" class="ltx_Math" display="inline" id="S3.2.p2.18.m18.1"><semantics id="S3.2.p2.18.m18.1a"><mi id="S3.2.p2.18.m18.1.1" xref="S3.2.p2.18.m18.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.18.m18.1b"><ci id="S3.2.p2.18.m18.1.1.cmml" xref="S3.2.p2.18.m18.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.18.m18.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.18.m18.1d">italic_W</annotation></semantics></math>. Set <math alttext="W_{i+1}:=\{v_{1},\ldots,v_{r}\}\cup W_{i}" class="ltx_Math" display="inline" id="S3.2.p2.19.m19.3"><semantics id="S3.2.p2.19.m19.3a"><mrow id="S3.2.p2.19.m19.3.3" xref="S3.2.p2.19.m19.3.3.cmml"><msub id="S3.2.p2.19.m19.3.3.4" xref="S3.2.p2.19.m19.3.3.4.cmml"><mi id="S3.2.p2.19.m19.3.3.4.2" xref="S3.2.p2.19.m19.3.3.4.2.cmml">W</mi><mrow id="S3.2.p2.19.m19.3.3.4.3" xref="S3.2.p2.19.m19.3.3.4.3.cmml"><mi id="S3.2.p2.19.m19.3.3.4.3.2" xref="S3.2.p2.19.m19.3.3.4.3.2.cmml">i</mi><mo id="S3.2.p2.19.m19.3.3.4.3.1" xref="S3.2.p2.19.m19.3.3.4.3.1.cmml">+</mo><mn id="S3.2.p2.19.m19.3.3.4.3.3" xref="S3.2.p2.19.m19.3.3.4.3.3.cmml">1</mn></mrow></msub><mo id="S3.2.p2.19.m19.3.3.3" lspace="0.278em" rspace="0.278em" xref="S3.2.p2.19.m19.3.3.3.cmml">:=</mo><mrow id="S3.2.p2.19.m19.3.3.2" xref="S3.2.p2.19.m19.3.3.2.cmml"><mrow id="S3.2.p2.19.m19.3.3.2.2.2" xref="S3.2.p2.19.m19.3.3.2.2.3.cmml"><mo id="S3.2.p2.19.m19.3.3.2.2.2.3" stretchy="false" xref="S3.2.p2.19.m19.3.3.2.2.3.cmml">{</mo><msub id="S3.2.p2.19.m19.2.2.1.1.1.1" xref="S3.2.p2.19.m19.2.2.1.1.1.1.cmml"><mi id="S3.2.p2.19.m19.2.2.1.1.1.1.2" xref="S3.2.p2.19.m19.2.2.1.1.1.1.2.cmml">v</mi><mn id="S3.2.p2.19.m19.2.2.1.1.1.1.3" xref="S3.2.p2.19.m19.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.2.p2.19.m19.3.3.2.2.2.4" xref="S3.2.p2.19.m19.3.3.2.2.3.cmml">,</mo><mi id="S3.2.p2.19.m19.1.1" mathvariant="normal" xref="S3.2.p2.19.m19.1.1.cmml">…</mi><mo id="S3.2.p2.19.m19.3.3.2.2.2.5" xref="S3.2.p2.19.m19.3.3.2.2.3.cmml">,</mo><msub id="S3.2.p2.19.m19.3.3.2.2.2.2" xref="S3.2.p2.19.m19.3.3.2.2.2.2.cmml"><mi id="S3.2.p2.19.m19.3.3.2.2.2.2.2" xref="S3.2.p2.19.m19.3.3.2.2.2.2.2.cmml">v</mi><mi id="S3.2.p2.19.m19.3.3.2.2.2.2.3" xref="S3.2.p2.19.m19.3.3.2.2.2.2.3.cmml">r</mi></msub><mo id="S3.2.p2.19.m19.3.3.2.2.2.6" stretchy="false" xref="S3.2.p2.19.m19.3.3.2.2.3.cmml">}</mo></mrow><mo id="S3.2.p2.19.m19.3.3.2.3" xref="S3.2.p2.19.m19.3.3.2.3.cmml">∪</mo><msub id="S3.2.p2.19.m19.3.3.2.4" xref="S3.2.p2.19.m19.3.3.2.4.cmml"><mi id="S3.2.p2.19.m19.3.3.2.4.2" xref="S3.2.p2.19.m19.3.3.2.4.2.cmml">W</mi><mi id="S3.2.p2.19.m19.3.3.2.4.3" xref="S3.2.p2.19.m19.3.3.2.4.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.19.m19.3b"><apply id="S3.2.p2.19.m19.3.3.cmml" xref="S3.2.p2.19.m19.3.3"><csymbol cd="latexml" id="S3.2.p2.19.m19.3.3.3.cmml" xref="S3.2.p2.19.m19.3.3.3">assign</csymbol><apply id="S3.2.p2.19.m19.3.3.4.cmml" xref="S3.2.p2.19.m19.3.3.4"><csymbol cd="ambiguous" id="S3.2.p2.19.m19.3.3.4.1.cmml" xref="S3.2.p2.19.m19.3.3.4">subscript</csymbol><ci id="S3.2.p2.19.m19.3.3.4.2.cmml" xref="S3.2.p2.19.m19.3.3.4.2">𝑊</ci><apply id="S3.2.p2.19.m19.3.3.4.3.cmml" xref="S3.2.p2.19.m19.3.3.4.3"><plus id="S3.2.p2.19.m19.3.3.4.3.1.cmml" xref="S3.2.p2.19.m19.3.3.4.3.1"></plus><ci id="S3.2.p2.19.m19.3.3.4.3.2.cmml" xref="S3.2.p2.19.m19.3.3.4.3.2">𝑖</ci><cn id="S3.2.p2.19.m19.3.3.4.3.3.cmml" type="integer" xref="S3.2.p2.19.m19.3.3.4.3.3">1</cn></apply></apply><apply id="S3.2.p2.19.m19.3.3.2.cmml" xref="S3.2.p2.19.m19.3.3.2"><union id="S3.2.p2.19.m19.3.3.2.3.cmml" xref="S3.2.p2.19.m19.3.3.2.3"></union><set id="S3.2.p2.19.m19.3.3.2.2.3.cmml" xref="S3.2.p2.19.m19.3.3.2.2.2"><apply id="S3.2.p2.19.m19.2.2.1.1.1.1.cmml" xref="S3.2.p2.19.m19.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.2.p2.19.m19.2.2.1.1.1.1.1.cmml" xref="S3.2.p2.19.m19.2.2.1.1.1.1">subscript</csymbol><ci id="S3.2.p2.19.m19.2.2.1.1.1.1.2.cmml" xref="S3.2.p2.19.m19.2.2.1.1.1.1.2">𝑣</ci><cn id="S3.2.p2.19.m19.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.2.p2.19.m19.2.2.1.1.1.1.3">1</cn></apply><ci id="S3.2.p2.19.m19.1.1.cmml" xref="S3.2.p2.19.m19.1.1">…</ci><apply id="S3.2.p2.19.m19.3.3.2.2.2.2.cmml" xref="S3.2.p2.19.m19.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S3.2.p2.19.m19.3.3.2.2.2.2.1.cmml" xref="S3.2.p2.19.m19.3.3.2.2.2.2">subscript</csymbol><ci id="S3.2.p2.19.m19.3.3.2.2.2.2.2.cmml" xref="S3.2.p2.19.m19.3.3.2.2.2.2.2">𝑣</ci><ci id="S3.2.p2.19.m19.3.3.2.2.2.2.3.cmml" xref="S3.2.p2.19.m19.3.3.2.2.2.2.3">𝑟</ci></apply></set><apply id="S3.2.p2.19.m19.3.3.2.4.cmml" xref="S3.2.p2.19.m19.3.3.2.4"><csymbol cd="ambiguous" id="S3.2.p2.19.m19.3.3.2.4.1.cmml" xref="S3.2.p2.19.m19.3.3.2.4">subscript</csymbol><ci id="S3.2.p2.19.m19.3.3.2.4.2.cmml" xref="S3.2.p2.19.m19.3.3.2.4.2">𝑊</ci><ci id="S3.2.p2.19.m19.3.3.2.4.3.cmml" xref="S3.2.p2.19.m19.3.3.2.4.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.19.m19.3c">W_{i+1}:=\{v_{1},\ldots,v_{r}\}\cup W_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.19.m19.3d">italic_W start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT := { italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT } ∪ italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which implies that <math alttext="s_{i+1}=r" class="ltx_Math" display="inline" id="S3.2.p2.20.m20.1"><semantics id="S3.2.p2.20.m20.1a"><mrow id="S3.2.p2.20.m20.1.1" xref="S3.2.p2.20.m20.1.1.cmml"><msub id="S3.2.p2.20.m20.1.1.2" xref="S3.2.p2.20.m20.1.1.2.cmml"><mi id="S3.2.p2.20.m20.1.1.2.2" xref="S3.2.p2.20.m20.1.1.2.2.cmml">s</mi><mrow id="S3.2.p2.20.m20.1.1.2.3" xref="S3.2.p2.20.m20.1.1.2.3.cmml"><mi id="S3.2.p2.20.m20.1.1.2.3.2" xref="S3.2.p2.20.m20.1.1.2.3.2.cmml">i</mi><mo id="S3.2.p2.20.m20.1.1.2.3.1" xref="S3.2.p2.20.m20.1.1.2.3.1.cmml">+</mo><mn id="S3.2.p2.20.m20.1.1.2.3.3" xref="S3.2.p2.20.m20.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.2.p2.20.m20.1.1.1" xref="S3.2.p2.20.m20.1.1.1.cmml">=</mo><mi id="S3.2.p2.20.m20.1.1.3" xref="S3.2.p2.20.m20.1.1.3.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.20.m20.1b"><apply id="S3.2.p2.20.m20.1.1.cmml" xref="S3.2.p2.20.m20.1.1"><eq id="S3.2.p2.20.m20.1.1.1.cmml" xref="S3.2.p2.20.m20.1.1.1"></eq><apply id="S3.2.p2.20.m20.1.1.2.cmml" xref="S3.2.p2.20.m20.1.1.2"><csymbol cd="ambiguous" id="S3.2.p2.20.m20.1.1.2.1.cmml" xref="S3.2.p2.20.m20.1.1.2">subscript</csymbol><ci id="S3.2.p2.20.m20.1.1.2.2.cmml" xref="S3.2.p2.20.m20.1.1.2.2">𝑠</ci><apply id="S3.2.p2.20.m20.1.1.2.3.cmml" xref="S3.2.p2.20.m20.1.1.2.3"><plus id="S3.2.p2.20.m20.1.1.2.3.1.cmml" xref="S3.2.p2.20.m20.1.1.2.3.1"></plus><ci id="S3.2.p2.20.m20.1.1.2.3.2.cmml" xref="S3.2.p2.20.m20.1.1.2.3.2">𝑖</ci><cn id="S3.2.p2.20.m20.1.1.2.3.3.cmml" type="integer" xref="S3.2.p2.20.m20.1.1.2.3.3">1</cn></apply></apply><ci id="S3.2.p2.20.m20.1.1.3.cmml" xref="S3.2.p2.20.m20.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.20.m20.1c">s_{i+1}=r</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.20.m20.1d">italic_s start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT = italic_r</annotation></semantics></math>. The paths <math alttext="P_{1}^{\prime},\ldots,P_{s_{i+1}}^{\prime}" class="ltx_Math" display="inline" id="S3.2.p2.21.m21.3"><semantics id="S3.2.p2.21.m21.3a"><mrow id="S3.2.p2.21.m21.3.3.2" xref="S3.2.p2.21.m21.3.3.3.cmml"><msubsup id="S3.2.p2.21.m21.2.2.1.1" xref="S3.2.p2.21.m21.2.2.1.1.cmml"><mi id="S3.2.p2.21.m21.2.2.1.1.2.2" xref="S3.2.p2.21.m21.2.2.1.1.2.2.cmml">P</mi><mn id="S3.2.p2.21.m21.2.2.1.1.2.3" xref="S3.2.p2.21.m21.2.2.1.1.2.3.cmml">1</mn><mo id="S3.2.p2.21.m21.2.2.1.1.3" xref="S3.2.p2.21.m21.2.2.1.1.3.cmml">′</mo></msubsup><mo id="S3.2.p2.21.m21.3.3.2.3" xref="S3.2.p2.21.m21.3.3.3.cmml">,</mo><mi id="S3.2.p2.21.m21.1.1" mathvariant="normal" xref="S3.2.p2.21.m21.1.1.cmml">…</mi><mo id="S3.2.p2.21.m21.3.3.2.4" xref="S3.2.p2.21.m21.3.3.3.cmml">,</mo><msubsup id="S3.2.p2.21.m21.3.3.2.2" xref="S3.2.p2.21.m21.3.3.2.2.cmml"><mi id="S3.2.p2.21.m21.3.3.2.2.2.2" xref="S3.2.p2.21.m21.3.3.2.2.2.2.cmml">P</mi><msub id="S3.2.p2.21.m21.3.3.2.2.2.3" xref="S3.2.p2.21.m21.3.3.2.2.2.3.cmml"><mi id="S3.2.p2.21.m21.3.3.2.2.2.3.2" xref="S3.2.p2.21.m21.3.3.2.2.2.3.2.cmml">s</mi><mrow id="S3.2.p2.21.m21.3.3.2.2.2.3.3" xref="S3.2.p2.21.m21.3.3.2.2.2.3.3.cmml"><mi id="S3.2.p2.21.m21.3.3.2.2.2.3.3.2" xref="S3.2.p2.21.m21.3.3.2.2.2.3.3.2.cmml">i</mi><mo id="S3.2.p2.21.m21.3.3.2.2.2.3.3.1" xref="S3.2.p2.21.m21.3.3.2.2.2.3.3.1.cmml">+</mo><mn id="S3.2.p2.21.m21.3.3.2.2.2.3.3.3" xref="S3.2.p2.21.m21.3.3.2.2.2.3.3.3.cmml">1</mn></mrow></msub><mo id="S3.2.p2.21.m21.3.3.2.2.3" xref="S3.2.p2.21.m21.3.3.2.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.21.m21.3b"><list id="S3.2.p2.21.m21.3.3.3.cmml" xref="S3.2.p2.21.m21.3.3.2"><apply id="S3.2.p2.21.m21.2.2.1.1.cmml" xref="S3.2.p2.21.m21.2.2.1.1"><csymbol cd="ambiguous" id="S3.2.p2.21.m21.2.2.1.1.1.cmml" xref="S3.2.p2.21.m21.2.2.1.1">superscript</csymbol><apply id="S3.2.p2.21.m21.2.2.1.1.2.cmml" xref="S3.2.p2.21.m21.2.2.1.1"><csymbol cd="ambiguous" id="S3.2.p2.21.m21.2.2.1.1.2.1.cmml" xref="S3.2.p2.21.m21.2.2.1.1">subscript</csymbol><ci id="S3.2.p2.21.m21.2.2.1.1.2.2.cmml" xref="S3.2.p2.21.m21.2.2.1.1.2.2">𝑃</ci><cn id="S3.2.p2.21.m21.2.2.1.1.2.3.cmml" type="integer" xref="S3.2.p2.21.m21.2.2.1.1.2.3">1</cn></apply><ci id="S3.2.p2.21.m21.2.2.1.1.3.cmml" xref="S3.2.p2.21.m21.2.2.1.1.3">′</ci></apply><ci id="S3.2.p2.21.m21.1.1.cmml" xref="S3.2.p2.21.m21.1.1">…</ci><apply id="S3.2.p2.21.m21.3.3.2.2.cmml" xref="S3.2.p2.21.m21.3.3.2.2"><csymbol cd="ambiguous" id="S3.2.p2.21.m21.3.3.2.2.1.cmml" xref="S3.2.p2.21.m21.3.3.2.2">superscript</csymbol><apply id="S3.2.p2.21.m21.3.3.2.2.2.cmml" xref="S3.2.p2.21.m21.3.3.2.2"><csymbol cd="ambiguous" id="S3.2.p2.21.m21.3.3.2.2.2.1.cmml" xref="S3.2.p2.21.m21.3.3.2.2">subscript</csymbol><ci id="S3.2.p2.21.m21.3.3.2.2.2.2.cmml" xref="S3.2.p2.21.m21.3.3.2.2.2.2">𝑃</ci><apply id="S3.2.p2.21.m21.3.3.2.2.2.3.cmml" xref="S3.2.p2.21.m21.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S3.2.p2.21.m21.3.3.2.2.2.3.1.cmml" xref="S3.2.p2.21.m21.3.3.2.2.2.3">subscript</csymbol><ci id="S3.2.p2.21.m21.3.3.2.2.2.3.2.cmml" xref="S3.2.p2.21.m21.3.3.2.2.2.3.2">𝑠</ci><apply id="S3.2.p2.21.m21.3.3.2.2.2.3.3.cmml" xref="S3.2.p2.21.m21.3.3.2.2.2.3.3"><plus id="S3.2.p2.21.m21.3.3.2.2.2.3.3.1.cmml" xref="S3.2.p2.21.m21.3.3.2.2.2.3.3.1"></plus><ci id="S3.2.p2.21.m21.3.3.2.2.2.3.3.2.cmml" xref="S3.2.p2.21.m21.3.3.2.2.2.3.3.2">𝑖</ci><cn id="S3.2.p2.21.m21.3.3.2.2.2.3.3.3.cmml" type="integer" xref="S3.2.p2.21.m21.3.3.2.2.2.3.3.3">1</cn></apply></apply></apply><ci id="S3.2.p2.21.m21.3.3.2.2.3.cmml" xref="S3.2.p2.21.m21.3.3.2.2.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.21.m21.3c">P_{1}^{\prime},\ldots,P_{s_{i+1}}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.21.m21.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> certify that <math alttext="W_{0},\ldots,W_{i+1}" class="ltx_Math" display="inline" id="S3.2.p2.22.m22.3"><semantics id="S3.2.p2.22.m22.3a"><mrow id="S3.2.p2.22.m22.3.3.2" xref="S3.2.p2.22.m22.3.3.3.cmml"><msub id="S3.2.p2.22.m22.2.2.1.1" xref="S3.2.p2.22.m22.2.2.1.1.cmml"><mi id="S3.2.p2.22.m22.2.2.1.1.2" xref="S3.2.p2.22.m22.2.2.1.1.2.cmml">W</mi><mn id="S3.2.p2.22.m22.2.2.1.1.3" xref="S3.2.p2.22.m22.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.2.p2.22.m22.3.3.2.3" xref="S3.2.p2.22.m22.3.3.3.cmml">,</mo><mi id="S3.2.p2.22.m22.1.1" mathvariant="normal" xref="S3.2.p2.22.m22.1.1.cmml">…</mi><mo id="S3.2.p2.22.m22.3.3.2.4" xref="S3.2.p2.22.m22.3.3.3.cmml">,</mo><msub id="S3.2.p2.22.m22.3.3.2.2" xref="S3.2.p2.22.m22.3.3.2.2.cmml"><mi id="S3.2.p2.22.m22.3.3.2.2.2" xref="S3.2.p2.22.m22.3.3.2.2.2.cmml">W</mi><mrow id="S3.2.p2.22.m22.3.3.2.2.3" xref="S3.2.p2.22.m22.3.3.2.2.3.cmml"><mi id="S3.2.p2.22.m22.3.3.2.2.3.2" xref="S3.2.p2.22.m22.3.3.2.2.3.2.cmml">i</mi><mo id="S3.2.p2.22.m22.3.3.2.2.3.1" xref="S3.2.p2.22.m22.3.3.2.2.3.1.cmml">+</mo><mn id="S3.2.p2.22.m22.3.3.2.2.3.3" xref="S3.2.p2.22.m22.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.22.m22.3b"><list id="S3.2.p2.22.m22.3.3.3.cmml" xref="S3.2.p2.22.m22.3.3.2"><apply id="S3.2.p2.22.m22.2.2.1.1.cmml" xref="S3.2.p2.22.m22.2.2.1.1"><csymbol cd="ambiguous" id="S3.2.p2.22.m22.2.2.1.1.1.cmml" xref="S3.2.p2.22.m22.2.2.1.1">subscript</csymbol><ci id="S3.2.p2.22.m22.2.2.1.1.2.cmml" xref="S3.2.p2.22.m22.2.2.1.1.2">𝑊</ci><cn id="S3.2.p2.22.m22.2.2.1.1.3.cmml" type="integer" xref="S3.2.p2.22.m22.2.2.1.1.3">0</cn></apply><ci id="S3.2.p2.22.m22.1.1.cmml" xref="S3.2.p2.22.m22.1.1">…</ci><apply id="S3.2.p2.22.m22.3.3.2.2.cmml" xref="S3.2.p2.22.m22.3.3.2.2"><csymbol cd="ambiguous" id="S3.2.p2.22.m22.3.3.2.2.1.cmml" xref="S3.2.p2.22.m22.3.3.2.2">subscript</csymbol><ci id="S3.2.p2.22.m22.3.3.2.2.2.cmml" xref="S3.2.p2.22.m22.3.3.2.2.2">𝑊</ci><apply id="S3.2.p2.22.m22.3.3.2.2.3.cmml" xref="S3.2.p2.22.m22.3.3.2.2.3"><plus id="S3.2.p2.22.m22.3.3.2.2.3.1.cmml" xref="S3.2.p2.22.m22.3.3.2.2.3.1"></plus><ci id="S3.2.p2.22.m22.3.3.2.2.3.2.cmml" xref="S3.2.p2.22.m22.3.3.2.2.3.2">𝑖</ci><cn id="S3.2.p2.22.m22.3.3.2.2.3.3.cmml" type="integer" xref="S3.2.p2.22.m22.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.22.m22.3c">W_{0},\ldots,W_{i+1}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.22.m22.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math> satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i4" title="Item (d) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(d)</span></a>.</p> </div> <div class="ltx_para" id="S3.3.p3"> <p class="ltx_p" id="S3.3.p3.4">If <math alttext="r=w" class="ltx_Math" display="inline" id="S3.3.p3.1.m1.1"><semantics id="S3.3.p3.1.m1.1a"><mrow id="S3.3.p3.1.m1.1.1" xref="S3.3.p3.1.m1.1.1.cmml"><mi id="S3.3.p3.1.m1.1.1.2" xref="S3.3.p3.1.m1.1.1.2.cmml">r</mi><mo id="S3.3.p3.1.m1.1.1.1" xref="S3.3.p3.1.m1.1.1.1.cmml">=</mo><mi id="S3.3.p3.1.m1.1.1.3" xref="S3.3.p3.1.m1.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.3.p3.1.m1.1b"><apply id="S3.3.p3.1.m1.1.1.cmml" xref="S3.3.p3.1.m1.1.1"><eq id="S3.3.p3.1.m1.1.1.1.cmml" xref="S3.3.p3.1.m1.1.1.1"></eq><ci id="S3.3.p3.1.m1.1.1.2.cmml" xref="S3.3.p3.1.m1.1.1.2">𝑟</ci><ci id="S3.3.p3.1.m1.1.1.3.cmml" xref="S3.3.p3.1.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p3.1.m1.1c">r=w</annotation><annotation encoding="application/x-llamapun" id="S3.3.p3.1.m1.1d">italic_r = italic_w</annotation></semantics></math> then <math alttext="W_{i+1}" class="ltx_Math" display="inline" id="S3.3.p3.2.m2.1"><semantics id="S3.3.p3.2.m2.1a"><msub id="S3.3.p3.2.m2.1.1" xref="S3.3.p3.2.m2.1.1.cmml"><mi id="S3.3.p3.2.m2.1.1.2" xref="S3.3.p3.2.m2.1.1.2.cmml">W</mi><mrow id="S3.3.p3.2.m2.1.1.3" xref="S3.3.p3.2.m2.1.1.3.cmml"><mi id="S3.3.p3.2.m2.1.1.3.2" xref="S3.3.p3.2.m2.1.1.3.2.cmml">i</mi><mo id="S3.3.p3.2.m2.1.1.3.1" xref="S3.3.p3.2.m2.1.1.3.1.cmml">+</mo><mn id="S3.3.p3.2.m2.1.1.3.3" xref="S3.3.p3.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.3.p3.2.m2.1b"><apply id="S3.3.p3.2.m2.1.1.cmml" xref="S3.3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.3.p3.2.m2.1.1.1.cmml" xref="S3.3.p3.2.m2.1.1">subscript</csymbol><ci id="S3.3.p3.2.m2.1.1.2.cmml" xref="S3.3.p3.2.m2.1.1.2">𝑊</ci><apply id="S3.3.p3.2.m2.1.1.3.cmml" xref="S3.3.p3.2.m2.1.1.3"><plus id="S3.3.p3.2.m2.1.1.3.1.cmml" xref="S3.3.p3.2.m2.1.1.3.1"></plus><ci id="S3.3.p3.2.m2.1.1.3.2.cmml" xref="S3.3.p3.2.m2.1.1.3.2">𝑖</ci><cn id="S3.3.p3.2.m2.1.1.3.3.cmml" type="integer" xref="S3.3.p3.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p3.2.m2.1c">W_{i+1}</annotation><annotation encoding="application/x-llamapun" id="S3.3.p3.2.m2.1d">italic_W start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math> also satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i2" title="Item (b) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(b)</span></a> so that we now a sequence <math alttext="W_{0},\ldots,W_{i+1}" class="ltx_Math" display="inline" id="S3.3.p3.3.m3.3"><semantics id="S3.3.p3.3.m3.3a"><mrow id="S3.3.p3.3.m3.3.3.2" xref="S3.3.p3.3.m3.3.3.3.cmml"><msub id="S3.3.p3.3.m3.2.2.1.1" xref="S3.3.p3.3.m3.2.2.1.1.cmml"><mi id="S3.3.p3.3.m3.2.2.1.1.2" xref="S3.3.p3.3.m3.2.2.1.1.2.cmml">W</mi><mn id="S3.3.p3.3.m3.2.2.1.1.3" xref="S3.3.p3.3.m3.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.3.p3.3.m3.3.3.2.3" xref="S3.3.p3.3.m3.3.3.3.cmml">,</mo><mi id="S3.3.p3.3.m3.1.1" mathvariant="normal" xref="S3.3.p3.3.m3.1.1.cmml">…</mi><mo id="S3.3.p3.3.m3.3.3.2.4" xref="S3.3.p3.3.m3.3.3.3.cmml">,</mo><msub id="S3.3.p3.3.m3.3.3.2.2" xref="S3.3.p3.3.m3.3.3.2.2.cmml"><mi id="S3.3.p3.3.m3.3.3.2.2.2" xref="S3.3.p3.3.m3.3.3.2.2.2.cmml">W</mi><mrow id="S3.3.p3.3.m3.3.3.2.2.3" xref="S3.3.p3.3.m3.3.3.2.2.3.cmml"><mi id="S3.3.p3.3.m3.3.3.2.2.3.2" xref="S3.3.p3.3.m3.3.3.2.2.3.2.cmml">i</mi><mo id="S3.3.p3.3.m3.3.3.2.2.3.1" xref="S3.3.p3.3.m3.3.3.2.2.3.1.cmml">+</mo><mn id="S3.3.p3.3.m3.3.3.2.2.3.3" xref="S3.3.p3.3.m3.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.3.p3.3.m3.3b"><list id="S3.3.p3.3.m3.3.3.3.cmml" xref="S3.3.p3.3.m3.3.3.2"><apply id="S3.3.p3.3.m3.2.2.1.1.cmml" xref="S3.3.p3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.3.p3.3.m3.2.2.1.1.1.cmml" xref="S3.3.p3.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.3.p3.3.m3.2.2.1.1.2.cmml" xref="S3.3.p3.3.m3.2.2.1.1.2">𝑊</ci><cn id="S3.3.p3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.3.p3.3.m3.2.2.1.1.3">0</cn></apply><ci id="S3.3.p3.3.m3.1.1.cmml" xref="S3.3.p3.3.m3.1.1">…</ci><apply id="S3.3.p3.3.m3.3.3.2.2.cmml" xref="S3.3.p3.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S3.3.p3.3.m3.3.3.2.2.1.cmml" xref="S3.3.p3.3.m3.3.3.2.2">subscript</csymbol><ci id="S3.3.p3.3.m3.3.3.2.2.2.cmml" xref="S3.3.p3.3.m3.3.3.2.2.2">𝑊</ci><apply id="S3.3.p3.3.m3.3.3.2.2.3.cmml" xref="S3.3.p3.3.m3.3.3.2.2.3"><plus id="S3.3.p3.3.m3.3.3.2.2.3.1.cmml" xref="S3.3.p3.3.m3.3.3.2.2.3.1"></plus><ci id="S3.3.p3.3.m3.3.3.2.2.3.2.cmml" xref="S3.3.p3.3.m3.3.3.2.2.3.2">𝑖</ci><cn id="S3.3.p3.3.m3.3.3.2.2.3.3.cmml" type="integer" xref="S3.3.p3.3.m3.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p3.3.m3.3c">W_{0},\ldots,W_{i+1}</annotation><annotation encoding="application/x-llamapun" id="S3.3.p3.3.m3.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math> that satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i1" title="Item (a) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(a)</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i2" title="Item (b) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(b)</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i4" title="Item (d) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(d)</span></a>. In this case we can continue to define <math alttext="W_{i+2}" class="ltx_Math" display="inline" id="S3.3.p3.4.m4.1"><semantics id="S3.3.p3.4.m4.1a"><msub id="S3.3.p3.4.m4.1.1" xref="S3.3.p3.4.m4.1.1.cmml"><mi id="S3.3.p3.4.m4.1.1.2" xref="S3.3.p3.4.m4.1.1.2.cmml">W</mi><mrow id="S3.3.p3.4.m4.1.1.3" xref="S3.3.p3.4.m4.1.1.3.cmml"><mi id="S3.3.p3.4.m4.1.1.3.2" xref="S3.3.p3.4.m4.1.1.3.2.cmml">i</mi><mo id="S3.3.p3.4.m4.1.1.3.1" xref="S3.3.p3.4.m4.1.1.3.1.cmml">+</mo><mn id="S3.3.p3.4.m4.1.1.3.3" xref="S3.3.p3.4.m4.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.3.p3.4.m4.1b"><apply id="S3.3.p3.4.m4.1.1.cmml" xref="S3.3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S3.3.p3.4.m4.1.1.1.cmml" xref="S3.3.p3.4.m4.1.1">subscript</csymbol><ci id="S3.3.p3.4.m4.1.1.2.cmml" xref="S3.3.p3.4.m4.1.1.2">𝑊</ci><apply id="S3.3.p3.4.m4.1.1.3.cmml" xref="S3.3.p3.4.m4.1.1.3"><plus id="S3.3.p3.4.m4.1.1.3.1.cmml" xref="S3.3.p3.4.m4.1.1.3.1"></plus><ci id="S3.3.p3.4.m4.1.1.3.2.cmml" xref="S3.3.p3.4.m4.1.1.3.2">𝑖</ci><cn id="S3.3.p3.4.m4.1.1.3.3.cmml" type="integer" xref="S3.3.p3.4.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p3.4.m4.1c">W_{i+2}</annotation><annotation encoding="application/x-llamapun" id="S3.3.p3.4.m4.1d">italic_W start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT</annotation></semantics></math> as above.</p> </div> <div class="ltx_para" id="S3.4.p4"> <p class="ltx_p" id="S3.4.p4.35">If <math alttext="r<w" class="ltx_Math" display="inline" id="S3.4.p4.1.m1.1"><semantics id="S3.4.p4.1.m1.1a"><mrow id="S3.4.p4.1.m1.1.1" xref="S3.4.p4.1.m1.1.1.cmml"><mi id="S3.4.p4.1.m1.1.1.2" xref="S3.4.p4.1.m1.1.1.2.cmml">r</mi><mo id="S3.4.p4.1.m1.1.1.1" xref="S3.4.p4.1.m1.1.1.1.cmml"><</mo><mi id="S3.4.p4.1.m1.1.1.3" xref="S3.4.p4.1.m1.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.1.m1.1b"><apply id="S3.4.p4.1.m1.1.1.cmml" xref="S3.4.p4.1.m1.1.1"><lt id="S3.4.p4.1.m1.1.1.1.cmml" xref="S3.4.p4.1.m1.1.1.1"></lt><ci id="S3.4.p4.1.m1.1.1.2.cmml" xref="S3.4.p4.1.m1.1.1.2">𝑟</ci><ci id="S3.4.p4.1.m1.1.1.3.cmml" xref="S3.4.p4.1.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.1.m1.1c">r<w</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.1.m1.1d">italic_r < italic_w</annotation></semantics></math> then we set <math alttext="\ell:=i" class="ltx_Math" display="inline" id="S3.4.p4.2.m2.1"><semantics id="S3.4.p4.2.m2.1a"><mrow id="S3.4.p4.2.m2.1.1" xref="S3.4.p4.2.m2.1.1.cmml"><mi id="S3.4.p4.2.m2.1.1.2" mathvariant="normal" xref="S3.4.p4.2.m2.1.1.2.cmml">ℓ</mi><mo id="S3.4.p4.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.4.p4.2.m2.1.1.1.cmml">:=</mo><mi id="S3.4.p4.2.m2.1.1.3" xref="S3.4.p4.2.m2.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.2.m2.1b"><apply id="S3.4.p4.2.m2.1.1.cmml" xref="S3.4.p4.2.m2.1.1"><csymbol cd="latexml" id="S3.4.p4.2.m2.1.1.1.cmml" xref="S3.4.p4.2.m2.1.1.1">assign</csymbol><ci id="S3.4.p4.2.m2.1.1.2.cmml" xref="S3.4.p4.2.m2.1.1.2">ℓ</ci><ci id="S3.4.p4.2.m2.1.1.3.cmml" xref="S3.4.p4.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.2.m2.1c">\ell:=i</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.2.m2.1d">roman_ℓ := italic_i</annotation></semantics></math>. Since <math alttext="r<w" class="ltx_Math" display="inline" id="S3.4.p4.3.m3.1"><semantics id="S3.4.p4.3.m3.1a"><mrow id="S3.4.p4.3.m3.1.1" xref="S3.4.p4.3.m3.1.1.cmml"><mi id="S3.4.p4.3.m3.1.1.2" xref="S3.4.p4.3.m3.1.1.2.cmml">r</mi><mo id="S3.4.p4.3.m3.1.1.1" xref="S3.4.p4.3.m3.1.1.1.cmml"><</mo><mi id="S3.4.p4.3.m3.1.1.3" xref="S3.4.p4.3.m3.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.3.m3.1b"><apply id="S3.4.p4.3.m3.1.1.cmml" xref="S3.4.p4.3.m3.1.1"><lt id="S3.4.p4.3.m3.1.1.1.cmml" xref="S3.4.p4.3.m3.1.1.1"></lt><ci id="S3.4.p4.3.m3.1.1.2.cmml" xref="S3.4.p4.3.m3.1.1.2">𝑟</ci><ci id="S3.4.p4.3.m3.1.1.3.cmml" xref="S3.4.p4.3.m3.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.3.m3.1c">r<w</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.3.m3.1d">italic_r < italic_w</annotation></semantics></math>, <math alttext="s_{\ell+1}=r<w" class="ltx_Math" display="inline" id="S3.4.p4.4.m4.1"><semantics id="S3.4.p4.4.m4.1a"><mrow id="S3.4.p4.4.m4.1.1" xref="S3.4.p4.4.m4.1.1.cmml"><msub id="S3.4.p4.4.m4.1.1.2" xref="S3.4.p4.4.m4.1.1.2.cmml"><mi id="S3.4.p4.4.m4.1.1.2.2" xref="S3.4.p4.4.m4.1.1.2.2.cmml">s</mi><mrow id="S3.4.p4.4.m4.1.1.2.3" xref="S3.4.p4.4.m4.1.1.2.3.cmml"><mi id="S3.4.p4.4.m4.1.1.2.3.2" mathvariant="normal" xref="S3.4.p4.4.m4.1.1.2.3.2.cmml">ℓ</mi><mo id="S3.4.p4.4.m4.1.1.2.3.1" xref="S3.4.p4.4.m4.1.1.2.3.1.cmml">+</mo><mn id="S3.4.p4.4.m4.1.1.2.3.3" xref="S3.4.p4.4.m4.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.4.p4.4.m4.1.1.3" xref="S3.4.p4.4.m4.1.1.3.cmml">=</mo><mi id="S3.4.p4.4.m4.1.1.4" xref="S3.4.p4.4.m4.1.1.4.cmml">r</mi><mo id="S3.4.p4.4.m4.1.1.5" xref="S3.4.p4.4.m4.1.1.5.cmml"><</mo><mi id="S3.4.p4.4.m4.1.1.6" xref="S3.4.p4.4.m4.1.1.6.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.4.m4.1b"><apply id="S3.4.p4.4.m4.1.1.cmml" xref="S3.4.p4.4.m4.1.1"><and id="S3.4.p4.4.m4.1.1a.cmml" xref="S3.4.p4.4.m4.1.1"></and><apply id="S3.4.p4.4.m4.1.1b.cmml" xref="S3.4.p4.4.m4.1.1"><eq id="S3.4.p4.4.m4.1.1.3.cmml" xref="S3.4.p4.4.m4.1.1.3"></eq><apply id="S3.4.p4.4.m4.1.1.2.cmml" xref="S3.4.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.4.p4.4.m4.1.1.2.1.cmml" xref="S3.4.p4.4.m4.1.1.2">subscript</csymbol><ci id="S3.4.p4.4.m4.1.1.2.2.cmml" xref="S3.4.p4.4.m4.1.1.2.2">𝑠</ci><apply id="S3.4.p4.4.m4.1.1.2.3.cmml" xref="S3.4.p4.4.m4.1.1.2.3"><plus id="S3.4.p4.4.m4.1.1.2.3.1.cmml" xref="S3.4.p4.4.m4.1.1.2.3.1"></plus><ci id="S3.4.p4.4.m4.1.1.2.3.2.cmml" xref="S3.4.p4.4.m4.1.1.2.3.2">ℓ</ci><cn id="S3.4.p4.4.m4.1.1.2.3.3.cmml" type="integer" xref="S3.4.p4.4.m4.1.1.2.3.3">1</cn></apply></apply><ci id="S3.4.p4.4.m4.1.1.4.cmml" xref="S3.4.p4.4.m4.1.1.4">𝑟</ci></apply><apply id="S3.4.p4.4.m4.1.1c.cmml" xref="S3.4.p4.4.m4.1.1"><lt id="S3.4.p4.4.m4.1.1.5.cmml" xref="S3.4.p4.4.m4.1.1.5"></lt><share href="https://arxiv.org/html/2503.17112v1#S3.4.p4.4.m4.1.1.4.cmml" id="S3.4.p4.4.m4.1.1d.cmml" xref="S3.4.p4.4.m4.1.1"></share><ci id="S3.4.p4.4.m4.1.1.6.cmml" xref="S3.4.p4.4.m4.1.1.6">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.4.m4.1c">s_{\ell+1}=r<w</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.4.m4.1d">italic_s start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT = italic_r < italic_w</annotation></semantics></math>, so this choice of <math alttext="\ell" class="ltx_Math" display="inline" id="S3.4.p4.5.m5.1"><semantics id="S3.4.p4.5.m5.1a"><mi id="S3.4.p4.5.m5.1.1" mathvariant="normal" xref="S3.4.p4.5.m5.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.5.m5.1b"><ci id="S3.4.p4.5.m5.1.1.cmml" xref="S3.4.p4.5.m5.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.5.m5.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.5.m5.1d">roman_ℓ</annotation></semantics></math> satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i3" title="Item (c) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(c)</span></a>. Since <math alttext="G" class="ltx_Math" display="inline" id="S3.4.p4.6.m6.1"><semantics id="S3.4.p4.6.m6.1a"><mi id="S3.4.p4.6.m6.1.1" xref="S3.4.p4.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.6.m6.1b"><ci id="S3.4.p4.6.m6.1.1.cmml" xref="S3.4.p4.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.6.m6.1d">italic_G</annotation></semantics></math> does not contain <math alttext="k:=r+1" class="ltx_Math" display="inline" id="S3.4.p4.7.m7.1"><semantics id="S3.4.p4.7.m7.1a"><mrow id="S3.4.p4.7.m7.1.1" xref="S3.4.p4.7.m7.1.1.cmml"><mi id="S3.4.p4.7.m7.1.1.2" xref="S3.4.p4.7.m7.1.1.2.cmml">k</mi><mo id="S3.4.p4.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.4.p4.7.m7.1.1.1.cmml">:=</mo><mrow id="S3.4.p4.7.m7.1.1.3" xref="S3.4.p4.7.m7.1.1.3.cmml"><mi id="S3.4.p4.7.m7.1.1.3.2" xref="S3.4.p4.7.m7.1.1.3.2.cmml">r</mi><mo id="S3.4.p4.7.m7.1.1.3.1" xref="S3.4.p4.7.m7.1.1.3.1.cmml">+</mo><mn id="S3.4.p4.7.m7.1.1.3.3" xref="S3.4.p4.7.m7.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.7.m7.1b"><apply id="S3.4.p4.7.m7.1.1.cmml" xref="S3.4.p4.7.m7.1.1"><csymbol cd="latexml" id="S3.4.p4.7.m7.1.1.1.cmml" xref="S3.4.p4.7.m7.1.1.1">assign</csymbol><ci id="S3.4.p4.7.m7.1.1.2.cmml" xref="S3.4.p4.7.m7.1.1.2">𝑘</ci><apply id="S3.4.p4.7.m7.1.1.3.cmml" xref="S3.4.p4.7.m7.1.1.3"><plus id="S3.4.p4.7.m7.1.1.3.1.cmml" xref="S3.4.p4.7.m7.1.1.3.1"></plus><ci id="S3.4.p4.7.m7.1.1.3.2.cmml" xref="S3.4.p4.7.m7.1.1.3.2">𝑟</ci><cn id="S3.4.p4.7.m7.1.1.3.3.cmml" type="integer" xref="S3.4.p4.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.7.m7.1c">k:=r+1</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.7.m7.1d">italic_k := italic_r + 1</annotation></semantics></math> pairwise vertex-disjoint <math alttext="(V(G)\setminus W_{i})" class="ltx_Math" display="inline" id="S3.4.p4.8.m8.2"><semantics id="S3.4.p4.8.m8.2a"><mrow id="S3.4.p4.8.m8.2.2.1" xref="S3.4.p4.8.m8.2.2.1.1.cmml"><mo id="S3.4.p4.8.m8.2.2.1.2" stretchy="false" xref="S3.4.p4.8.m8.2.2.1.1.cmml">(</mo><mrow id="S3.4.p4.8.m8.2.2.1.1" xref="S3.4.p4.8.m8.2.2.1.1.cmml"><mrow id="S3.4.p4.8.m8.2.2.1.1.2" xref="S3.4.p4.8.m8.2.2.1.1.2.cmml"><mi id="S3.4.p4.8.m8.2.2.1.1.2.2" xref="S3.4.p4.8.m8.2.2.1.1.2.2.cmml">V</mi><mo id="S3.4.p4.8.m8.2.2.1.1.2.1" xref="S3.4.p4.8.m8.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S3.4.p4.8.m8.2.2.1.1.2.3.2" xref="S3.4.p4.8.m8.2.2.1.1.2.cmml"><mo id="S3.4.p4.8.m8.2.2.1.1.2.3.2.1" stretchy="false" xref="S3.4.p4.8.m8.2.2.1.1.2.cmml">(</mo><mi id="S3.4.p4.8.m8.1.1" xref="S3.4.p4.8.m8.1.1.cmml">G</mi><mo id="S3.4.p4.8.m8.2.2.1.1.2.3.2.2" stretchy="false" xref="S3.4.p4.8.m8.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.4.p4.8.m8.2.2.1.1.1" xref="S3.4.p4.8.m8.2.2.1.1.1.cmml">∖</mo><msub id="S3.4.p4.8.m8.2.2.1.1.3" xref="S3.4.p4.8.m8.2.2.1.1.3.cmml"><mi id="S3.4.p4.8.m8.2.2.1.1.3.2" xref="S3.4.p4.8.m8.2.2.1.1.3.2.cmml">W</mi><mi id="S3.4.p4.8.m8.2.2.1.1.3.3" xref="S3.4.p4.8.m8.2.2.1.1.3.3.cmml">i</mi></msub></mrow><mo id="S3.4.p4.8.m8.2.2.1.3" stretchy="false" xref="S3.4.p4.8.m8.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.8.m8.2b"><apply id="S3.4.p4.8.m8.2.2.1.1.cmml" xref="S3.4.p4.8.m8.2.2.1"><setdiff id="S3.4.p4.8.m8.2.2.1.1.1.cmml" xref="S3.4.p4.8.m8.2.2.1.1.1"></setdiff><apply id="S3.4.p4.8.m8.2.2.1.1.2.cmml" xref="S3.4.p4.8.m8.2.2.1.1.2"><times id="S3.4.p4.8.m8.2.2.1.1.2.1.cmml" xref="S3.4.p4.8.m8.2.2.1.1.2.1"></times><ci id="S3.4.p4.8.m8.2.2.1.1.2.2.cmml" xref="S3.4.p4.8.m8.2.2.1.1.2.2">𝑉</ci><ci id="S3.4.p4.8.m8.1.1.cmml" xref="S3.4.p4.8.m8.1.1">𝐺</ci></apply><apply id="S3.4.p4.8.m8.2.2.1.1.3.cmml" xref="S3.4.p4.8.m8.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.4.p4.8.m8.2.2.1.1.3.1.cmml" xref="S3.4.p4.8.m8.2.2.1.1.3">subscript</csymbol><ci id="S3.4.p4.8.m8.2.2.1.1.3.2.cmml" xref="S3.4.p4.8.m8.2.2.1.1.3.2">𝑊</ci><ci id="S3.4.p4.8.m8.2.2.1.1.3.3.cmml" xref="S3.4.p4.8.m8.2.2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.8.m8.2c">(V(G)\setminus W_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.8.m8.2d">( italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>-<math alttext="W" class="ltx_Math" display="inline" id="S3.4.p4.9.m9.1"><semantics id="S3.4.p4.9.m9.1a"><mi id="S3.4.p4.9.m9.1.1" xref="S3.4.p4.9.m9.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.9.m9.1b"><ci id="S3.4.p4.9.m9.1.1.cmml" xref="S3.4.p4.9.m9.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.9.m9.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.9.m9.1d">italic_W</annotation></semantics></math> paths, <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm3" title="Theorem 3 (Menger’s Theorem). ‣ 2 Preliminaries ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">3</span></a> implies that there exists <math alttext="Z\subseteq V(G)" class="ltx_Math" display="inline" id="S3.4.p4.10.m10.1"><semantics id="S3.4.p4.10.m10.1a"><mrow id="S3.4.p4.10.m10.1.2" xref="S3.4.p4.10.m10.1.2.cmml"><mi id="S3.4.p4.10.m10.1.2.2" xref="S3.4.p4.10.m10.1.2.2.cmml">Z</mi><mo id="S3.4.p4.10.m10.1.2.1" xref="S3.4.p4.10.m10.1.2.1.cmml">⊆</mo><mrow id="S3.4.p4.10.m10.1.2.3" xref="S3.4.p4.10.m10.1.2.3.cmml"><mi id="S3.4.p4.10.m10.1.2.3.2" xref="S3.4.p4.10.m10.1.2.3.2.cmml">V</mi><mo id="S3.4.p4.10.m10.1.2.3.1" xref="S3.4.p4.10.m10.1.2.3.1.cmml">⁢</mo><mrow id="S3.4.p4.10.m10.1.2.3.3.2" xref="S3.4.p4.10.m10.1.2.3.cmml"><mo id="S3.4.p4.10.m10.1.2.3.3.2.1" stretchy="false" xref="S3.4.p4.10.m10.1.2.3.cmml">(</mo><mi id="S3.4.p4.10.m10.1.1" xref="S3.4.p4.10.m10.1.1.cmml">G</mi><mo id="S3.4.p4.10.m10.1.2.3.3.2.2" stretchy="false" xref="S3.4.p4.10.m10.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.10.m10.1b"><apply id="S3.4.p4.10.m10.1.2.cmml" xref="S3.4.p4.10.m10.1.2"><subset id="S3.4.p4.10.m10.1.2.1.cmml" xref="S3.4.p4.10.m10.1.2.1"></subset><ci id="S3.4.p4.10.m10.1.2.2.cmml" xref="S3.4.p4.10.m10.1.2.2">𝑍</ci><apply id="S3.4.p4.10.m10.1.2.3.cmml" xref="S3.4.p4.10.m10.1.2.3"><times id="S3.4.p4.10.m10.1.2.3.1.cmml" xref="S3.4.p4.10.m10.1.2.3.1"></times><ci id="S3.4.p4.10.m10.1.2.3.2.cmml" xref="S3.4.p4.10.m10.1.2.3.2">𝑉</ci><ci id="S3.4.p4.10.m10.1.1.cmml" xref="S3.4.p4.10.m10.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.10.m10.1c">Z\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.10.m10.1d">italic_Z ⊆ italic_V ( italic_G )</annotation></semantics></math> with <math alttext="|Z|\leq r" class="ltx_Math" display="inline" id="S3.4.p4.11.m11.1"><semantics id="S3.4.p4.11.m11.1a"><mrow id="S3.4.p4.11.m11.1.2" xref="S3.4.p4.11.m11.1.2.cmml"><mrow id="S3.4.p4.11.m11.1.2.2.2" xref="S3.4.p4.11.m11.1.2.2.1.cmml"><mo id="S3.4.p4.11.m11.1.2.2.2.1" stretchy="false" xref="S3.4.p4.11.m11.1.2.2.1.1.cmml">|</mo><mi id="S3.4.p4.11.m11.1.1" xref="S3.4.p4.11.m11.1.1.cmml">Z</mi><mo id="S3.4.p4.11.m11.1.2.2.2.2" stretchy="false" xref="S3.4.p4.11.m11.1.2.2.1.1.cmml">|</mo></mrow><mo id="S3.4.p4.11.m11.1.2.1" xref="S3.4.p4.11.m11.1.2.1.cmml">≤</mo><mi id="S3.4.p4.11.m11.1.2.3" xref="S3.4.p4.11.m11.1.2.3.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.11.m11.1b"><apply id="S3.4.p4.11.m11.1.2.cmml" xref="S3.4.p4.11.m11.1.2"><leq id="S3.4.p4.11.m11.1.2.1.cmml" xref="S3.4.p4.11.m11.1.2.1"></leq><apply id="S3.4.p4.11.m11.1.2.2.1.cmml" xref="S3.4.p4.11.m11.1.2.2.2"><abs id="S3.4.p4.11.m11.1.2.2.1.1.cmml" xref="S3.4.p4.11.m11.1.2.2.2.1"></abs><ci id="S3.4.p4.11.m11.1.1.cmml" xref="S3.4.p4.11.m11.1.1">𝑍</ci></apply><ci id="S3.4.p4.11.m11.1.2.3.cmml" xref="S3.4.p4.11.m11.1.2.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.11.m11.1c">|Z|\leq r</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.11.m11.1d">| italic_Z | ≤ italic_r</annotation></semantics></math> such that <math alttext="G-Z" class="ltx_Math" display="inline" id="S3.4.p4.12.m12.1"><semantics id="S3.4.p4.12.m12.1a"><mrow id="S3.4.p4.12.m12.1.1" xref="S3.4.p4.12.m12.1.1.cmml"><mi id="S3.4.p4.12.m12.1.1.2" xref="S3.4.p4.12.m12.1.1.2.cmml">G</mi><mo id="S3.4.p4.12.m12.1.1.1" xref="S3.4.p4.12.m12.1.1.1.cmml">−</mo><mi id="S3.4.p4.12.m12.1.1.3" xref="S3.4.p4.12.m12.1.1.3.cmml">Z</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.12.m12.1b"><apply id="S3.4.p4.12.m12.1.1.cmml" xref="S3.4.p4.12.m12.1.1"><minus id="S3.4.p4.12.m12.1.1.1.cmml" xref="S3.4.p4.12.m12.1.1.1"></minus><ci id="S3.4.p4.12.m12.1.1.2.cmml" xref="S3.4.p4.12.m12.1.1.2">𝐺</ci><ci id="S3.4.p4.12.m12.1.1.3.cmml" xref="S3.4.p4.12.m12.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.12.m12.1c">G-Z</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.12.m12.1d">italic_G - italic_Z</annotation></semantics></math> has no <math alttext="(V(G)\setminus W_{\ell})" class="ltx_Math" display="inline" id="S3.4.p4.13.m13.2"><semantics id="S3.4.p4.13.m13.2a"><mrow id="S3.4.p4.13.m13.2.2.1" xref="S3.4.p4.13.m13.2.2.1.1.cmml"><mo id="S3.4.p4.13.m13.2.2.1.2" stretchy="false" xref="S3.4.p4.13.m13.2.2.1.1.cmml">(</mo><mrow id="S3.4.p4.13.m13.2.2.1.1" xref="S3.4.p4.13.m13.2.2.1.1.cmml"><mrow id="S3.4.p4.13.m13.2.2.1.1.2" xref="S3.4.p4.13.m13.2.2.1.1.2.cmml"><mi id="S3.4.p4.13.m13.2.2.1.1.2.2" xref="S3.4.p4.13.m13.2.2.1.1.2.2.cmml">V</mi><mo id="S3.4.p4.13.m13.2.2.1.1.2.1" xref="S3.4.p4.13.m13.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S3.4.p4.13.m13.2.2.1.1.2.3.2" xref="S3.4.p4.13.m13.2.2.1.1.2.cmml"><mo id="S3.4.p4.13.m13.2.2.1.1.2.3.2.1" stretchy="false" xref="S3.4.p4.13.m13.2.2.1.1.2.cmml">(</mo><mi id="S3.4.p4.13.m13.1.1" xref="S3.4.p4.13.m13.1.1.cmml">G</mi><mo id="S3.4.p4.13.m13.2.2.1.1.2.3.2.2" stretchy="false" xref="S3.4.p4.13.m13.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.4.p4.13.m13.2.2.1.1.1" xref="S3.4.p4.13.m13.2.2.1.1.1.cmml">∖</mo><msub id="S3.4.p4.13.m13.2.2.1.1.3" xref="S3.4.p4.13.m13.2.2.1.1.3.cmml"><mi id="S3.4.p4.13.m13.2.2.1.1.3.2" xref="S3.4.p4.13.m13.2.2.1.1.3.2.cmml">W</mi><mi id="S3.4.p4.13.m13.2.2.1.1.3.3" mathvariant="normal" xref="S3.4.p4.13.m13.2.2.1.1.3.3.cmml">ℓ</mi></msub></mrow><mo id="S3.4.p4.13.m13.2.2.1.3" stretchy="false" xref="S3.4.p4.13.m13.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.13.m13.2b"><apply id="S3.4.p4.13.m13.2.2.1.1.cmml" xref="S3.4.p4.13.m13.2.2.1"><setdiff id="S3.4.p4.13.m13.2.2.1.1.1.cmml" xref="S3.4.p4.13.m13.2.2.1.1.1"></setdiff><apply id="S3.4.p4.13.m13.2.2.1.1.2.cmml" xref="S3.4.p4.13.m13.2.2.1.1.2"><times id="S3.4.p4.13.m13.2.2.1.1.2.1.cmml" xref="S3.4.p4.13.m13.2.2.1.1.2.1"></times><ci id="S3.4.p4.13.m13.2.2.1.1.2.2.cmml" xref="S3.4.p4.13.m13.2.2.1.1.2.2">𝑉</ci><ci id="S3.4.p4.13.m13.1.1.cmml" xref="S3.4.p4.13.m13.1.1">𝐺</ci></apply><apply id="S3.4.p4.13.m13.2.2.1.1.3.cmml" xref="S3.4.p4.13.m13.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.4.p4.13.m13.2.2.1.1.3.1.cmml" xref="S3.4.p4.13.m13.2.2.1.1.3">subscript</csymbol><ci id="S3.4.p4.13.m13.2.2.1.1.3.2.cmml" xref="S3.4.p4.13.m13.2.2.1.1.3.2">𝑊</ci><ci id="S3.4.p4.13.m13.2.2.1.1.3.3.cmml" xref="S3.4.p4.13.m13.2.2.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.13.m13.2c">(V(G)\setminus W_{\ell})</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.13.m13.2d">( italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT )</annotation></semantics></math>-<math alttext="W" class="ltx_Math" display="inline" id="S3.4.p4.14.m14.1"><semantics id="S3.4.p4.14.m14.1a"><mi id="S3.4.p4.14.m14.1.1" xref="S3.4.p4.14.m14.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.14.m14.1b"><ci id="S3.4.p4.14.m14.1.1.cmml" xref="S3.4.p4.14.m14.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.14.m14.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.14.m14.1d">italic_W</annotation></semantics></math> path. Since the paths <math alttext="P_{1}^{\prime},\ldots,P_{r}^{\prime}" class="ltx_Math" display="inline" id="S3.4.p4.15.m15.3"><semantics id="S3.4.p4.15.m15.3a"><mrow id="S3.4.p4.15.m15.3.3.2" xref="S3.4.p4.15.m15.3.3.3.cmml"><msubsup id="S3.4.p4.15.m15.2.2.1.1" xref="S3.4.p4.15.m15.2.2.1.1.cmml"><mi id="S3.4.p4.15.m15.2.2.1.1.2.2" xref="S3.4.p4.15.m15.2.2.1.1.2.2.cmml">P</mi><mn id="S3.4.p4.15.m15.2.2.1.1.2.3" xref="S3.4.p4.15.m15.2.2.1.1.2.3.cmml">1</mn><mo id="S3.4.p4.15.m15.2.2.1.1.3" xref="S3.4.p4.15.m15.2.2.1.1.3.cmml">′</mo></msubsup><mo id="S3.4.p4.15.m15.3.3.2.3" xref="S3.4.p4.15.m15.3.3.3.cmml">,</mo><mi id="S3.4.p4.15.m15.1.1" mathvariant="normal" xref="S3.4.p4.15.m15.1.1.cmml">…</mi><mo id="S3.4.p4.15.m15.3.3.2.4" xref="S3.4.p4.15.m15.3.3.3.cmml">,</mo><msubsup id="S3.4.p4.15.m15.3.3.2.2" xref="S3.4.p4.15.m15.3.3.2.2.cmml"><mi id="S3.4.p4.15.m15.3.3.2.2.2.2" xref="S3.4.p4.15.m15.3.3.2.2.2.2.cmml">P</mi><mi id="S3.4.p4.15.m15.3.3.2.2.2.3" xref="S3.4.p4.15.m15.3.3.2.2.2.3.cmml">r</mi><mo id="S3.4.p4.15.m15.3.3.2.2.3" xref="S3.4.p4.15.m15.3.3.2.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.15.m15.3b"><list id="S3.4.p4.15.m15.3.3.3.cmml" xref="S3.4.p4.15.m15.3.3.2"><apply id="S3.4.p4.15.m15.2.2.1.1.cmml" xref="S3.4.p4.15.m15.2.2.1.1"><csymbol cd="ambiguous" id="S3.4.p4.15.m15.2.2.1.1.1.cmml" xref="S3.4.p4.15.m15.2.2.1.1">superscript</csymbol><apply id="S3.4.p4.15.m15.2.2.1.1.2.cmml" xref="S3.4.p4.15.m15.2.2.1.1"><csymbol cd="ambiguous" id="S3.4.p4.15.m15.2.2.1.1.2.1.cmml" xref="S3.4.p4.15.m15.2.2.1.1">subscript</csymbol><ci id="S3.4.p4.15.m15.2.2.1.1.2.2.cmml" xref="S3.4.p4.15.m15.2.2.1.1.2.2">𝑃</ci><cn id="S3.4.p4.15.m15.2.2.1.1.2.3.cmml" type="integer" xref="S3.4.p4.15.m15.2.2.1.1.2.3">1</cn></apply><ci id="S3.4.p4.15.m15.2.2.1.1.3.cmml" xref="S3.4.p4.15.m15.2.2.1.1.3">′</ci></apply><ci id="S3.4.p4.15.m15.1.1.cmml" xref="S3.4.p4.15.m15.1.1">…</ci><apply id="S3.4.p4.15.m15.3.3.2.2.cmml" xref="S3.4.p4.15.m15.3.3.2.2"><csymbol cd="ambiguous" id="S3.4.p4.15.m15.3.3.2.2.1.cmml" xref="S3.4.p4.15.m15.3.3.2.2">superscript</csymbol><apply id="S3.4.p4.15.m15.3.3.2.2.2.cmml" xref="S3.4.p4.15.m15.3.3.2.2"><csymbol cd="ambiguous" id="S3.4.p4.15.m15.3.3.2.2.2.1.cmml" xref="S3.4.p4.15.m15.3.3.2.2">subscript</csymbol><ci id="S3.4.p4.15.m15.3.3.2.2.2.2.cmml" xref="S3.4.p4.15.m15.3.3.2.2.2.2">𝑃</ci><ci id="S3.4.p4.15.m15.3.3.2.2.2.3.cmml" xref="S3.4.p4.15.m15.3.3.2.2.2.3">𝑟</ci></apply><ci id="S3.4.p4.15.m15.3.3.2.2.3.cmml" xref="S3.4.p4.15.m15.3.3.2.2.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.15.m15.3c">P_{1}^{\prime},\ldots,P_{r}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.15.m15.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are pairwise vertex-disjoint <math alttext="(V(G)\setminus W_{\ell})" class="ltx_Math" display="inline" id="S3.4.p4.16.m16.2"><semantics id="S3.4.p4.16.m16.2a"><mrow id="S3.4.p4.16.m16.2.2.1" xref="S3.4.p4.16.m16.2.2.1.1.cmml"><mo id="S3.4.p4.16.m16.2.2.1.2" stretchy="false" xref="S3.4.p4.16.m16.2.2.1.1.cmml">(</mo><mrow id="S3.4.p4.16.m16.2.2.1.1" xref="S3.4.p4.16.m16.2.2.1.1.cmml"><mrow id="S3.4.p4.16.m16.2.2.1.1.2" xref="S3.4.p4.16.m16.2.2.1.1.2.cmml"><mi id="S3.4.p4.16.m16.2.2.1.1.2.2" xref="S3.4.p4.16.m16.2.2.1.1.2.2.cmml">V</mi><mo id="S3.4.p4.16.m16.2.2.1.1.2.1" xref="S3.4.p4.16.m16.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S3.4.p4.16.m16.2.2.1.1.2.3.2" xref="S3.4.p4.16.m16.2.2.1.1.2.cmml"><mo id="S3.4.p4.16.m16.2.2.1.1.2.3.2.1" stretchy="false" xref="S3.4.p4.16.m16.2.2.1.1.2.cmml">(</mo><mi id="S3.4.p4.16.m16.1.1" xref="S3.4.p4.16.m16.1.1.cmml">G</mi><mo id="S3.4.p4.16.m16.2.2.1.1.2.3.2.2" stretchy="false" xref="S3.4.p4.16.m16.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.4.p4.16.m16.2.2.1.1.1" xref="S3.4.p4.16.m16.2.2.1.1.1.cmml">∖</mo><msub id="S3.4.p4.16.m16.2.2.1.1.3" xref="S3.4.p4.16.m16.2.2.1.1.3.cmml"><mi id="S3.4.p4.16.m16.2.2.1.1.3.2" xref="S3.4.p4.16.m16.2.2.1.1.3.2.cmml">W</mi><mi id="S3.4.p4.16.m16.2.2.1.1.3.3" mathvariant="normal" xref="S3.4.p4.16.m16.2.2.1.1.3.3.cmml">ℓ</mi></msub></mrow><mo id="S3.4.p4.16.m16.2.2.1.3" stretchy="false" xref="S3.4.p4.16.m16.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.16.m16.2b"><apply id="S3.4.p4.16.m16.2.2.1.1.cmml" xref="S3.4.p4.16.m16.2.2.1"><setdiff id="S3.4.p4.16.m16.2.2.1.1.1.cmml" xref="S3.4.p4.16.m16.2.2.1.1.1"></setdiff><apply id="S3.4.p4.16.m16.2.2.1.1.2.cmml" xref="S3.4.p4.16.m16.2.2.1.1.2"><times id="S3.4.p4.16.m16.2.2.1.1.2.1.cmml" xref="S3.4.p4.16.m16.2.2.1.1.2.1"></times><ci id="S3.4.p4.16.m16.2.2.1.1.2.2.cmml" xref="S3.4.p4.16.m16.2.2.1.1.2.2">𝑉</ci><ci id="S3.4.p4.16.m16.1.1.cmml" xref="S3.4.p4.16.m16.1.1">𝐺</ci></apply><apply id="S3.4.p4.16.m16.2.2.1.1.3.cmml" xref="S3.4.p4.16.m16.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.4.p4.16.m16.2.2.1.1.3.1.cmml" xref="S3.4.p4.16.m16.2.2.1.1.3">subscript</csymbol><ci id="S3.4.p4.16.m16.2.2.1.1.3.2.cmml" xref="S3.4.p4.16.m16.2.2.1.1.3.2">𝑊</ci><ci id="S3.4.p4.16.m16.2.2.1.1.3.3.cmml" xref="S3.4.p4.16.m16.2.2.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.16.m16.2c">(V(G)\setminus W_{\ell})</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.16.m16.2d">( italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT )</annotation></semantics></math>-<math alttext="W" class="ltx_Math" display="inline" id="S3.4.p4.17.m17.1"><semantics id="S3.4.p4.17.m17.1a"><mi id="S3.4.p4.17.m17.1.1" xref="S3.4.p4.17.m17.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.17.m17.1b"><ci id="S3.4.p4.17.m17.1.1.cmml" xref="S3.4.p4.17.m17.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.17.m17.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.17.m17.1d">italic_W</annotation></semantics></math> paths, <math alttext="Z" class="ltx_Math" display="inline" id="S3.4.p4.18.m18.1"><semantics id="S3.4.p4.18.m18.1a"><mi id="S3.4.p4.18.m18.1.1" xref="S3.4.p4.18.m18.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.18.m18.1b"><ci id="S3.4.p4.18.m18.1.1.cmml" xref="S3.4.p4.18.m18.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.18.m18.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.18.m18.1d">italic_Z</annotation></semantics></math> must contain at least one vertex from each of these paths, so <math alttext="|Z|\geq r" class="ltx_Math" display="inline" id="S3.4.p4.19.m19.1"><semantics id="S3.4.p4.19.m19.1a"><mrow id="S3.4.p4.19.m19.1.2" xref="S3.4.p4.19.m19.1.2.cmml"><mrow id="S3.4.p4.19.m19.1.2.2.2" xref="S3.4.p4.19.m19.1.2.2.1.cmml"><mo id="S3.4.p4.19.m19.1.2.2.2.1" stretchy="false" xref="S3.4.p4.19.m19.1.2.2.1.1.cmml">|</mo><mi id="S3.4.p4.19.m19.1.1" xref="S3.4.p4.19.m19.1.1.cmml">Z</mi><mo id="S3.4.p4.19.m19.1.2.2.2.2" stretchy="false" xref="S3.4.p4.19.m19.1.2.2.1.1.cmml">|</mo></mrow><mo id="S3.4.p4.19.m19.1.2.1" xref="S3.4.p4.19.m19.1.2.1.cmml">≥</mo><mi id="S3.4.p4.19.m19.1.2.3" xref="S3.4.p4.19.m19.1.2.3.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.19.m19.1b"><apply id="S3.4.p4.19.m19.1.2.cmml" xref="S3.4.p4.19.m19.1.2"><geq id="S3.4.p4.19.m19.1.2.1.cmml" xref="S3.4.p4.19.m19.1.2.1"></geq><apply id="S3.4.p4.19.m19.1.2.2.1.cmml" xref="S3.4.p4.19.m19.1.2.2.2"><abs id="S3.4.p4.19.m19.1.2.2.1.1.cmml" xref="S3.4.p4.19.m19.1.2.2.2.1"></abs><ci id="S3.4.p4.19.m19.1.1.cmml" xref="S3.4.p4.19.m19.1.1">𝑍</ci></apply><ci id="S3.4.p4.19.m19.1.2.3.cmml" xref="S3.4.p4.19.m19.1.2.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.19.m19.1c">|Z|\geq r</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.19.m19.1d">| italic_Z | ≥ italic_r</annotation></semantics></math>. Therefore, <math alttext="r\leq|Z|\leq r" class="ltx_Math" display="inline" id="S3.4.p4.20.m20.1"><semantics id="S3.4.p4.20.m20.1a"><mrow id="S3.4.p4.20.m20.1.2" xref="S3.4.p4.20.m20.1.2.cmml"><mi id="S3.4.p4.20.m20.1.2.2" xref="S3.4.p4.20.m20.1.2.2.cmml">r</mi><mo id="S3.4.p4.20.m20.1.2.3" xref="S3.4.p4.20.m20.1.2.3.cmml">≤</mo><mrow id="S3.4.p4.20.m20.1.2.4.2" xref="S3.4.p4.20.m20.1.2.4.1.cmml"><mo id="S3.4.p4.20.m20.1.2.4.2.1" stretchy="false" xref="S3.4.p4.20.m20.1.2.4.1.1.cmml">|</mo><mi id="S3.4.p4.20.m20.1.1" xref="S3.4.p4.20.m20.1.1.cmml">Z</mi><mo id="S3.4.p4.20.m20.1.2.4.2.2" stretchy="false" xref="S3.4.p4.20.m20.1.2.4.1.1.cmml">|</mo></mrow><mo id="S3.4.p4.20.m20.1.2.5" xref="S3.4.p4.20.m20.1.2.5.cmml">≤</mo><mi id="S3.4.p4.20.m20.1.2.6" xref="S3.4.p4.20.m20.1.2.6.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.20.m20.1b"><apply id="S3.4.p4.20.m20.1.2.cmml" xref="S3.4.p4.20.m20.1.2"><and id="S3.4.p4.20.m20.1.2a.cmml" xref="S3.4.p4.20.m20.1.2"></and><apply id="S3.4.p4.20.m20.1.2b.cmml" xref="S3.4.p4.20.m20.1.2"><leq id="S3.4.p4.20.m20.1.2.3.cmml" xref="S3.4.p4.20.m20.1.2.3"></leq><ci id="S3.4.p4.20.m20.1.2.2.cmml" xref="S3.4.p4.20.m20.1.2.2">𝑟</ci><apply id="S3.4.p4.20.m20.1.2.4.1.cmml" xref="S3.4.p4.20.m20.1.2.4.2"><abs id="S3.4.p4.20.m20.1.2.4.1.1.cmml" xref="S3.4.p4.20.m20.1.2.4.2.1"></abs><ci id="S3.4.p4.20.m20.1.1.cmml" xref="S3.4.p4.20.m20.1.1">𝑍</ci></apply></apply><apply id="S3.4.p4.20.m20.1.2c.cmml" xref="S3.4.p4.20.m20.1.2"><leq id="S3.4.p4.20.m20.1.2.5.cmml" xref="S3.4.p4.20.m20.1.2.5"></leq><share href="https://arxiv.org/html/2503.17112v1#S3.4.p4.20.m20.1.2.4.cmml" id="S3.4.p4.20.m20.1.2d.cmml" xref="S3.4.p4.20.m20.1.2"></share><ci id="S3.4.p4.20.m20.1.2.6.cmml" xref="S3.4.p4.20.m20.1.2.6">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.20.m20.1c">r\leq|Z|\leq r</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.20.m20.1d">italic_r ≤ | italic_Z | ≤ italic_r</annotation></semantics></math>, so <math alttext="|Z|=r=s_{\ell+1}" class="ltx_Math" display="inline" id="S3.4.p4.21.m21.1"><semantics id="S3.4.p4.21.m21.1a"><mrow id="S3.4.p4.21.m21.1.2" xref="S3.4.p4.21.m21.1.2.cmml"><mrow id="S3.4.p4.21.m21.1.2.2.2" xref="S3.4.p4.21.m21.1.2.2.1.cmml"><mo id="S3.4.p4.21.m21.1.2.2.2.1" stretchy="false" xref="S3.4.p4.21.m21.1.2.2.1.1.cmml">|</mo><mi id="S3.4.p4.21.m21.1.1" xref="S3.4.p4.21.m21.1.1.cmml">Z</mi><mo id="S3.4.p4.21.m21.1.2.2.2.2" stretchy="false" xref="S3.4.p4.21.m21.1.2.2.1.1.cmml">|</mo></mrow><mo id="S3.4.p4.21.m21.1.2.3" xref="S3.4.p4.21.m21.1.2.3.cmml">=</mo><mi id="S3.4.p4.21.m21.1.2.4" xref="S3.4.p4.21.m21.1.2.4.cmml">r</mi><mo id="S3.4.p4.21.m21.1.2.5" xref="S3.4.p4.21.m21.1.2.5.cmml">=</mo><msub id="S3.4.p4.21.m21.1.2.6" xref="S3.4.p4.21.m21.1.2.6.cmml"><mi id="S3.4.p4.21.m21.1.2.6.2" xref="S3.4.p4.21.m21.1.2.6.2.cmml">s</mi><mrow id="S3.4.p4.21.m21.1.2.6.3" xref="S3.4.p4.21.m21.1.2.6.3.cmml"><mi id="S3.4.p4.21.m21.1.2.6.3.2" mathvariant="normal" xref="S3.4.p4.21.m21.1.2.6.3.2.cmml">ℓ</mi><mo id="S3.4.p4.21.m21.1.2.6.3.1" xref="S3.4.p4.21.m21.1.2.6.3.1.cmml">+</mo><mn id="S3.4.p4.21.m21.1.2.6.3.3" xref="S3.4.p4.21.m21.1.2.6.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.21.m21.1b"><apply id="S3.4.p4.21.m21.1.2.cmml" xref="S3.4.p4.21.m21.1.2"><and id="S3.4.p4.21.m21.1.2a.cmml" xref="S3.4.p4.21.m21.1.2"></and><apply id="S3.4.p4.21.m21.1.2b.cmml" xref="S3.4.p4.21.m21.1.2"><eq id="S3.4.p4.21.m21.1.2.3.cmml" xref="S3.4.p4.21.m21.1.2.3"></eq><apply id="S3.4.p4.21.m21.1.2.2.1.cmml" xref="S3.4.p4.21.m21.1.2.2.2"><abs id="S3.4.p4.21.m21.1.2.2.1.1.cmml" xref="S3.4.p4.21.m21.1.2.2.2.1"></abs><ci id="S3.4.p4.21.m21.1.1.cmml" xref="S3.4.p4.21.m21.1.1">𝑍</ci></apply><ci id="S3.4.p4.21.m21.1.2.4.cmml" xref="S3.4.p4.21.m21.1.2.4">𝑟</ci></apply><apply id="S3.4.p4.21.m21.1.2c.cmml" xref="S3.4.p4.21.m21.1.2"><eq id="S3.4.p4.21.m21.1.2.5.cmml" xref="S3.4.p4.21.m21.1.2.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S3.4.p4.21.m21.1.2.4.cmml" id="S3.4.p4.21.m21.1.2d.cmml" xref="S3.4.p4.21.m21.1.2"></share><apply id="S3.4.p4.21.m21.1.2.6.cmml" xref="S3.4.p4.21.m21.1.2.6"><csymbol cd="ambiguous" id="S3.4.p4.21.m21.1.2.6.1.cmml" xref="S3.4.p4.21.m21.1.2.6">subscript</csymbol><ci id="S3.4.p4.21.m21.1.2.6.2.cmml" xref="S3.4.p4.21.m21.1.2.6.2">𝑠</ci><apply id="S3.4.p4.21.m21.1.2.6.3.cmml" xref="S3.4.p4.21.m21.1.2.6.3"><plus id="S3.4.p4.21.m21.1.2.6.3.1.cmml" xref="S3.4.p4.21.m21.1.2.6.3.1"></plus><ci id="S3.4.p4.21.m21.1.2.6.3.2.cmml" xref="S3.4.p4.21.m21.1.2.6.3.2">ℓ</ci><cn id="S3.4.p4.21.m21.1.2.6.3.3.cmml" type="integer" xref="S3.4.p4.21.m21.1.2.6.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.21.m21.1c">|Z|=r=s_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.21.m21.1d">| italic_Z | = italic_r = italic_s start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="V(P_{j}^{\prime})\subseteq W_{\ell+1}" class="ltx_Math" display="inline" id="S3.4.p4.22.m22.1"><semantics id="S3.4.p4.22.m22.1a"><mrow id="S3.4.p4.22.m22.1.1" xref="S3.4.p4.22.m22.1.1.cmml"><mrow id="S3.4.p4.22.m22.1.1.1" xref="S3.4.p4.22.m22.1.1.1.cmml"><mi id="S3.4.p4.22.m22.1.1.1.3" xref="S3.4.p4.22.m22.1.1.1.3.cmml">V</mi><mo id="S3.4.p4.22.m22.1.1.1.2" xref="S3.4.p4.22.m22.1.1.1.2.cmml">⁢</mo><mrow id="S3.4.p4.22.m22.1.1.1.1.1" xref="S3.4.p4.22.m22.1.1.1.1.1.1.cmml"><mo id="S3.4.p4.22.m22.1.1.1.1.1.2" stretchy="false" xref="S3.4.p4.22.m22.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.4.p4.22.m22.1.1.1.1.1.1" xref="S3.4.p4.22.m22.1.1.1.1.1.1.cmml"><mi id="S3.4.p4.22.m22.1.1.1.1.1.1.2.2" xref="S3.4.p4.22.m22.1.1.1.1.1.1.2.2.cmml">P</mi><mi id="S3.4.p4.22.m22.1.1.1.1.1.1.2.3" xref="S3.4.p4.22.m22.1.1.1.1.1.1.2.3.cmml">j</mi><mo id="S3.4.p4.22.m22.1.1.1.1.1.1.3" xref="S3.4.p4.22.m22.1.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="S3.4.p4.22.m22.1.1.1.1.1.3" stretchy="false" xref="S3.4.p4.22.m22.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.4.p4.22.m22.1.1.2" xref="S3.4.p4.22.m22.1.1.2.cmml">⊆</mo><msub id="S3.4.p4.22.m22.1.1.3" xref="S3.4.p4.22.m22.1.1.3.cmml"><mi id="S3.4.p4.22.m22.1.1.3.2" xref="S3.4.p4.22.m22.1.1.3.2.cmml">W</mi><mrow id="S3.4.p4.22.m22.1.1.3.3" xref="S3.4.p4.22.m22.1.1.3.3.cmml"><mi id="S3.4.p4.22.m22.1.1.3.3.2" mathvariant="normal" xref="S3.4.p4.22.m22.1.1.3.3.2.cmml">ℓ</mi><mo id="S3.4.p4.22.m22.1.1.3.3.1" xref="S3.4.p4.22.m22.1.1.3.3.1.cmml">+</mo><mn id="S3.4.p4.22.m22.1.1.3.3.3" xref="S3.4.p4.22.m22.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.22.m22.1b"><apply id="S3.4.p4.22.m22.1.1.cmml" xref="S3.4.p4.22.m22.1.1"><subset id="S3.4.p4.22.m22.1.1.2.cmml" xref="S3.4.p4.22.m22.1.1.2"></subset><apply id="S3.4.p4.22.m22.1.1.1.cmml" xref="S3.4.p4.22.m22.1.1.1"><times id="S3.4.p4.22.m22.1.1.1.2.cmml" xref="S3.4.p4.22.m22.1.1.1.2"></times><ci id="S3.4.p4.22.m22.1.1.1.3.cmml" xref="S3.4.p4.22.m22.1.1.1.3">𝑉</ci><apply id="S3.4.p4.22.m22.1.1.1.1.1.1.cmml" xref="S3.4.p4.22.m22.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.4.p4.22.m22.1.1.1.1.1.1.1.cmml" xref="S3.4.p4.22.m22.1.1.1.1.1">superscript</csymbol><apply id="S3.4.p4.22.m22.1.1.1.1.1.1.2.cmml" xref="S3.4.p4.22.m22.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.4.p4.22.m22.1.1.1.1.1.1.2.1.cmml" xref="S3.4.p4.22.m22.1.1.1.1.1">subscript</csymbol><ci id="S3.4.p4.22.m22.1.1.1.1.1.1.2.2.cmml" xref="S3.4.p4.22.m22.1.1.1.1.1.1.2.2">𝑃</ci><ci id="S3.4.p4.22.m22.1.1.1.1.1.1.2.3.cmml" xref="S3.4.p4.22.m22.1.1.1.1.1.1.2.3">𝑗</ci></apply><ci id="S3.4.p4.22.m22.1.1.1.1.1.1.3.cmml" xref="S3.4.p4.22.m22.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S3.4.p4.22.m22.1.1.3.cmml" xref="S3.4.p4.22.m22.1.1.3"><csymbol cd="ambiguous" id="S3.4.p4.22.m22.1.1.3.1.cmml" xref="S3.4.p4.22.m22.1.1.3">subscript</csymbol><ci id="S3.4.p4.22.m22.1.1.3.2.cmml" xref="S3.4.p4.22.m22.1.1.3.2">𝑊</ci><apply id="S3.4.p4.22.m22.1.1.3.3.cmml" xref="S3.4.p4.22.m22.1.1.3.3"><plus id="S3.4.p4.22.m22.1.1.3.3.1.cmml" xref="S3.4.p4.22.m22.1.1.3.3.1"></plus><ci id="S3.4.p4.22.m22.1.1.3.3.2.cmml" xref="S3.4.p4.22.m22.1.1.3.3.2">ℓ</ci><cn id="S3.4.p4.22.m22.1.1.3.3.3.cmml" type="integer" xref="S3.4.p4.22.m22.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.22.m22.1c">V(P_{j}^{\prime})\subseteq W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.22.m22.1d">italic_V ( italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⊆ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> for each <math alttext="j\in\{1,\ldots,s_{\ell+1}\}" class="ltx_Math" display="inline" id="S3.4.p4.23.m23.3"><semantics id="S3.4.p4.23.m23.3a"><mrow id="S3.4.p4.23.m23.3.3" xref="S3.4.p4.23.m23.3.3.cmml"><mi id="S3.4.p4.23.m23.3.3.3" xref="S3.4.p4.23.m23.3.3.3.cmml">j</mi><mo id="S3.4.p4.23.m23.3.3.2" xref="S3.4.p4.23.m23.3.3.2.cmml">∈</mo><mrow id="S3.4.p4.23.m23.3.3.1.1" xref="S3.4.p4.23.m23.3.3.1.2.cmml"><mo id="S3.4.p4.23.m23.3.3.1.1.2" stretchy="false" xref="S3.4.p4.23.m23.3.3.1.2.cmml">{</mo><mn id="S3.4.p4.23.m23.1.1" xref="S3.4.p4.23.m23.1.1.cmml">1</mn><mo id="S3.4.p4.23.m23.3.3.1.1.3" xref="S3.4.p4.23.m23.3.3.1.2.cmml">,</mo><mi id="S3.4.p4.23.m23.2.2" mathvariant="normal" xref="S3.4.p4.23.m23.2.2.cmml">…</mi><mo id="S3.4.p4.23.m23.3.3.1.1.4" xref="S3.4.p4.23.m23.3.3.1.2.cmml">,</mo><msub id="S3.4.p4.23.m23.3.3.1.1.1" xref="S3.4.p4.23.m23.3.3.1.1.1.cmml"><mi id="S3.4.p4.23.m23.3.3.1.1.1.2" xref="S3.4.p4.23.m23.3.3.1.1.1.2.cmml">s</mi><mrow id="S3.4.p4.23.m23.3.3.1.1.1.3" xref="S3.4.p4.23.m23.3.3.1.1.1.3.cmml"><mi id="S3.4.p4.23.m23.3.3.1.1.1.3.2" mathvariant="normal" xref="S3.4.p4.23.m23.3.3.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.4.p4.23.m23.3.3.1.1.1.3.1" xref="S3.4.p4.23.m23.3.3.1.1.1.3.1.cmml">+</mo><mn id="S3.4.p4.23.m23.3.3.1.1.1.3.3" xref="S3.4.p4.23.m23.3.3.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.4.p4.23.m23.3.3.1.1.5" stretchy="false" xref="S3.4.p4.23.m23.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.23.m23.3b"><apply id="S3.4.p4.23.m23.3.3.cmml" xref="S3.4.p4.23.m23.3.3"><in id="S3.4.p4.23.m23.3.3.2.cmml" xref="S3.4.p4.23.m23.3.3.2"></in><ci id="S3.4.p4.23.m23.3.3.3.cmml" xref="S3.4.p4.23.m23.3.3.3">𝑗</ci><set id="S3.4.p4.23.m23.3.3.1.2.cmml" xref="S3.4.p4.23.m23.3.3.1.1"><cn id="S3.4.p4.23.m23.1.1.cmml" type="integer" xref="S3.4.p4.23.m23.1.1">1</cn><ci id="S3.4.p4.23.m23.2.2.cmml" xref="S3.4.p4.23.m23.2.2">…</ci><apply id="S3.4.p4.23.m23.3.3.1.1.1.cmml" xref="S3.4.p4.23.m23.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.4.p4.23.m23.3.3.1.1.1.1.cmml" xref="S3.4.p4.23.m23.3.3.1.1.1">subscript</csymbol><ci id="S3.4.p4.23.m23.3.3.1.1.1.2.cmml" xref="S3.4.p4.23.m23.3.3.1.1.1.2">𝑠</ci><apply id="S3.4.p4.23.m23.3.3.1.1.1.3.cmml" xref="S3.4.p4.23.m23.3.3.1.1.1.3"><plus id="S3.4.p4.23.m23.3.3.1.1.1.3.1.cmml" xref="S3.4.p4.23.m23.3.3.1.1.1.3.1"></plus><ci id="S3.4.p4.23.m23.3.3.1.1.1.3.2.cmml" xref="S3.4.p4.23.m23.3.3.1.1.1.3.2">ℓ</ci><cn id="S3.4.p4.23.m23.3.3.1.1.1.3.3.cmml" type="integer" xref="S3.4.p4.23.m23.3.3.1.1.1.3.3">1</cn></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.23.m23.3c">j\in\{1,\ldots,s_{\ell+1}\}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.23.m23.3d">italic_j ∈ { 1 , … , italic_s start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT }</annotation></semantics></math>, <math alttext="Z\subseteq W_{\ell+1}" class="ltx_Math" display="inline" id="S3.4.p4.24.m24.1"><semantics id="S3.4.p4.24.m24.1a"><mrow id="S3.4.p4.24.m24.1.1" xref="S3.4.p4.24.m24.1.1.cmml"><mi id="S3.4.p4.24.m24.1.1.2" xref="S3.4.p4.24.m24.1.1.2.cmml">Z</mi><mo id="S3.4.p4.24.m24.1.1.1" xref="S3.4.p4.24.m24.1.1.1.cmml">⊆</mo><msub id="S3.4.p4.24.m24.1.1.3" xref="S3.4.p4.24.m24.1.1.3.cmml"><mi id="S3.4.p4.24.m24.1.1.3.2" xref="S3.4.p4.24.m24.1.1.3.2.cmml">W</mi><mrow id="S3.4.p4.24.m24.1.1.3.3" xref="S3.4.p4.24.m24.1.1.3.3.cmml"><mi id="S3.4.p4.24.m24.1.1.3.3.2" mathvariant="normal" xref="S3.4.p4.24.m24.1.1.3.3.2.cmml">ℓ</mi><mo id="S3.4.p4.24.m24.1.1.3.3.1" xref="S3.4.p4.24.m24.1.1.3.3.1.cmml">+</mo><mn id="S3.4.p4.24.m24.1.1.3.3.3" xref="S3.4.p4.24.m24.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.24.m24.1b"><apply id="S3.4.p4.24.m24.1.1.cmml" xref="S3.4.p4.24.m24.1.1"><subset id="S3.4.p4.24.m24.1.1.1.cmml" xref="S3.4.p4.24.m24.1.1.1"></subset><ci id="S3.4.p4.24.m24.1.1.2.cmml" xref="S3.4.p4.24.m24.1.1.2">𝑍</ci><apply id="S3.4.p4.24.m24.1.1.3.cmml" xref="S3.4.p4.24.m24.1.1.3"><csymbol cd="ambiguous" id="S3.4.p4.24.m24.1.1.3.1.cmml" xref="S3.4.p4.24.m24.1.1.3">subscript</csymbol><ci id="S3.4.p4.24.m24.1.1.3.2.cmml" xref="S3.4.p4.24.m24.1.1.3.2">𝑊</ci><apply id="S3.4.p4.24.m24.1.1.3.3.cmml" xref="S3.4.p4.24.m24.1.1.3.3"><plus id="S3.4.p4.24.m24.1.1.3.3.1.cmml" xref="S3.4.p4.24.m24.1.1.3.3.1"></plus><ci id="S3.4.p4.24.m24.1.1.3.3.2.cmml" xref="S3.4.p4.24.m24.1.1.3.3.2">ℓ</ci><cn id="S3.4.p4.24.m24.1.1.3.3.3.cmml" type="integer" xref="S3.4.p4.24.m24.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.24.m24.1c">Z\subseteq W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.24.m24.1d">italic_Z ⊆ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, this choice of <math alttext="Z" class="ltx_Math" display="inline" id="S3.4.p4.25.m25.1"><semantics id="S3.4.p4.25.m25.1a"><mi id="S3.4.p4.25.m25.1.1" xref="S3.4.p4.25.m25.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.25.m25.1b"><ci id="S3.4.p4.25.m25.1.1.cmml" xref="S3.4.p4.25.m25.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.25.m25.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.25.m25.1d">italic_Z</annotation></semantics></math> satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i5" title="Item (e) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(e)</span></a>. Therefore, the sequence <math alttext="W_{0},\ldots,W_{\ell+1}" class="ltx_Math" display="inline" id="S3.4.p4.26.m26.3"><semantics id="S3.4.p4.26.m26.3a"><mrow id="S3.4.p4.26.m26.3.3.2" xref="S3.4.p4.26.m26.3.3.3.cmml"><msub id="S3.4.p4.26.m26.2.2.1.1" xref="S3.4.p4.26.m26.2.2.1.1.cmml"><mi id="S3.4.p4.26.m26.2.2.1.1.2" xref="S3.4.p4.26.m26.2.2.1.1.2.cmml">W</mi><mn id="S3.4.p4.26.m26.2.2.1.1.3" xref="S3.4.p4.26.m26.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.4.p4.26.m26.3.3.2.3" xref="S3.4.p4.26.m26.3.3.3.cmml">,</mo><mi id="S3.4.p4.26.m26.1.1" mathvariant="normal" xref="S3.4.p4.26.m26.1.1.cmml">…</mi><mo id="S3.4.p4.26.m26.3.3.2.4" xref="S3.4.p4.26.m26.3.3.3.cmml">,</mo><msub id="S3.4.p4.26.m26.3.3.2.2" xref="S3.4.p4.26.m26.3.3.2.2.cmml"><mi id="S3.4.p4.26.m26.3.3.2.2.2" xref="S3.4.p4.26.m26.3.3.2.2.2.cmml">W</mi><mrow id="S3.4.p4.26.m26.3.3.2.2.3" xref="S3.4.p4.26.m26.3.3.2.2.3.cmml"><mi id="S3.4.p4.26.m26.3.3.2.2.3.2" mathvariant="normal" xref="S3.4.p4.26.m26.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.4.p4.26.m26.3.3.2.2.3.1" xref="S3.4.p4.26.m26.3.3.2.2.3.1.cmml">+</mo><mn id="S3.4.p4.26.m26.3.3.2.2.3.3" xref="S3.4.p4.26.m26.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.26.m26.3b"><list id="S3.4.p4.26.m26.3.3.3.cmml" xref="S3.4.p4.26.m26.3.3.2"><apply id="S3.4.p4.26.m26.2.2.1.1.cmml" xref="S3.4.p4.26.m26.2.2.1.1"><csymbol cd="ambiguous" id="S3.4.p4.26.m26.2.2.1.1.1.cmml" xref="S3.4.p4.26.m26.2.2.1.1">subscript</csymbol><ci id="S3.4.p4.26.m26.2.2.1.1.2.cmml" xref="S3.4.p4.26.m26.2.2.1.1.2">𝑊</ci><cn id="S3.4.p4.26.m26.2.2.1.1.3.cmml" type="integer" xref="S3.4.p4.26.m26.2.2.1.1.3">0</cn></apply><ci id="S3.4.p4.26.m26.1.1.cmml" xref="S3.4.p4.26.m26.1.1">…</ci><apply id="S3.4.p4.26.m26.3.3.2.2.cmml" xref="S3.4.p4.26.m26.3.3.2.2"><csymbol cd="ambiguous" id="S3.4.p4.26.m26.3.3.2.2.1.cmml" xref="S3.4.p4.26.m26.3.3.2.2">subscript</csymbol><ci id="S3.4.p4.26.m26.3.3.2.2.2.cmml" xref="S3.4.p4.26.m26.3.3.2.2.2">𝑊</ci><apply id="S3.4.p4.26.m26.3.3.2.2.3.cmml" xref="S3.4.p4.26.m26.3.3.2.2.3"><plus id="S3.4.p4.26.m26.3.3.2.2.3.1.cmml" xref="S3.4.p4.26.m26.3.3.2.2.3.1"></plus><ci id="S3.4.p4.26.m26.3.3.2.2.3.2.cmml" xref="S3.4.p4.26.m26.3.3.2.2.3.2">ℓ</ci><cn id="S3.4.p4.26.m26.3.3.2.2.3.3.cmml" type="integer" xref="S3.4.p4.26.m26.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.26.m26.3c">W_{0},\ldots,W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.26.m26.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i1" title="Item (a) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(a)</span></a> (a condition on <math alttext="W_{0}" class="ltx_Math" display="inline" id="S3.4.p4.27.m27.1"><semantics id="S3.4.p4.27.m27.1a"><msub id="S3.4.p4.27.m27.1.1" xref="S3.4.p4.27.m27.1.1.cmml"><mi id="S3.4.p4.27.m27.1.1.2" xref="S3.4.p4.27.m27.1.1.2.cmml">W</mi><mn id="S3.4.p4.27.m27.1.1.3" xref="S3.4.p4.27.m27.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.4.p4.27.m27.1b"><apply id="S3.4.p4.27.m27.1.1.cmml" xref="S3.4.p4.27.m27.1.1"><csymbol cd="ambiguous" id="S3.4.p4.27.m27.1.1.1.cmml" xref="S3.4.p4.27.m27.1.1">subscript</csymbol><ci id="S3.4.p4.27.m27.1.1.2.cmml" xref="S3.4.p4.27.m27.1.1.2">𝑊</ci><cn id="S3.4.p4.27.m27.1.1.3.cmml" type="integer" xref="S3.4.p4.27.m27.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.27.m27.1c">W_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.27.m27.1d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>), <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i2" title="Item (b) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(b)</span></a> (conditions on <math alttext="W_{1},\ldots,W_{\ell}" class="ltx_Math" display="inline" id="S3.4.p4.28.m28.3"><semantics id="S3.4.p4.28.m28.3a"><mrow id="S3.4.p4.28.m28.3.3.2" xref="S3.4.p4.28.m28.3.3.3.cmml"><msub id="S3.4.p4.28.m28.2.2.1.1" xref="S3.4.p4.28.m28.2.2.1.1.cmml"><mi id="S3.4.p4.28.m28.2.2.1.1.2" xref="S3.4.p4.28.m28.2.2.1.1.2.cmml">W</mi><mn id="S3.4.p4.28.m28.2.2.1.1.3" xref="S3.4.p4.28.m28.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.4.p4.28.m28.3.3.2.3" xref="S3.4.p4.28.m28.3.3.3.cmml">,</mo><mi id="S3.4.p4.28.m28.1.1" mathvariant="normal" xref="S3.4.p4.28.m28.1.1.cmml">…</mi><mo id="S3.4.p4.28.m28.3.3.2.4" xref="S3.4.p4.28.m28.3.3.3.cmml">,</mo><msub id="S3.4.p4.28.m28.3.3.2.2" xref="S3.4.p4.28.m28.3.3.2.2.cmml"><mi id="S3.4.p4.28.m28.3.3.2.2.2" xref="S3.4.p4.28.m28.3.3.2.2.2.cmml">W</mi><mi id="S3.4.p4.28.m28.3.3.2.2.3" mathvariant="normal" xref="S3.4.p4.28.m28.3.3.2.2.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.28.m28.3b"><list id="S3.4.p4.28.m28.3.3.3.cmml" xref="S3.4.p4.28.m28.3.3.2"><apply id="S3.4.p4.28.m28.2.2.1.1.cmml" xref="S3.4.p4.28.m28.2.2.1.1"><csymbol cd="ambiguous" id="S3.4.p4.28.m28.2.2.1.1.1.cmml" xref="S3.4.p4.28.m28.2.2.1.1">subscript</csymbol><ci id="S3.4.p4.28.m28.2.2.1.1.2.cmml" xref="S3.4.p4.28.m28.2.2.1.1.2">𝑊</ci><cn id="S3.4.p4.28.m28.2.2.1.1.3.cmml" type="integer" xref="S3.4.p4.28.m28.2.2.1.1.3">1</cn></apply><ci id="S3.4.p4.28.m28.1.1.cmml" xref="S3.4.p4.28.m28.1.1">…</ci><apply id="S3.4.p4.28.m28.3.3.2.2.cmml" xref="S3.4.p4.28.m28.3.3.2.2"><csymbol cd="ambiguous" id="S3.4.p4.28.m28.3.3.2.2.1.cmml" xref="S3.4.p4.28.m28.3.3.2.2">subscript</csymbol><ci id="S3.4.p4.28.m28.3.3.2.2.2.cmml" xref="S3.4.p4.28.m28.3.3.2.2.2">𝑊</ci><ci id="S3.4.p4.28.m28.3.3.2.2.3.cmml" xref="S3.4.p4.28.m28.3.3.2.2.3">ℓ</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.28.m28.3c">W_{1},\ldots,W_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.28.m28.3d">italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>), <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i3" title="Item (c) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(c)</span></a> (a condition <math alttext="W_{\ell+1}" class="ltx_Math" display="inline" id="S3.4.p4.29.m29.1"><semantics id="S3.4.p4.29.m29.1a"><msub id="S3.4.p4.29.m29.1.1" xref="S3.4.p4.29.m29.1.1.cmml"><mi id="S3.4.p4.29.m29.1.1.2" xref="S3.4.p4.29.m29.1.1.2.cmml">W</mi><mrow id="S3.4.p4.29.m29.1.1.3" xref="S3.4.p4.29.m29.1.1.3.cmml"><mi id="S3.4.p4.29.m29.1.1.3.2" mathvariant="normal" xref="S3.4.p4.29.m29.1.1.3.2.cmml">ℓ</mi><mo id="S3.4.p4.29.m29.1.1.3.1" xref="S3.4.p4.29.m29.1.1.3.1.cmml">+</mo><mn id="S3.4.p4.29.m29.1.1.3.3" xref="S3.4.p4.29.m29.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.4.p4.29.m29.1b"><apply id="S3.4.p4.29.m29.1.1.cmml" xref="S3.4.p4.29.m29.1.1"><csymbol cd="ambiguous" id="S3.4.p4.29.m29.1.1.1.cmml" xref="S3.4.p4.29.m29.1.1">subscript</csymbol><ci id="S3.4.p4.29.m29.1.1.2.cmml" xref="S3.4.p4.29.m29.1.1.2">𝑊</ci><apply id="S3.4.p4.29.m29.1.1.3.cmml" xref="S3.4.p4.29.m29.1.1.3"><plus id="S3.4.p4.29.m29.1.1.3.1.cmml" xref="S3.4.p4.29.m29.1.1.3.1"></plus><ci id="S3.4.p4.29.m29.1.1.3.2.cmml" xref="S3.4.p4.29.m29.1.1.3.2">ℓ</ci><cn id="S3.4.p4.29.m29.1.1.3.3.cmml" type="integer" xref="S3.4.p4.29.m29.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.29.m29.1c">W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.29.m29.1d">italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>), <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i4" title="Item (d) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(d)</span></a> (conditions on <math alttext="W_{0},\ldots,W_{\ell+1}" class="ltx_Math" display="inline" id="S3.4.p4.30.m30.3"><semantics id="S3.4.p4.30.m30.3a"><mrow id="S3.4.p4.30.m30.3.3.2" xref="S3.4.p4.30.m30.3.3.3.cmml"><msub id="S3.4.p4.30.m30.2.2.1.1" xref="S3.4.p4.30.m30.2.2.1.1.cmml"><mi id="S3.4.p4.30.m30.2.2.1.1.2" xref="S3.4.p4.30.m30.2.2.1.1.2.cmml">W</mi><mn id="S3.4.p4.30.m30.2.2.1.1.3" xref="S3.4.p4.30.m30.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.4.p4.30.m30.3.3.2.3" xref="S3.4.p4.30.m30.3.3.3.cmml">,</mo><mi id="S3.4.p4.30.m30.1.1" mathvariant="normal" xref="S3.4.p4.30.m30.1.1.cmml">…</mi><mo id="S3.4.p4.30.m30.3.3.2.4" xref="S3.4.p4.30.m30.3.3.3.cmml">,</mo><msub id="S3.4.p4.30.m30.3.3.2.2" xref="S3.4.p4.30.m30.3.3.2.2.cmml"><mi id="S3.4.p4.30.m30.3.3.2.2.2" xref="S3.4.p4.30.m30.3.3.2.2.2.cmml">W</mi><mrow id="S3.4.p4.30.m30.3.3.2.2.3" xref="S3.4.p4.30.m30.3.3.2.2.3.cmml"><mi id="S3.4.p4.30.m30.3.3.2.2.3.2" mathvariant="normal" xref="S3.4.p4.30.m30.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.4.p4.30.m30.3.3.2.2.3.1" xref="S3.4.p4.30.m30.3.3.2.2.3.1.cmml">+</mo><mn id="S3.4.p4.30.m30.3.3.2.2.3.3" xref="S3.4.p4.30.m30.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.30.m30.3b"><list id="S3.4.p4.30.m30.3.3.3.cmml" xref="S3.4.p4.30.m30.3.3.2"><apply id="S3.4.p4.30.m30.2.2.1.1.cmml" xref="S3.4.p4.30.m30.2.2.1.1"><csymbol cd="ambiguous" id="S3.4.p4.30.m30.2.2.1.1.1.cmml" xref="S3.4.p4.30.m30.2.2.1.1">subscript</csymbol><ci id="S3.4.p4.30.m30.2.2.1.1.2.cmml" xref="S3.4.p4.30.m30.2.2.1.1.2">𝑊</ci><cn id="S3.4.p4.30.m30.2.2.1.1.3.cmml" type="integer" xref="S3.4.p4.30.m30.2.2.1.1.3">0</cn></apply><ci id="S3.4.p4.30.m30.1.1.cmml" xref="S3.4.p4.30.m30.1.1">…</ci><apply id="S3.4.p4.30.m30.3.3.2.2.cmml" xref="S3.4.p4.30.m30.3.3.2.2"><csymbol cd="ambiguous" id="S3.4.p4.30.m30.3.3.2.2.1.cmml" xref="S3.4.p4.30.m30.3.3.2.2">subscript</csymbol><ci id="S3.4.p4.30.m30.3.3.2.2.2.cmml" xref="S3.4.p4.30.m30.3.3.2.2.2">𝑊</ci><apply id="S3.4.p4.30.m30.3.3.2.2.3.cmml" xref="S3.4.p4.30.m30.3.3.2.2.3"><plus id="S3.4.p4.30.m30.3.3.2.2.3.1.cmml" xref="S3.4.p4.30.m30.3.3.2.2.3.1"></plus><ci id="S3.4.p4.30.m30.3.3.2.2.3.2.cmml" xref="S3.4.p4.30.m30.3.3.2.2.3.2">ℓ</ci><cn id="S3.4.p4.30.m30.3.3.2.2.3.3.cmml" type="integer" xref="S3.4.p4.30.m30.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.30.m30.3c">W_{0},\ldots,W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.30.m30.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>), and <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.I1.i5" title="Item (e) ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">(e)</span></a> (a condition on <math alttext="W_{\ell+1}" class="ltx_Math" display="inline" id="S3.4.p4.31.m31.1"><semantics id="S3.4.p4.31.m31.1a"><msub id="S3.4.p4.31.m31.1.1" xref="S3.4.p4.31.m31.1.1.cmml"><mi id="S3.4.p4.31.m31.1.1.2" xref="S3.4.p4.31.m31.1.1.2.cmml">W</mi><mrow id="S3.4.p4.31.m31.1.1.3" xref="S3.4.p4.31.m31.1.1.3.cmml"><mi id="S3.4.p4.31.m31.1.1.3.2" mathvariant="normal" xref="S3.4.p4.31.m31.1.1.3.2.cmml">ℓ</mi><mo id="S3.4.p4.31.m31.1.1.3.1" xref="S3.4.p4.31.m31.1.1.3.1.cmml">+</mo><mn id="S3.4.p4.31.m31.1.1.3.3" xref="S3.4.p4.31.m31.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.4.p4.31.m31.1b"><apply id="S3.4.p4.31.m31.1.1.cmml" xref="S3.4.p4.31.m31.1.1"><csymbol cd="ambiguous" id="S3.4.p4.31.m31.1.1.1.cmml" xref="S3.4.p4.31.m31.1.1">subscript</csymbol><ci id="S3.4.p4.31.m31.1.1.2.cmml" xref="S3.4.p4.31.m31.1.1.2">𝑊</ci><apply id="S3.4.p4.31.m31.1.1.3.cmml" xref="S3.4.p4.31.m31.1.1.3"><plus id="S3.4.p4.31.m31.1.1.3.1.cmml" xref="S3.4.p4.31.m31.1.1.3.1"></plus><ci id="S3.4.p4.31.m31.1.1.3.2.cmml" xref="S3.4.p4.31.m31.1.1.3.2">ℓ</ci><cn id="S3.4.p4.31.m31.1.1.3.3.cmml" type="integer" xref="S3.4.p4.31.m31.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.31.m31.1c">W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.31.m31.1d">italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>). Thus, <math alttext="W_{0},\ldots,W_{\ell+1}" class="ltx_Math" display="inline" id="S3.4.p4.32.m32.3"><semantics id="S3.4.p4.32.m32.3a"><mrow id="S3.4.p4.32.m32.3.3.2" xref="S3.4.p4.32.m32.3.3.3.cmml"><msub id="S3.4.p4.32.m32.2.2.1.1" xref="S3.4.p4.32.m32.2.2.1.1.cmml"><mi id="S3.4.p4.32.m32.2.2.1.1.2" xref="S3.4.p4.32.m32.2.2.1.1.2.cmml">W</mi><mn id="S3.4.p4.32.m32.2.2.1.1.3" xref="S3.4.p4.32.m32.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.4.p4.32.m32.3.3.2.3" xref="S3.4.p4.32.m32.3.3.3.cmml">,</mo><mi id="S3.4.p4.32.m32.1.1" mathvariant="normal" xref="S3.4.p4.32.m32.1.1.cmml">…</mi><mo id="S3.4.p4.32.m32.3.3.2.4" xref="S3.4.p4.32.m32.3.3.3.cmml">,</mo><msub id="S3.4.p4.32.m32.3.3.2.2" xref="S3.4.p4.32.m32.3.3.2.2.cmml"><mi id="S3.4.p4.32.m32.3.3.2.2.2" xref="S3.4.p4.32.m32.3.3.2.2.2.cmml">W</mi><mrow id="S3.4.p4.32.m32.3.3.2.2.3" xref="S3.4.p4.32.m32.3.3.2.2.3.cmml"><mi id="S3.4.p4.32.m32.3.3.2.2.3.2" mathvariant="normal" xref="S3.4.p4.32.m32.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.4.p4.32.m32.3.3.2.2.3.1" xref="S3.4.p4.32.m32.3.3.2.2.3.1.cmml">+</mo><mn id="S3.4.p4.32.m32.3.3.2.2.3.3" xref="S3.4.p4.32.m32.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p4.32.m32.3b"><list id="S3.4.p4.32.m32.3.3.3.cmml" xref="S3.4.p4.32.m32.3.3.2"><apply id="S3.4.p4.32.m32.2.2.1.1.cmml" xref="S3.4.p4.32.m32.2.2.1.1"><csymbol cd="ambiguous" id="S3.4.p4.32.m32.2.2.1.1.1.cmml" xref="S3.4.p4.32.m32.2.2.1.1">subscript</csymbol><ci id="S3.4.p4.32.m32.2.2.1.1.2.cmml" xref="S3.4.p4.32.m32.2.2.1.1.2">𝑊</ci><cn id="S3.4.p4.32.m32.2.2.1.1.3.cmml" type="integer" xref="S3.4.p4.32.m32.2.2.1.1.3">0</cn></apply><ci id="S3.4.p4.32.m32.1.1.cmml" xref="S3.4.p4.32.m32.1.1">…</ci><apply id="S3.4.p4.32.m32.3.3.2.2.cmml" xref="S3.4.p4.32.m32.3.3.2.2"><csymbol cd="ambiguous" id="S3.4.p4.32.m32.3.3.2.2.1.cmml" xref="S3.4.p4.32.m32.3.3.2.2">subscript</csymbol><ci id="S3.4.p4.32.m32.3.3.2.2.2.cmml" xref="S3.4.p4.32.m32.3.3.2.2.2">𝑊</ci><apply id="S3.4.p4.32.m32.3.3.2.2.3.cmml" xref="S3.4.p4.32.m32.3.3.2.2.3"><plus id="S3.4.p4.32.m32.3.3.2.2.3.1.cmml" xref="S3.4.p4.32.m32.3.3.2.2.3.1"></plus><ci id="S3.4.p4.32.m32.3.3.2.2.3.2.cmml" xref="S3.4.p4.32.m32.3.3.2.2.3.2">ℓ</ci><cn id="S3.4.p4.32.m32.3.3.2.2.3.3.cmml" type="integer" xref="S3.4.p4.32.m32.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.32.m32.3c">W_{0},\ldots,W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.32.m32.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> is a <math alttext="W" class="ltx_Math" display="inline" id="S3.4.p4.33.m33.1"><semantics id="S3.4.p4.33.m33.1a"><mi id="S3.4.p4.33.m33.1.1" xref="S3.4.p4.33.m33.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.33.m33.1b"><ci id="S3.4.p4.33.m33.1.1.cmml" xref="S3.4.p4.33.m33.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.33.m33.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.33.m33.1d">italic_W</annotation></semantics></math>-sequence of width <math alttext="w" class="ltx_Math" display="inline" id="S3.4.p4.34.m34.1"><semantics id="S3.4.p4.34.m34.1a"><mi id="S3.4.p4.34.m34.1.1" xref="S3.4.p4.34.m34.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.34.m34.1b"><ci id="S3.4.p4.34.m34.1.1.cmml" xref="S3.4.p4.34.m34.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.34.m34.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.34.m34.1d">italic_w</annotation></semantics></math> in <math alttext="G" class="ltx_Math" display="inline" id="S3.4.p4.35.m35.1"><semantics id="S3.4.p4.35.m35.1a"><mi id="S3.4.p4.35.m35.1.1" xref="S3.4.p4.35.m35.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.4.p4.35.m35.1b"><ci id="S3.4.p4.35.m35.1.1.cmml" xref="S3.4.p4.35.m35.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p4.35.m35.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.4.p4.35.m35.1d">italic_G</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.11">Observe that, for any <math alttext="W" 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">W</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">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_W</annotation></semantics></math>-sequence <math alttext="W_{0},\ldots,W_{\ell+1}" class="ltx_Math" display="inline" id="S3.p2.2.m2.3"><semantics id="S3.p2.2.m2.3a"><mrow id="S3.p2.2.m2.3.3.2" xref="S3.p2.2.m2.3.3.3.cmml"><msub id="S3.p2.2.m2.2.2.1.1" xref="S3.p2.2.m2.2.2.1.1.cmml"><mi id="S3.p2.2.m2.2.2.1.1.2" xref="S3.p2.2.m2.2.2.1.1.2.cmml">W</mi><mn id="S3.p2.2.m2.2.2.1.1.3" xref="S3.p2.2.m2.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.p2.2.m2.3.3.2.3" xref="S3.p2.2.m2.3.3.3.cmml">,</mo><mi id="S3.p2.2.m2.1.1" mathvariant="normal" xref="S3.p2.2.m2.1.1.cmml">…</mi><mo id="S3.p2.2.m2.3.3.2.4" xref="S3.p2.2.m2.3.3.3.cmml">,</mo><msub id="S3.p2.2.m2.3.3.2.2" xref="S3.p2.2.m2.3.3.2.2.cmml"><mi id="S3.p2.2.m2.3.3.2.2.2" xref="S3.p2.2.m2.3.3.2.2.2.cmml">W</mi><mrow id="S3.p2.2.m2.3.3.2.2.3" xref="S3.p2.2.m2.3.3.2.2.3.cmml"><mi id="S3.p2.2.m2.3.3.2.2.3.2" mathvariant="normal" xref="S3.p2.2.m2.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.p2.2.m2.3.3.2.2.3.1" xref="S3.p2.2.m2.3.3.2.2.3.1.cmml">+</mo><mn id="S3.p2.2.m2.3.3.2.2.3.3" xref="S3.p2.2.m2.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.3b"><list id="S3.p2.2.m2.3.3.3.cmml" xref="S3.p2.2.m2.3.3.2"><apply id="S3.p2.2.m2.2.2.1.1.cmml" xref="S3.p2.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3.p2.2.m2.2.2.1.1.1.cmml" xref="S3.p2.2.m2.2.2.1.1">subscript</csymbol><ci id="S3.p2.2.m2.2.2.1.1.2.cmml" xref="S3.p2.2.m2.2.2.1.1.2">𝑊</ci><cn id="S3.p2.2.m2.2.2.1.1.3.cmml" type="integer" xref="S3.p2.2.m2.2.2.1.1.3">0</cn></apply><ci id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1">…</ci><apply id="S3.p2.2.m2.3.3.2.2.cmml" xref="S3.p2.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S3.p2.2.m2.3.3.2.2.1.cmml" xref="S3.p2.2.m2.3.3.2.2">subscript</csymbol><ci id="S3.p2.2.m2.3.3.2.2.2.cmml" xref="S3.p2.2.m2.3.3.2.2.2">𝑊</ci><apply id="S3.p2.2.m2.3.3.2.2.3.cmml" xref="S3.p2.2.m2.3.3.2.2.3"><plus id="S3.p2.2.m2.3.3.2.2.3.1.cmml" xref="S3.p2.2.m2.3.3.2.2.3.1"></plus><ci id="S3.p2.2.m2.3.3.2.2.3.2.cmml" xref="S3.p2.2.m2.3.3.2.2.3.2">ℓ</ci><cn id="S3.p2.2.m2.3.3.2.2.3.3.cmml" type="integer" xref="S3.p2.2.m2.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.3c">W_{0},\ldots,W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, we have the bounds <math alttext="(\ell+1)|W|\leq|W_{\ell+1}|<(\ell+2)|W|" class="ltx_Math" display="inline" id="S3.p2.3.m3.5"><semantics id="S3.p2.3.m3.5a"><mrow id="S3.p2.3.m3.5.5" xref="S3.p2.3.m3.5.5.cmml"><mrow id="S3.p2.3.m3.3.3.1" xref="S3.p2.3.m3.3.3.1.cmml"><mrow id="S3.p2.3.m3.3.3.1.1.1" xref="S3.p2.3.m3.3.3.1.1.1.1.cmml"><mo id="S3.p2.3.m3.3.3.1.1.1.2" stretchy="false" xref="S3.p2.3.m3.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.p2.3.m3.3.3.1.1.1.1" xref="S3.p2.3.m3.3.3.1.1.1.1.cmml"><mi id="S3.p2.3.m3.3.3.1.1.1.1.2" mathvariant="normal" xref="S3.p2.3.m3.3.3.1.1.1.1.2.cmml">ℓ</mi><mo id="S3.p2.3.m3.3.3.1.1.1.1.1" xref="S3.p2.3.m3.3.3.1.1.1.1.1.cmml">+</mo><mn id="S3.p2.3.m3.3.3.1.1.1.1.3" xref="S3.p2.3.m3.3.3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.3.m3.3.3.1.1.1.3" stretchy="false" xref="S3.p2.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S3.p2.3.m3.3.3.1.2" xref="S3.p2.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="S3.p2.3.m3.3.3.1.3.2" xref="S3.p2.3.m3.3.3.1.3.1.cmml"><mo id="S3.p2.3.m3.3.3.1.3.2.1" stretchy="false" xref="S3.p2.3.m3.3.3.1.3.1.1.cmml">|</mo><mi id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml">W</mi><mo id="S3.p2.3.m3.3.3.1.3.2.2" stretchy="false" xref="S3.p2.3.m3.3.3.1.3.1.1.cmml">|</mo></mrow></mrow><mo id="S3.p2.3.m3.5.5.5" xref="S3.p2.3.m3.5.5.5.cmml">≤</mo><mrow id="S3.p2.3.m3.4.4.2.1" xref="S3.p2.3.m3.4.4.2.2.cmml"><mo id="S3.p2.3.m3.4.4.2.1.2" stretchy="false" xref="S3.p2.3.m3.4.4.2.2.1.cmml">|</mo><msub id="S3.p2.3.m3.4.4.2.1.1" xref="S3.p2.3.m3.4.4.2.1.1.cmml"><mi id="S3.p2.3.m3.4.4.2.1.1.2" xref="S3.p2.3.m3.4.4.2.1.1.2.cmml">W</mi><mrow id="S3.p2.3.m3.4.4.2.1.1.3" xref="S3.p2.3.m3.4.4.2.1.1.3.cmml"><mi id="S3.p2.3.m3.4.4.2.1.1.3.2" mathvariant="normal" xref="S3.p2.3.m3.4.4.2.1.1.3.2.cmml">ℓ</mi><mo id="S3.p2.3.m3.4.4.2.1.1.3.1" xref="S3.p2.3.m3.4.4.2.1.1.3.1.cmml">+</mo><mn id="S3.p2.3.m3.4.4.2.1.1.3.3" xref="S3.p2.3.m3.4.4.2.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.p2.3.m3.4.4.2.1.3" stretchy="false" xref="S3.p2.3.m3.4.4.2.2.1.cmml">|</mo></mrow><mo id="S3.p2.3.m3.5.5.6" xref="S3.p2.3.m3.5.5.6.cmml"><</mo><mrow id="S3.p2.3.m3.5.5.3" xref="S3.p2.3.m3.5.5.3.cmml"><mrow id="S3.p2.3.m3.5.5.3.1.1" xref="S3.p2.3.m3.5.5.3.1.1.1.cmml"><mo id="S3.p2.3.m3.5.5.3.1.1.2" stretchy="false" xref="S3.p2.3.m3.5.5.3.1.1.1.cmml">(</mo><mrow id="S3.p2.3.m3.5.5.3.1.1.1" xref="S3.p2.3.m3.5.5.3.1.1.1.cmml"><mi id="S3.p2.3.m3.5.5.3.1.1.1.2" mathvariant="normal" xref="S3.p2.3.m3.5.5.3.1.1.1.2.cmml">ℓ</mi><mo id="S3.p2.3.m3.5.5.3.1.1.1.1" xref="S3.p2.3.m3.5.5.3.1.1.1.1.cmml">+</mo><mn id="S3.p2.3.m3.5.5.3.1.1.1.3" xref="S3.p2.3.m3.5.5.3.1.1.1.3.cmml">2</mn></mrow><mo id="S3.p2.3.m3.5.5.3.1.1.3" stretchy="false" xref="S3.p2.3.m3.5.5.3.1.1.1.cmml">)</mo></mrow><mo id="S3.p2.3.m3.5.5.3.2" xref="S3.p2.3.m3.5.5.3.2.cmml">⁢</mo><mrow id="S3.p2.3.m3.5.5.3.3.2" xref="S3.p2.3.m3.5.5.3.3.1.cmml"><mo id="S3.p2.3.m3.5.5.3.3.2.1" stretchy="false" xref="S3.p2.3.m3.5.5.3.3.1.1.cmml">|</mo><mi id="S3.p2.3.m3.2.2" xref="S3.p2.3.m3.2.2.cmml">W</mi><mo id="S3.p2.3.m3.5.5.3.3.2.2" stretchy="false" xref="S3.p2.3.m3.5.5.3.3.1.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.5b"><apply id="S3.p2.3.m3.5.5.cmml" xref="S3.p2.3.m3.5.5"><and id="S3.p2.3.m3.5.5a.cmml" xref="S3.p2.3.m3.5.5"></and><apply id="S3.p2.3.m3.5.5b.cmml" xref="S3.p2.3.m3.5.5"><leq id="S3.p2.3.m3.5.5.5.cmml" xref="S3.p2.3.m3.5.5.5"></leq><apply id="S3.p2.3.m3.3.3.1.cmml" xref="S3.p2.3.m3.3.3.1"><times id="S3.p2.3.m3.3.3.1.2.cmml" xref="S3.p2.3.m3.3.3.1.2"></times><apply id="S3.p2.3.m3.3.3.1.1.1.1.cmml" xref="S3.p2.3.m3.3.3.1.1.1"><plus id="S3.p2.3.m3.3.3.1.1.1.1.1.cmml" xref="S3.p2.3.m3.3.3.1.1.1.1.1"></plus><ci id="S3.p2.3.m3.3.3.1.1.1.1.2.cmml" xref="S3.p2.3.m3.3.3.1.1.1.1.2">ℓ</ci><cn id="S3.p2.3.m3.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.p2.3.m3.3.3.1.1.1.1.3">1</cn></apply><apply id="S3.p2.3.m3.3.3.1.3.1.cmml" xref="S3.p2.3.m3.3.3.1.3.2"><abs id="S3.p2.3.m3.3.3.1.3.1.1.cmml" xref="S3.p2.3.m3.3.3.1.3.2.1"></abs><ci id="S3.p2.3.m3.1.1.cmml" xref="S3.p2.3.m3.1.1">𝑊</ci></apply></apply><apply id="S3.p2.3.m3.4.4.2.2.cmml" xref="S3.p2.3.m3.4.4.2.1"><abs id="S3.p2.3.m3.4.4.2.2.1.cmml" xref="S3.p2.3.m3.4.4.2.1.2"></abs><apply id="S3.p2.3.m3.4.4.2.1.1.cmml" xref="S3.p2.3.m3.4.4.2.1.1"><csymbol cd="ambiguous" id="S3.p2.3.m3.4.4.2.1.1.1.cmml" xref="S3.p2.3.m3.4.4.2.1.1">subscript</csymbol><ci id="S3.p2.3.m3.4.4.2.1.1.2.cmml" xref="S3.p2.3.m3.4.4.2.1.1.2">𝑊</ci><apply id="S3.p2.3.m3.4.4.2.1.1.3.cmml" xref="S3.p2.3.m3.4.4.2.1.1.3"><plus id="S3.p2.3.m3.4.4.2.1.1.3.1.cmml" xref="S3.p2.3.m3.4.4.2.1.1.3.1"></plus><ci id="S3.p2.3.m3.4.4.2.1.1.3.2.cmml" xref="S3.p2.3.m3.4.4.2.1.1.3.2">ℓ</ci><cn id="S3.p2.3.m3.4.4.2.1.1.3.3.cmml" type="integer" xref="S3.p2.3.m3.4.4.2.1.1.3.3">1</cn></apply></apply></apply></apply><apply id="S3.p2.3.m3.5.5c.cmml" xref="S3.p2.3.m3.5.5"><lt id="S3.p2.3.m3.5.5.6.cmml" xref="S3.p2.3.m3.5.5.6"></lt><share href="https://arxiv.org/html/2503.17112v1#S3.p2.3.m3.4.4.2.cmml" id="S3.p2.3.m3.5.5d.cmml" xref="S3.p2.3.m3.5.5"></share><apply id="S3.p2.3.m3.5.5.3.cmml" xref="S3.p2.3.m3.5.5.3"><times id="S3.p2.3.m3.5.5.3.2.cmml" xref="S3.p2.3.m3.5.5.3.2"></times><apply id="S3.p2.3.m3.5.5.3.1.1.1.cmml" xref="S3.p2.3.m3.5.5.3.1.1"><plus id="S3.p2.3.m3.5.5.3.1.1.1.1.cmml" xref="S3.p2.3.m3.5.5.3.1.1.1.1"></plus><ci id="S3.p2.3.m3.5.5.3.1.1.1.2.cmml" xref="S3.p2.3.m3.5.5.3.1.1.1.2">ℓ</ci><cn id="S3.p2.3.m3.5.5.3.1.1.1.3.cmml" type="integer" xref="S3.p2.3.m3.5.5.3.1.1.1.3">2</cn></apply><apply id="S3.p2.3.m3.5.5.3.3.1.cmml" xref="S3.p2.3.m3.5.5.3.3.2"><abs id="S3.p2.3.m3.5.5.3.3.1.1.cmml" xref="S3.p2.3.m3.5.5.3.3.2.1"></abs><ci id="S3.p2.3.m3.2.2.cmml" xref="S3.p2.3.m3.2.2">𝑊</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.5c">(\ell+1)|W|\leq|W_{\ell+1}|<(\ell+2)|W|</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.5d">( roman_ℓ + 1 ) | italic_W | ≤ | italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT | < ( roman_ℓ + 2 ) | italic_W |</annotation></semantics></math>, so <math alttext="1/(\ell+2)<|W|/|W_{\ell+1}|\leq 1/(\ell+1)" class="ltx_Math" display="inline" id="S3.p2.4.m4.4"><semantics id="S3.p2.4.m4.4a"><mrow id="S3.p2.4.m4.4.4" xref="S3.p2.4.m4.4.4.cmml"><mrow id="S3.p2.4.m4.2.2.1" xref="S3.p2.4.m4.2.2.1.cmml"><mn id="S3.p2.4.m4.2.2.1.3" xref="S3.p2.4.m4.2.2.1.3.cmml">1</mn><mo id="S3.p2.4.m4.2.2.1.2" xref="S3.p2.4.m4.2.2.1.2.cmml">/</mo><mrow id="S3.p2.4.m4.2.2.1.1.1" xref="S3.p2.4.m4.2.2.1.1.1.1.cmml"><mo id="S3.p2.4.m4.2.2.1.1.1.2" stretchy="false" xref="S3.p2.4.m4.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.p2.4.m4.2.2.1.1.1.1" xref="S3.p2.4.m4.2.2.1.1.1.1.cmml"><mi id="S3.p2.4.m4.2.2.1.1.1.1.2" mathvariant="normal" xref="S3.p2.4.m4.2.2.1.1.1.1.2.cmml">ℓ</mi><mo id="S3.p2.4.m4.2.2.1.1.1.1.1" xref="S3.p2.4.m4.2.2.1.1.1.1.1.cmml">+</mo><mn id="S3.p2.4.m4.2.2.1.1.1.1.3" xref="S3.p2.4.m4.2.2.1.1.1.1.3.cmml">2</mn></mrow><mo id="S3.p2.4.m4.2.2.1.1.1.3" stretchy="false" xref="S3.p2.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p2.4.m4.4.4.5" xref="S3.p2.4.m4.4.4.5.cmml"><</mo><mrow id="S3.p2.4.m4.3.3.2" xref="S3.p2.4.m4.3.3.2.cmml"><mrow id="S3.p2.4.m4.3.3.2.3.2" xref="S3.p2.4.m4.3.3.2.3.1.cmml"><mo id="S3.p2.4.m4.3.3.2.3.2.1" stretchy="false" xref="S3.p2.4.m4.3.3.2.3.1.1.cmml">|</mo><mi id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml">W</mi><mo id="S3.p2.4.m4.3.3.2.3.2.2" stretchy="false" xref="S3.p2.4.m4.3.3.2.3.1.1.cmml">|</mo></mrow><mo id="S3.p2.4.m4.3.3.2.2" xref="S3.p2.4.m4.3.3.2.2.cmml">/</mo><mrow id="S3.p2.4.m4.3.3.2.1.1" xref="S3.p2.4.m4.3.3.2.1.2.cmml"><mo id="S3.p2.4.m4.3.3.2.1.1.2" stretchy="false" xref="S3.p2.4.m4.3.3.2.1.2.1.cmml">|</mo><msub id="S3.p2.4.m4.3.3.2.1.1.1" xref="S3.p2.4.m4.3.3.2.1.1.1.cmml"><mi id="S3.p2.4.m4.3.3.2.1.1.1.2" xref="S3.p2.4.m4.3.3.2.1.1.1.2.cmml">W</mi><mrow id="S3.p2.4.m4.3.3.2.1.1.1.3" xref="S3.p2.4.m4.3.3.2.1.1.1.3.cmml"><mi id="S3.p2.4.m4.3.3.2.1.1.1.3.2" mathvariant="normal" xref="S3.p2.4.m4.3.3.2.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.p2.4.m4.3.3.2.1.1.1.3.1" xref="S3.p2.4.m4.3.3.2.1.1.1.3.1.cmml">+</mo><mn id="S3.p2.4.m4.3.3.2.1.1.1.3.3" xref="S3.p2.4.m4.3.3.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.p2.4.m4.3.3.2.1.1.3" stretchy="false" xref="S3.p2.4.m4.3.3.2.1.2.1.cmml">|</mo></mrow></mrow><mo id="S3.p2.4.m4.4.4.6" xref="S3.p2.4.m4.4.4.6.cmml">≤</mo><mrow id="S3.p2.4.m4.4.4.3" xref="S3.p2.4.m4.4.4.3.cmml"><mn id="S3.p2.4.m4.4.4.3.3" xref="S3.p2.4.m4.4.4.3.3.cmml">1</mn><mo id="S3.p2.4.m4.4.4.3.2" xref="S3.p2.4.m4.4.4.3.2.cmml">/</mo><mrow id="S3.p2.4.m4.4.4.3.1.1" xref="S3.p2.4.m4.4.4.3.1.1.1.cmml"><mo id="S3.p2.4.m4.4.4.3.1.1.2" stretchy="false" xref="S3.p2.4.m4.4.4.3.1.1.1.cmml">(</mo><mrow id="S3.p2.4.m4.4.4.3.1.1.1" xref="S3.p2.4.m4.4.4.3.1.1.1.cmml"><mi id="S3.p2.4.m4.4.4.3.1.1.1.2" mathvariant="normal" xref="S3.p2.4.m4.4.4.3.1.1.1.2.cmml">ℓ</mi><mo id="S3.p2.4.m4.4.4.3.1.1.1.1" xref="S3.p2.4.m4.4.4.3.1.1.1.1.cmml">+</mo><mn id="S3.p2.4.m4.4.4.3.1.1.1.3" xref="S3.p2.4.m4.4.4.3.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.4.m4.4.4.3.1.1.3" stretchy="false" xref="S3.p2.4.m4.4.4.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.4.m4.4b"><apply id="S3.p2.4.m4.4.4.cmml" xref="S3.p2.4.m4.4.4"><and id="S3.p2.4.m4.4.4a.cmml" xref="S3.p2.4.m4.4.4"></and><apply id="S3.p2.4.m4.4.4b.cmml" xref="S3.p2.4.m4.4.4"><lt id="S3.p2.4.m4.4.4.5.cmml" xref="S3.p2.4.m4.4.4.5"></lt><apply id="S3.p2.4.m4.2.2.1.cmml" xref="S3.p2.4.m4.2.2.1"><divide id="S3.p2.4.m4.2.2.1.2.cmml" xref="S3.p2.4.m4.2.2.1.2"></divide><cn id="S3.p2.4.m4.2.2.1.3.cmml" type="integer" xref="S3.p2.4.m4.2.2.1.3">1</cn><apply id="S3.p2.4.m4.2.2.1.1.1.1.cmml" xref="S3.p2.4.m4.2.2.1.1.1"><plus id="S3.p2.4.m4.2.2.1.1.1.1.1.cmml" xref="S3.p2.4.m4.2.2.1.1.1.1.1"></plus><ci id="S3.p2.4.m4.2.2.1.1.1.1.2.cmml" xref="S3.p2.4.m4.2.2.1.1.1.1.2">ℓ</ci><cn id="S3.p2.4.m4.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.p2.4.m4.2.2.1.1.1.1.3">2</cn></apply></apply><apply id="S3.p2.4.m4.3.3.2.cmml" xref="S3.p2.4.m4.3.3.2"><divide id="S3.p2.4.m4.3.3.2.2.cmml" xref="S3.p2.4.m4.3.3.2.2"></divide><apply id="S3.p2.4.m4.3.3.2.3.1.cmml" xref="S3.p2.4.m4.3.3.2.3.2"><abs id="S3.p2.4.m4.3.3.2.3.1.1.cmml" xref="S3.p2.4.m4.3.3.2.3.2.1"></abs><ci id="S3.p2.4.m4.1.1.cmml" xref="S3.p2.4.m4.1.1">𝑊</ci></apply><apply id="S3.p2.4.m4.3.3.2.1.2.cmml" xref="S3.p2.4.m4.3.3.2.1.1"><abs id="S3.p2.4.m4.3.3.2.1.2.1.cmml" xref="S3.p2.4.m4.3.3.2.1.1.2"></abs><apply id="S3.p2.4.m4.3.3.2.1.1.1.cmml" xref="S3.p2.4.m4.3.3.2.1.1.1"><csymbol cd="ambiguous" id="S3.p2.4.m4.3.3.2.1.1.1.1.cmml" xref="S3.p2.4.m4.3.3.2.1.1.1">subscript</csymbol><ci id="S3.p2.4.m4.3.3.2.1.1.1.2.cmml" xref="S3.p2.4.m4.3.3.2.1.1.1.2">𝑊</ci><apply id="S3.p2.4.m4.3.3.2.1.1.1.3.cmml" xref="S3.p2.4.m4.3.3.2.1.1.1.3"><plus id="S3.p2.4.m4.3.3.2.1.1.1.3.1.cmml" xref="S3.p2.4.m4.3.3.2.1.1.1.3.1"></plus><ci id="S3.p2.4.m4.3.3.2.1.1.1.3.2.cmml" xref="S3.p2.4.m4.3.3.2.1.1.1.3.2">ℓ</ci><cn id="S3.p2.4.m4.3.3.2.1.1.1.3.3.cmml" type="integer" xref="S3.p2.4.m4.3.3.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply><apply id="S3.p2.4.m4.4.4c.cmml" xref="S3.p2.4.m4.4.4"><leq id="S3.p2.4.m4.4.4.6.cmml" xref="S3.p2.4.m4.4.4.6"></leq><share href="https://arxiv.org/html/2503.17112v1#S3.p2.4.m4.3.3.2.cmml" id="S3.p2.4.m4.4.4d.cmml" xref="S3.p2.4.m4.4.4"></share><apply id="S3.p2.4.m4.4.4.3.cmml" xref="S3.p2.4.m4.4.4.3"><divide id="S3.p2.4.m4.4.4.3.2.cmml" xref="S3.p2.4.m4.4.4.3.2"></divide><cn id="S3.p2.4.m4.4.4.3.3.cmml" type="integer" xref="S3.p2.4.m4.4.4.3.3">1</cn><apply id="S3.p2.4.m4.4.4.3.1.1.1.cmml" xref="S3.p2.4.m4.4.4.3.1.1"><plus id="S3.p2.4.m4.4.4.3.1.1.1.1.cmml" xref="S3.p2.4.m4.4.4.3.1.1.1.1"></plus><ci id="S3.p2.4.m4.4.4.3.1.1.1.2.cmml" xref="S3.p2.4.m4.4.4.3.1.1.1.2">ℓ</ci><cn id="S3.p2.4.m4.4.4.3.1.1.1.3.cmml" type="integer" xref="S3.p2.4.m4.4.4.3.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.m4.4c">1/(\ell+2)<|W|/|W_{\ell+1}|\leq 1/(\ell+1)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m4.4d">1 / ( roman_ℓ + 2 ) < | italic_W | / | italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT | ≤ 1 / ( roman_ℓ + 1 )</annotation></semantics></math>. The following lemma shows that, for any separation <math alttext="(A,B)" class="ltx_Math" display="inline" id="S3.p2.5.m5.2"><semantics id="S3.p2.5.m5.2a"><mrow id="S3.p2.5.m5.2.3.2" xref="S3.p2.5.m5.2.3.1.cmml"><mo id="S3.p2.5.m5.2.3.2.1" stretchy="false" xref="S3.p2.5.m5.2.3.1.cmml">(</mo><mi id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml">A</mi><mo id="S3.p2.5.m5.2.3.2.2" xref="S3.p2.5.m5.2.3.1.cmml">,</mo><mi id="S3.p2.5.m5.2.2" xref="S3.p2.5.m5.2.2.cmml">B</mi><mo id="S3.p2.5.m5.2.3.2.3" stretchy="false" xref="S3.p2.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.2b"><interval closure="open" id="S3.p2.5.m5.2.3.1.cmml" xref="S3.p2.5.m5.2.3.2"><ci id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1">𝐴</ci><ci id="S3.p2.5.m5.2.2.cmml" xref="S3.p2.5.m5.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.2d">( italic_A , italic_B )</annotation></semantics></math> of <math alttext="G[W_{\ell+1}]" class="ltx_Math" display="inline" id="S3.p2.6.m6.1"><semantics id="S3.p2.6.m6.1a"><mrow id="S3.p2.6.m6.1.1" xref="S3.p2.6.m6.1.1.cmml"><mi id="S3.p2.6.m6.1.1.3" xref="S3.p2.6.m6.1.1.3.cmml">G</mi><mo id="S3.p2.6.m6.1.1.2" xref="S3.p2.6.m6.1.1.2.cmml">⁢</mo><mrow id="S3.p2.6.m6.1.1.1.1" xref="S3.p2.6.m6.1.1.1.2.cmml"><mo id="S3.p2.6.m6.1.1.1.1.2" stretchy="false" xref="S3.p2.6.m6.1.1.1.2.1.cmml">[</mo><msub id="S3.p2.6.m6.1.1.1.1.1" xref="S3.p2.6.m6.1.1.1.1.1.cmml"><mi id="S3.p2.6.m6.1.1.1.1.1.2" xref="S3.p2.6.m6.1.1.1.1.1.2.cmml">W</mi><mrow id="S3.p2.6.m6.1.1.1.1.1.3" xref="S3.p2.6.m6.1.1.1.1.1.3.cmml"><mi id="S3.p2.6.m6.1.1.1.1.1.3.2" mathvariant="normal" xref="S3.p2.6.m6.1.1.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.p2.6.m6.1.1.1.1.1.3.1" xref="S3.p2.6.m6.1.1.1.1.1.3.1.cmml">+</mo><mn id="S3.p2.6.m6.1.1.1.1.1.3.3" xref="S3.p2.6.m6.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.p2.6.m6.1.1.1.1.3" stretchy="false" xref="S3.p2.6.m6.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.6.m6.1b"><apply id="S3.p2.6.m6.1.1.cmml" xref="S3.p2.6.m6.1.1"><times id="S3.p2.6.m6.1.1.2.cmml" xref="S3.p2.6.m6.1.1.2"></times><ci id="S3.p2.6.m6.1.1.3.cmml" xref="S3.p2.6.m6.1.1.3">𝐺</ci><apply id="S3.p2.6.m6.1.1.1.2.cmml" xref="S3.p2.6.m6.1.1.1.1"><csymbol cd="latexml" id="S3.p2.6.m6.1.1.1.2.1.cmml" xref="S3.p2.6.m6.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.p2.6.m6.1.1.1.1.1.cmml" xref="S3.p2.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p2.6.m6.1.1.1.1.1.1.cmml" xref="S3.p2.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S3.p2.6.m6.1.1.1.1.1.2.cmml" xref="S3.p2.6.m6.1.1.1.1.1.2">𝑊</ci><apply id="S3.p2.6.m6.1.1.1.1.1.3.cmml" xref="S3.p2.6.m6.1.1.1.1.1.3"><plus id="S3.p2.6.m6.1.1.1.1.1.3.1.cmml" xref="S3.p2.6.m6.1.1.1.1.1.3.1"></plus><ci id="S3.p2.6.m6.1.1.1.1.1.3.2.cmml" xref="S3.p2.6.m6.1.1.1.1.1.3.2">ℓ</ci><cn id="S3.p2.6.m6.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.p2.6.m6.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.6.m6.1c">G[W_{\ell+1}]</annotation><annotation encoding="application/x-llamapun" id="S3.p2.6.m6.1d">italic_G [ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math>, the size of the intersection of <math alttext="A\setminus B" class="ltx_Math" display="inline" id="S3.p2.7.m7.1"><semantics id="S3.p2.7.m7.1a"><mrow id="S3.p2.7.m7.1.1" xref="S3.p2.7.m7.1.1.cmml"><mi id="S3.p2.7.m7.1.1.2" xref="S3.p2.7.m7.1.1.2.cmml">A</mi><mo id="S3.p2.7.m7.1.1.1" xref="S3.p2.7.m7.1.1.1.cmml">∖</mo><mi id="S3.p2.7.m7.1.1.3" xref="S3.p2.7.m7.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.7.m7.1b"><apply id="S3.p2.7.m7.1.1.cmml" xref="S3.p2.7.m7.1.1"><setdiff id="S3.p2.7.m7.1.1.1.cmml" xref="S3.p2.7.m7.1.1.1"></setdiff><ci id="S3.p2.7.m7.1.1.2.cmml" xref="S3.p2.7.m7.1.1.2">𝐴</ci><ci id="S3.p2.7.m7.1.1.3.cmml" xref="S3.p2.7.m7.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m7.1c">A\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.p2.7.m7.1d">italic_A ∖ italic_B</annotation></semantics></math> with <math alttext="W\cup Z" class="ltx_Math" display="inline" id="S3.p2.8.m8.1"><semantics id="S3.p2.8.m8.1a"><mrow id="S3.p2.8.m8.1.1" xref="S3.p2.8.m8.1.1.cmml"><mi id="S3.p2.8.m8.1.1.2" xref="S3.p2.8.m8.1.1.2.cmml">W</mi><mo id="S3.p2.8.m8.1.1.1" xref="S3.p2.8.m8.1.1.1.cmml">∪</mo><mi id="S3.p2.8.m8.1.1.3" xref="S3.p2.8.m8.1.1.3.cmml">Z</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.8.m8.1b"><apply id="S3.p2.8.m8.1.1.cmml" xref="S3.p2.8.m8.1.1"><union id="S3.p2.8.m8.1.1.1.cmml" xref="S3.p2.8.m8.1.1.1"></union><ci id="S3.p2.8.m8.1.1.2.cmml" xref="S3.p2.8.m8.1.1.2">𝑊</ci><ci id="S3.p2.8.m8.1.1.3.cmml" xref="S3.p2.8.m8.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.8.m8.1c">W\cup Z</annotation><annotation encoding="application/x-llamapun" id="S3.p2.8.m8.1d">italic_W ∪ italic_Z</annotation></semantics></math> is bounded by the order <math alttext="|A\cap B|" class="ltx_Math" display="inline" id="S3.p2.9.m9.1"><semantics id="S3.p2.9.m9.1a"><mrow id="S3.p2.9.m9.1.1.1" xref="S3.p2.9.m9.1.1.2.cmml"><mo id="S3.p2.9.m9.1.1.1.2" stretchy="false" xref="S3.p2.9.m9.1.1.2.1.cmml">|</mo><mrow id="S3.p2.9.m9.1.1.1.1" xref="S3.p2.9.m9.1.1.1.1.cmml"><mi id="S3.p2.9.m9.1.1.1.1.2" xref="S3.p2.9.m9.1.1.1.1.2.cmml">A</mi><mo id="S3.p2.9.m9.1.1.1.1.1" xref="S3.p2.9.m9.1.1.1.1.1.cmml">∩</mo><mi id="S3.p2.9.m9.1.1.1.1.3" xref="S3.p2.9.m9.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.p2.9.m9.1.1.1.3" stretchy="false" xref="S3.p2.9.m9.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.9.m9.1b"><apply id="S3.p2.9.m9.1.1.2.cmml" xref="S3.p2.9.m9.1.1.1"><abs id="S3.p2.9.m9.1.1.2.1.cmml" xref="S3.p2.9.m9.1.1.1.2"></abs><apply id="S3.p2.9.m9.1.1.1.1.cmml" xref="S3.p2.9.m9.1.1.1.1"><intersect id="S3.p2.9.m9.1.1.1.1.1.cmml" xref="S3.p2.9.m9.1.1.1.1.1"></intersect><ci id="S3.p2.9.m9.1.1.1.1.2.cmml" xref="S3.p2.9.m9.1.1.1.1.2">𝐴</ci><ci id="S3.p2.9.m9.1.1.1.1.3.cmml" xref="S3.p2.9.m9.1.1.1.1.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.9.m9.1c">|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.p2.9.m9.1d">| italic_A ∩ italic_B |</annotation></semantics></math> of <math alttext="(A,B)" class="ltx_Math" display="inline" id="S3.p2.10.m10.2"><semantics id="S3.p2.10.m10.2a"><mrow id="S3.p2.10.m10.2.3.2" xref="S3.p2.10.m10.2.3.1.cmml"><mo id="S3.p2.10.m10.2.3.2.1" stretchy="false" xref="S3.p2.10.m10.2.3.1.cmml">(</mo><mi id="S3.p2.10.m10.1.1" xref="S3.p2.10.m10.1.1.cmml">A</mi><mo id="S3.p2.10.m10.2.3.2.2" xref="S3.p2.10.m10.2.3.1.cmml">,</mo><mi id="S3.p2.10.m10.2.2" xref="S3.p2.10.m10.2.2.cmml">B</mi><mo id="S3.p2.10.m10.2.3.2.3" stretchy="false" xref="S3.p2.10.m10.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.10.m10.2b"><interval closure="open" id="S3.p2.10.m10.2.3.1.cmml" xref="S3.p2.10.m10.2.3.2"><ci id="S3.p2.10.m10.1.1.cmml" xref="S3.p2.10.m10.1.1">𝐴</ci><ci id="S3.p2.10.m10.2.2.cmml" xref="S3.p2.10.m10.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.10.m10.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.10.m10.2d">( italic_A , italic_B )</annotation></semantics></math> and the ratio <math alttext="|A\setminus B|\cdot|W|/|W_{\ell+1}|" class="ltx_Math" display="inline" id="S3.p2.11.m11.3"><semantics id="S3.p2.11.m11.3a"><mrow id="S3.p2.11.m11.3.3" xref="S3.p2.11.m11.3.3.cmml"><mrow id="S3.p2.11.m11.2.2.1" xref="S3.p2.11.m11.2.2.1.cmml"><mrow id="S3.p2.11.m11.2.2.1.1.1" xref="S3.p2.11.m11.2.2.1.1.2.cmml"><mo id="S3.p2.11.m11.2.2.1.1.1.2" stretchy="false" xref="S3.p2.11.m11.2.2.1.1.2.1.cmml">|</mo><mrow id="S3.p2.11.m11.2.2.1.1.1.1" xref="S3.p2.11.m11.2.2.1.1.1.1.cmml"><mi id="S3.p2.11.m11.2.2.1.1.1.1.2" xref="S3.p2.11.m11.2.2.1.1.1.1.2.cmml">A</mi><mo id="S3.p2.11.m11.2.2.1.1.1.1.1" xref="S3.p2.11.m11.2.2.1.1.1.1.1.cmml">∖</mo><mi id="S3.p2.11.m11.2.2.1.1.1.1.3" xref="S3.p2.11.m11.2.2.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.p2.11.m11.2.2.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.p2.11.m11.2.2.1.1.2.1.cmml">|</mo></mrow><mo id="S3.p2.11.m11.2.2.1.2" rspace="0.222em" xref="S3.p2.11.m11.2.2.1.2.cmml">⋅</mo><mrow id="S3.p2.11.m11.2.2.1.3.2" xref="S3.p2.11.m11.2.2.1.3.1.cmml"><mo id="S3.p2.11.m11.2.2.1.3.2.1" stretchy="false" xref="S3.p2.11.m11.2.2.1.3.1.1.cmml">|</mo><mi id="S3.p2.11.m11.1.1" xref="S3.p2.11.m11.1.1.cmml">W</mi><mo id="S3.p2.11.m11.2.2.1.3.2.2" stretchy="false" xref="S3.p2.11.m11.2.2.1.3.1.1.cmml">|</mo></mrow></mrow><mo id="S3.p2.11.m11.3.3.3" xref="S3.p2.11.m11.3.3.3.cmml">/</mo><mrow id="S3.p2.11.m11.3.3.2.1" xref="S3.p2.11.m11.3.3.2.2.cmml"><mo id="S3.p2.11.m11.3.3.2.1.2" stretchy="false" xref="S3.p2.11.m11.3.3.2.2.1.cmml">|</mo><msub id="S3.p2.11.m11.3.3.2.1.1" xref="S3.p2.11.m11.3.3.2.1.1.cmml"><mi id="S3.p2.11.m11.3.3.2.1.1.2" xref="S3.p2.11.m11.3.3.2.1.1.2.cmml">W</mi><mrow id="S3.p2.11.m11.3.3.2.1.1.3" xref="S3.p2.11.m11.3.3.2.1.1.3.cmml"><mi id="S3.p2.11.m11.3.3.2.1.1.3.2" mathvariant="normal" xref="S3.p2.11.m11.3.3.2.1.1.3.2.cmml">ℓ</mi><mo id="S3.p2.11.m11.3.3.2.1.1.3.1" xref="S3.p2.11.m11.3.3.2.1.1.3.1.cmml">+</mo><mn id="S3.p2.11.m11.3.3.2.1.1.3.3" xref="S3.p2.11.m11.3.3.2.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.p2.11.m11.3.3.2.1.3" stretchy="false" xref="S3.p2.11.m11.3.3.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.11.m11.3b"><apply id="S3.p2.11.m11.3.3.cmml" xref="S3.p2.11.m11.3.3"><divide id="S3.p2.11.m11.3.3.3.cmml" xref="S3.p2.11.m11.3.3.3"></divide><apply id="S3.p2.11.m11.2.2.1.cmml" xref="S3.p2.11.m11.2.2.1"><ci id="S3.p2.11.m11.2.2.1.2.cmml" xref="S3.p2.11.m11.2.2.1.2">⋅</ci><apply id="S3.p2.11.m11.2.2.1.1.2.cmml" xref="S3.p2.11.m11.2.2.1.1.1"><abs id="S3.p2.11.m11.2.2.1.1.2.1.cmml" xref="S3.p2.11.m11.2.2.1.1.1.2"></abs><apply id="S3.p2.11.m11.2.2.1.1.1.1.cmml" xref="S3.p2.11.m11.2.2.1.1.1.1"><setdiff id="S3.p2.11.m11.2.2.1.1.1.1.1.cmml" xref="S3.p2.11.m11.2.2.1.1.1.1.1"></setdiff><ci id="S3.p2.11.m11.2.2.1.1.1.1.2.cmml" xref="S3.p2.11.m11.2.2.1.1.1.1.2">𝐴</ci><ci id="S3.p2.11.m11.2.2.1.1.1.1.3.cmml" xref="S3.p2.11.m11.2.2.1.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.p2.11.m11.2.2.1.3.1.cmml" xref="S3.p2.11.m11.2.2.1.3.2"><abs id="S3.p2.11.m11.2.2.1.3.1.1.cmml" xref="S3.p2.11.m11.2.2.1.3.2.1"></abs><ci id="S3.p2.11.m11.1.1.cmml" xref="S3.p2.11.m11.1.1">𝑊</ci></apply></apply><apply id="S3.p2.11.m11.3.3.2.2.cmml" xref="S3.p2.11.m11.3.3.2.1"><abs id="S3.p2.11.m11.3.3.2.2.1.cmml" xref="S3.p2.11.m11.3.3.2.1.2"></abs><apply id="S3.p2.11.m11.3.3.2.1.1.cmml" xref="S3.p2.11.m11.3.3.2.1.1"><csymbol cd="ambiguous" id="S3.p2.11.m11.3.3.2.1.1.1.cmml" xref="S3.p2.11.m11.3.3.2.1.1">subscript</csymbol><ci id="S3.p2.11.m11.3.3.2.1.1.2.cmml" xref="S3.p2.11.m11.3.3.2.1.1.2">𝑊</ci><apply id="S3.p2.11.m11.3.3.2.1.1.3.cmml" xref="S3.p2.11.m11.3.3.2.1.1.3"><plus id="S3.p2.11.m11.3.3.2.1.1.3.1.cmml" xref="S3.p2.11.m11.3.3.2.1.1.3.1"></plus><ci id="S3.p2.11.m11.3.3.2.1.1.3.2.cmml" xref="S3.p2.11.m11.3.3.2.1.1.3.2">ℓ</ci><cn id="S3.p2.11.m11.3.3.2.1.1.3.3.cmml" type="integer" xref="S3.p2.11.m11.3.3.2.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.11.m11.3c">|A\setminus B|\cdot|W|/|W_{\ell+1}|</annotation><annotation encoding="application/x-llamapun" id="S3.p2.11.m11.3d">| italic_A ∖ italic_B | ⋅ | italic_W | / | italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT |</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="Thmthm7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm7.1.1.1">Lemma 7</span></span><span class="ltx_text ltx_font_bold" id="Thmthm7.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm7.p1"> <p class="ltx_p" id="Thmthm7.p1.18"><span class="ltx_text ltx_font_italic" id="Thmthm7.p1.18.18">Let <math alttext="G" class="ltx_Math" display="inline" id="Thmthm7.p1.1.1.m1.1"><semantics id="Thmthm7.p1.1.1.m1.1a"><mi id="Thmthm7.p1.1.1.m1.1.1" xref="Thmthm7.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.1.1.m1.1b"><ci id="Thmthm7.p1.1.1.m1.1.1.cmml" xref="Thmthm7.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be a graph, let <math alttext="W\subseteq V(G)" class="ltx_Math" display="inline" id="Thmthm7.p1.2.2.m2.1"><semantics id="Thmthm7.p1.2.2.m2.1a"><mrow id="Thmthm7.p1.2.2.m2.1.2" xref="Thmthm7.p1.2.2.m2.1.2.cmml"><mi id="Thmthm7.p1.2.2.m2.1.2.2" xref="Thmthm7.p1.2.2.m2.1.2.2.cmml">W</mi><mo id="Thmthm7.p1.2.2.m2.1.2.1" xref="Thmthm7.p1.2.2.m2.1.2.1.cmml">⊆</mo><mrow id="Thmthm7.p1.2.2.m2.1.2.3" xref="Thmthm7.p1.2.2.m2.1.2.3.cmml"><mi id="Thmthm7.p1.2.2.m2.1.2.3.2" xref="Thmthm7.p1.2.2.m2.1.2.3.2.cmml">V</mi><mo id="Thmthm7.p1.2.2.m2.1.2.3.1" xref="Thmthm7.p1.2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmthm7.p1.2.2.m2.1.2.3.3.2" xref="Thmthm7.p1.2.2.m2.1.2.3.cmml"><mo id="Thmthm7.p1.2.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmthm7.p1.2.2.m2.1.2.3.cmml">(</mo><mi id="Thmthm7.p1.2.2.m2.1.1" xref="Thmthm7.p1.2.2.m2.1.1.cmml">G</mi><mo id="Thmthm7.p1.2.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmthm7.p1.2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.2.2.m2.1b"><apply id="Thmthm7.p1.2.2.m2.1.2.cmml" xref="Thmthm7.p1.2.2.m2.1.2"><subset id="Thmthm7.p1.2.2.m2.1.2.1.cmml" xref="Thmthm7.p1.2.2.m2.1.2.1"></subset><ci id="Thmthm7.p1.2.2.m2.1.2.2.cmml" xref="Thmthm7.p1.2.2.m2.1.2.2">𝑊</ci><apply id="Thmthm7.p1.2.2.m2.1.2.3.cmml" xref="Thmthm7.p1.2.2.m2.1.2.3"><times id="Thmthm7.p1.2.2.m2.1.2.3.1.cmml" xref="Thmthm7.p1.2.2.m2.1.2.3.1"></times><ci id="Thmthm7.p1.2.2.m2.1.2.3.2.cmml" xref="Thmthm7.p1.2.2.m2.1.2.3.2">𝑉</ci><ci id="Thmthm7.p1.2.2.m2.1.1.cmml" xref="Thmthm7.p1.2.2.m2.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.2.2.m2.1c">W\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.2.2.m2.1d">italic_W ⊆ italic_V ( italic_G )</annotation></semantics></math>, let <math alttext="W_{0},\ldots,W_{\ell+1}" class="ltx_Math" display="inline" id="Thmthm7.p1.3.3.m3.3"><semantics id="Thmthm7.p1.3.3.m3.3a"><mrow id="Thmthm7.p1.3.3.m3.3.3.2" xref="Thmthm7.p1.3.3.m3.3.3.3.cmml"><msub id="Thmthm7.p1.3.3.m3.2.2.1.1" xref="Thmthm7.p1.3.3.m3.2.2.1.1.cmml"><mi id="Thmthm7.p1.3.3.m3.2.2.1.1.2" xref="Thmthm7.p1.3.3.m3.2.2.1.1.2.cmml">W</mi><mn id="Thmthm7.p1.3.3.m3.2.2.1.1.3" xref="Thmthm7.p1.3.3.m3.2.2.1.1.3.cmml">0</mn></msub><mo id="Thmthm7.p1.3.3.m3.3.3.2.3" xref="Thmthm7.p1.3.3.m3.3.3.3.cmml">,</mo><mi id="Thmthm7.p1.3.3.m3.1.1" mathvariant="normal" xref="Thmthm7.p1.3.3.m3.1.1.cmml">…</mi><mo id="Thmthm7.p1.3.3.m3.3.3.2.4" xref="Thmthm7.p1.3.3.m3.3.3.3.cmml">,</mo><msub id="Thmthm7.p1.3.3.m3.3.3.2.2" xref="Thmthm7.p1.3.3.m3.3.3.2.2.cmml"><mi id="Thmthm7.p1.3.3.m3.3.3.2.2.2" xref="Thmthm7.p1.3.3.m3.3.3.2.2.2.cmml">W</mi><mrow id="Thmthm7.p1.3.3.m3.3.3.2.2.3" xref="Thmthm7.p1.3.3.m3.3.3.2.2.3.cmml"><mi id="Thmthm7.p1.3.3.m3.3.3.2.2.3.2" mathvariant="normal" xref="Thmthm7.p1.3.3.m3.3.3.2.2.3.2.cmml">ℓ</mi><mo id="Thmthm7.p1.3.3.m3.3.3.2.2.3.1" xref="Thmthm7.p1.3.3.m3.3.3.2.2.3.1.cmml">+</mo><mn id="Thmthm7.p1.3.3.m3.3.3.2.2.3.3" xref="Thmthm7.p1.3.3.m3.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.3.3.m3.3b"><list id="Thmthm7.p1.3.3.m3.3.3.3.cmml" xref="Thmthm7.p1.3.3.m3.3.3.2"><apply id="Thmthm7.p1.3.3.m3.2.2.1.1.cmml" xref="Thmthm7.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="Thmthm7.p1.3.3.m3.2.2.1.1.1.cmml" xref="Thmthm7.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="Thmthm7.p1.3.3.m3.2.2.1.1.2.cmml" xref="Thmthm7.p1.3.3.m3.2.2.1.1.2">𝑊</ci><cn id="Thmthm7.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="Thmthm7.p1.3.3.m3.2.2.1.1.3">0</cn></apply><ci id="Thmthm7.p1.3.3.m3.1.1.cmml" xref="Thmthm7.p1.3.3.m3.1.1">…</ci><apply id="Thmthm7.p1.3.3.m3.3.3.2.2.cmml" xref="Thmthm7.p1.3.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="Thmthm7.p1.3.3.m3.3.3.2.2.1.cmml" xref="Thmthm7.p1.3.3.m3.3.3.2.2">subscript</csymbol><ci id="Thmthm7.p1.3.3.m3.3.3.2.2.2.cmml" xref="Thmthm7.p1.3.3.m3.3.3.2.2.2">𝑊</ci><apply id="Thmthm7.p1.3.3.m3.3.3.2.2.3.cmml" xref="Thmthm7.p1.3.3.m3.3.3.2.2.3"><plus id="Thmthm7.p1.3.3.m3.3.3.2.2.3.1.cmml" xref="Thmthm7.p1.3.3.m3.3.3.2.2.3.1"></plus><ci id="Thmthm7.p1.3.3.m3.3.3.2.2.3.2.cmml" xref="Thmthm7.p1.3.3.m3.3.3.2.2.3.2">ℓ</ci><cn id="Thmthm7.p1.3.3.m3.3.3.2.2.3.3.cmml" type="integer" xref="Thmthm7.p1.3.3.m3.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.3.3.m3.3c">W_{0},\ldots,W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.3.3.m3.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> be a <math alttext="W" class="ltx_Math" display="inline" id="Thmthm7.p1.4.4.m4.1"><semantics id="Thmthm7.p1.4.4.m4.1a"><mi id="Thmthm7.p1.4.4.m4.1.1" xref="Thmthm7.p1.4.4.m4.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.4.4.m4.1b"><ci id="Thmthm7.p1.4.4.m4.1.1.cmml" xref="Thmthm7.p1.4.4.m4.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.4.4.m4.1c">W</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.4.4.m4.1d">italic_W</annotation></semantics></math>-sequence of width <math alttext="|W|" class="ltx_Math" display="inline" id="Thmthm7.p1.5.5.m5.1"><semantics id="Thmthm7.p1.5.5.m5.1a"><mrow id="Thmthm7.p1.5.5.m5.1.2.2" xref="Thmthm7.p1.5.5.m5.1.2.1.cmml"><mo id="Thmthm7.p1.5.5.m5.1.2.2.1" stretchy="false" xref="Thmthm7.p1.5.5.m5.1.2.1.1.cmml">|</mo><mi id="Thmthm7.p1.5.5.m5.1.1" xref="Thmthm7.p1.5.5.m5.1.1.cmml">W</mi><mo id="Thmthm7.p1.5.5.m5.1.2.2.2" stretchy="false" xref="Thmthm7.p1.5.5.m5.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.5.5.m5.1b"><apply id="Thmthm7.p1.5.5.m5.1.2.1.cmml" xref="Thmthm7.p1.5.5.m5.1.2.2"><abs id="Thmthm7.p1.5.5.m5.1.2.1.1.cmml" xref="Thmthm7.p1.5.5.m5.1.2.2.1"></abs><ci id="Thmthm7.p1.5.5.m5.1.1.cmml" xref="Thmthm7.p1.5.5.m5.1.1">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.5.5.m5.1c">|W|</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.5.5.m5.1d">| italic_W |</annotation></semantics></math> in <math alttext="G" class="ltx_Math" display="inline" id="Thmthm7.p1.6.6.m6.1"><semantics id="Thmthm7.p1.6.6.m6.1a"><mi id="Thmthm7.p1.6.6.m6.1.1" xref="Thmthm7.p1.6.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.6.6.m6.1b"><ci id="Thmthm7.p1.6.6.m6.1.1.cmml" xref="Thmthm7.p1.6.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.6.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.6.6.m6.1d">italic_G</annotation></semantics></math> with <math alttext="\ell\geq 1" class="ltx_Math" display="inline" id="Thmthm7.p1.7.7.m7.1"><semantics id="Thmthm7.p1.7.7.m7.1a"><mrow id="Thmthm7.p1.7.7.m7.1.1" xref="Thmthm7.p1.7.7.m7.1.1.cmml"><mi id="Thmthm7.p1.7.7.m7.1.1.2" mathvariant="normal" xref="Thmthm7.p1.7.7.m7.1.1.2.cmml">ℓ</mi><mo id="Thmthm7.p1.7.7.m7.1.1.1" xref="Thmthm7.p1.7.7.m7.1.1.1.cmml">≥</mo><mn id="Thmthm7.p1.7.7.m7.1.1.3" xref="Thmthm7.p1.7.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.7.7.m7.1b"><apply id="Thmthm7.p1.7.7.m7.1.1.cmml" xref="Thmthm7.p1.7.7.m7.1.1"><geq id="Thmthm7.p1.7.7.m7.1.1.1.cmml" xref="Thmthm7.p1.7.7.m7.1.1.1"></geq><ci id="Thmthm7.p1.7.7.m7.1.1.2.cmml" xref="Thmthm7.p1.7.7.m7.1.1.2">ℓ</ci><cn id="Thmthm7.p1.7.7.m7.1.1.3.cmml" type="integer" xref="Thmthm7.p1.7.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.7.7.m7.1c">\ell\geq 1</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.7.7.m7.1d">roman_ℓ ≥ 1</annotation></semantics></math>, let <math alttext="\Delta_{\ell+1}:=W_{\ell+1}\setminus W_{\ell}" class="ltx_Math" display="inline" id="Thmthm7.p1.8.8.m8.1"><semantics id="Thmthm7.p1.8.8.m8.1a"><mrow id="Thmthm7.p1.8.8.m8.1.1" xref="Thmthm7.p1.8.8.m8.1.1.cmml"><msub id="Thmthm7.p1.8.8.m8.1.1.2" xref="Thmthm7.p1.8.8.m8.1.1.2.cmml"><mi id="Thmthm7.p1.8.8.m8.1.1.2.2" mathvariant="normal" xref="Thmthm7.p1.8.8.m8.1.1.2.2.cmml">Δ</mi><mrow id="Thmthm7.p1.8.8.m8.1.1.2.3" xref="Thmthm7.p1.8.8.m8.1.1.2.3.cmml"><mi id="Thmthm7.p1.8.8.m8.1.1.2.3.2" mathvariant="normal" xref="Thmthm7.p1.8.8.m8.1.1.2.3.2.cmml">ℓ</mi><mo id="Thmthm7.p1.8.8.m8.1.1.2.3.1" xref="Thmthm7.p1.8.8.m8.1.1.2.3.1.cmml">+</mo><mn id="Thmthm7.p1.8.8.m8.1.1.2.3.3" xref="Thmthm7.p1.8.8.m8.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="Thmthm7.p1.8.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="Thmthm7.p1.8.8.m8.1.1.1.cmml">:=</mo><mrow id="Thmthm7.p1.8.8.m8.1.1.3" xref="Thmthm7.p1.8.8.m8.1.1.3.cmml"><msub id="Thmthm7.p1.8.8.m8.1.1.3.2" xref="Thmthm7.p1.8.8.m8.1.1.3.2.cmml"><mi id="Thmthm7.p1.8.8.m8.1.1.3.2.2" xref="Thmthm7.p1.8.8.m8.1.1.3.2.2.cmml">W</mi><mrow id="Thmthm7.p1.8.8.m8.1.1.3.2.3" xref="Thmthm7.p1.8.8.m8.1.1.3.2.3.cmml"><mi id="Thmthm7.p1.8.8.m8.1.1.3.2.3.2" mathvariant="normal" xref="Thmthm7.p1.8.8.m8.1.1.3.2.3.2.cmml">ℓ</mi><mo id="Thmthm7.p1.8.8.m8.1.1.3.2.3.1" xref="Thmthm7.p1.8.8.m8.1.1.3.2.3.1.cmml">+</mo><mn id="Thmthm7.p1.8.8.m8.1.1.3.2.3.3" xref="Thmthm7.p1.8.8.m8.1.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="Thmthm7.p1.8.8.m8.1.1.3.1" xref="Thmthm7.p1.8.8.m8.1.1.3.1.cmml">∖</mo><msub id="Thmthm7.p1.8.8.m8.1.1.3.3" xref="Thmthm7.p1.8.8.m8.1.1.3.3.cmml"><mi id="Thmthm7.p1.8.8.m8.1.1.3.3.2" xref="Thmthm7.p1.8.8.m8.1.1.3.3.2.cmml">W</mi><mi id="Thmthm7.p1.8.8.m8.1.1.3.3.3" mathvariant="normal" xref="Thmthm7.p1.8.8.m8.1.1.3.3.3.cmml">ℓ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.8.8.m8.1b"><apply id="Thmthm7.p1.8.8.m8.1.1.cmml" xref="Thmthm7.p1.8.8.m8.1.1"><csymbol cd="latexml" id="Thmthm7.p1.8.8.m8.1.1.1.cmml" xref="Thmthm7.p1.8.8.m8.1.1.1">assign</csymbol><apply id="Thmthm7.p1.8.8.m8.1.1.2.cmml" xref="Thmthm7.p1.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="Thmthm7.p1.8.8.m8.1.1.2.1.cmml" xref="Thmthm7.p1.8.8.m8.1.1.2">subscript</csymbol><ci id="Thmthm7.p1.8.8.m8.1.1.2.2.cmml" xref="Thmthm7.p1.8.8.m8.1.1.2.2">Δ</ci><apply id="Thmthm7.p1.8.8.m8.1.1.2.3.cmml" xref="Thmthm7.p1.8.8.m8.1.1.2.3"><plus id="Thmthm7.p1.8.8.m8.1.1.2.3.1.cmml" xref="Thmthm7.p1.8.8.m8.1.1.2.3.1"></plus><ci id="Thmthm7.p1.8.8.m8.1.1.2.3.2.cmml" xref="Thmthm7.p1.8.8.m8.1.1.2.3.2">ℓ</ci><cn id="Thmthm7.p1.8.8.m8.1.1.2.3.3.cmml" type="integer" xref="Thmthm7.p1.8.8.m8.1.1.2.3.3">1</cn></apply></apply><apply id="Thmthm7.p1.8.8.m8.1.1.3.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3"><setdiff id="Thmthm7.p1.8.8.m8.1.1.3.1.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.1"></setdiff><apply id="Thmthm7.p1.8.8.m8.1.1.3.2.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="Thmthm7.p1.8.8.m8.1.1.3.2.1.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.2">subscript</csymbol><ci id="Thmthm7.p1.8.8.m8.1.1.3.2.2.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.2.2">𝑊</ci><apply id="Thmthm7.p1.8.8.m8.1.1.3.2.3.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.2.3"><plus id="Thmthm7.p1.8.8.m8.1.1.3.2.3.1.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.2.3.1"></plus><ci id="Thmthm7.p1.8.8.m8.1.1.3.2.3.2.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.2.3.2">ℓ</ci><cn id="Thmthm7.p1.8.8.m8.1.1.3.2.3.3.cmml" type="integer" xref="Thmthm7.p1.8.8.m8.1.1.3.2.3.3">1</cn></apply></apply><apply id="Thmthm7.p1.8.8.m8.1.1.3.3.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="Thmthm7.p1.8.8.m8.1.1.3.3.1.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.3">subscript</csymbol><ci id="Thmthm7.p1.8.8.m8.1.1.3.3.2.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.3.2">𝑊</ci><ci id="Thmthm7.p1.8.8.m8.1.1.3.3.3.cmml" xref="Thmthm7.p1.8.8.m8.1.1.3.3.3">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.8.8.m8.1c">\Delta_{\ell+1}:=W_{\ell+1}\setminus W_{\ell}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.8.8.m8.1d">roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT := italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∖ italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>, and let <math alttext="Z\subseteq W_{\ell+1}" class="ltx_Math" display="inline" id="Thmthm7.p1.9.9.m9.1"><semantics id="Thmthm7.p1.9.9.m9.1a"><mrow id="Thmthm7.p1.9.9.m9.1.1" xref="Thmthm7.p1.9.9.m9.1.1.cmml"><mi id="Thmthm7.p1.9.9.m9.1.1.2" xref="Thmthm7.p1.9.9.m9.1.1.2.cmml">Z</mi><mo id="Thmthm7.p1.9.9.m9.1.1.1" xref="Thmthm7.p1.9.9.m9.1.1.1.cmml">⊆</mo><msub id="Thmthm7.p1.9.9.m9.1.1.3" xref="Thmthm7.p1.9.9.m9.1.1.3.cmml"><mi id="Thmthm7.p1.9.9.m9.1.1.3.2" xref="Thmthm7.p1.9.9.m9.1.1.3.2.cmml">W</mi><mrow id="Thmthm7.p1.9.9.m9.1.1.3.3" xref="Thmthm7.p1.9.9.m9.1.1.3.3.cmml"><mi id="Thmthm7.p1.9.9.m9.1.1.3.3.2" mathvariant="normal" xref="Thmthm7.p1.9.9.m9.1.1.3.3.2.cmml">ℓ</mi><mo id="Thmthm7.p1.9.9.m9.1.1.3.3.1" xref="Thmthm7.p1.9.9.m9.1.1.3.3.1.cmml">+</mo><mn id="Thmthm7.p1.9.9.m9.1.1.3.3.3" xref="Thmthm7.p1.9.9.m9.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.9.9.m9.1b"><apply id="Thmthm7.p1.9.9.m9.1.1.cmml" xref="Thmthm7.p1.9.9.m9.1.1"><subset id="Thmthm7.p1.9.9.m9.1.1.1.cmml" xref="Thmthm7.p1.9.9.m9.1.1.1"></subset><ci id="Thmthm7.p1.9.9.m9.1.1.2.cmml" xref="Thmthm7.p1.9.9.m9.1.1.2">𝑍</ci><apply id="Thmthm7.p1.9.9.m9.1.1.3.cmml" xref="Thmthm7.p1.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="Thmthm7.p1.9.9.m9.1.1.3.1.cmml" xref="Thmthm7.p1.9.9.m9.1.1.3">subscript</csymbol><ci id="Thmthm7.p1.9.9.m9.1.1.3.2.cmml" xref="Thmthm7.p1.9.9.m9.1.1.3.2">𝑊</ci><apply id="Thmthm7.p1.9.9.m9.1.1.3.3.cmml" xref="Thmthm7.p1.9.9.m9.1.1.3.3"><plus id="Thmthm7.p1.9.9.m9.1.1.3.3.1.cmml" xref="Thmthm7.p1.9.9.m9.1.1.3.3.1"></plus><ci id="Thmthm7.p1.9.9.m9.1.1.3.3.2.cmml" xref="Thmthm7.p1.9.9.m9.1.1.3.3.2">ℓ</ci><cn id="Thmthm7.p1.9.9.m9.1.1.3.3.3.cmml" type="integer" xref="Thmthm7.p1.9.9.m9.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.9.9.m9.1c">Z\subseteq W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.9.9.m9.1d">italic_Z ⊆ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="|Z|=|\Delta_{\ell+1}|" class="ltx_Math" display="inline" id="Thmthm7.p1.10.10.m10.2"><semantics id="Thmthm7.p1.10.10.m10.2a"><mrow id="Thmthm7.p1.10.10.m10.2.2" xref="Thmthm7.p1.10.10.m10.2.2.cmml"><mrow id="Thmthm7.p1.10.10.m10.2.2.3.2" xref="Thmthm7.p1.10.10.m10.2.2.3.1.cmml"><mo id="Thmthm7.p1.10.10.m10.2.2.3.2.1" stretchy="false" xref="Thmthm7.p1.10.10.m10.2.2.3.1.1.cmml">|</mo><mi id="Thmthm7.p1.10.10.m10.1.1" xref="Thmthm7.p1.10.10.m10.1.1.cmml">Z</mi><mo id="Thmthm7.p1.10.10.m10.2.2.3.2.2" stretchy="false" xref="Thmthm7.p1.10.10.m10.2.2.3.1.1.cmml">|</mo></mrow><mo id="Thmthm7.p1.10.10.m10.2.2.2" xref="Thmthm7.p1.10.10.m10.2.2.2.cmml">=</mo><mrow id="Thmthm7.p1.10.10.m10.2.2.1.1" xref="Thmthm7.p1.10.10.m10.2.2.1.2.cmml"><mo id="Thmthm7.p1.10.10.m10.2.2.1.1.2" stretchy="false" xref="Thmthm7.p1.10.10.m10.2.2.1.2.1.cmml">|</mo><msub id="Thmthm7.p1.10.10.m10.2.2.1.1.1" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.cmml"><mi id="Thmthm7.p1.10.10.m10.2.2.1.1.1.2" mathvariant="normal" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.2.cmml">Δ</mi><mrow id="Thmthm7.p1.10.10.m10.2.2.1.1.1.3" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.cmml"><mi id="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.2" mathvariant="normal" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.2.cmml">ℓ</mi><mo id="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.1" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.1.cmml">+</mo><mn id="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.3" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmthm7.p1.10.10.m10.2.2.1.1.3" stretchy="false" xref="Thmthm7.p1.10.10.m10.2.2.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.10.10.m10.2b"><apply id="Thmthm7.p1.10.10.m10.2.2.cmml" xref="Thmthm7.p1.10.10.m10.2.2"><eq id="Thmthm7.p1.10.10.m10.2.2.2.cmml" xref="Thmthm7.p1.10.10.m10.2.2.2"></eq><apply id="Thmthm7.p1.10.10.m10.2.2.3.1.cmml" xref="Thmthm7.p1.10.10.m10.2.2.3.2"><abs id="Thmthm7.p1.10.10.m10.2.2.3.1.1.cmml" xref="Thmthm7.p1.10.10.m10.2.2.3.2.1"></abs><ci id="Thmthm7.p1.10.10.m10.1.1.cmml" xref="Thmthm7.p1.10.10.m10.1.1">𝑍</ci></apply><apply id="Thmthm7.p1.10.10.m10.2.2.1.2.cmml" xref="Thmthm7.p1.10.10.m10.2.2.1.1"><abs id="Thmthm7.p1.10.10.m10.2.2.1.2.1.cmml" xref="Thmthm7.p1.10.10.m10.2.2.1.1.2"></abs><apply id="Thmthm7.p1.10.10.m10.2.2.1.1.1.cmml" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmthm7.p1.10.10.m10.2.2.1.1.1.1.cmml" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1">subscript</csymbol><ci id="Thmthm7.p1.10.10.m10.2.2.1.1.1.2.cmml" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.2">Δ</ci><apply id="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.cmml" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.3"><plus id="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.1.cmml" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.1"></plus><ci id="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.2.cmml" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.2">ℓ</ci><cn id="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.3.cmml" type="integer" xref="Thmthm7.p1.10.10.m10.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.10.10.m10.2c">|Z|=|\Delta_{\ell+1}|</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.10.10.m10.2d">| italic_Z | = | roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT |</annotation></semantics></math>, separate <math alttext="V(G)\setminus W_{\ell}" class="ltx_Math" display="inline" id="Thmthm7.p1.11.11.m11.1"><semantics id="Thmthm7.p1.11.11.m11.1a"><mrow id="Thmthm7.p1.11.11.m11.1.2" xref="Thmthm7.p1.11.11.m11.1.2.cmml"><mrow id="Thmthm7.p1.11.11.m11.1.2.2" xref="Thmthm7.p1.11.11.m11.1.2.2.cmml"><mi id="Thmthm7.p1.11.11.m11.1.2.2.2" xref="Thmthm7.p1.11.11.m11.1.2.2.2.cmml">V</mi><mo id="Thmthm7.p1.11.11.m11.1.2.2.1" xref="Thmthm7.p1.11.11.m11.1.2.2.1.cmml">⁢</mo><mrow id="Thmthm7.p1.11.11.m11.1.2.2.3.2" xref="Thmthm7.p1.11.11.m11.1.2.2.cmml"><mo id="Thmthm7.p1.11.11.m11.1.2.2.3.2.1" stretchy="false" xref="Thmthm7.p1.11.11.m11.1.2.2.cmml">(</mo><mi id="Thmthm7.p1.11.11.m11.1.1" xref="Thmthm7.p1.11.11.m11.1.1.cmml">G</mi><mo id="Thmthm7.p1.11.11.m11.1.2.2.3.2.2" stretchy="false" xref="Thmthm7.p1.11.11.m11.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmthm7.p1.11.11.m11.1.2.1" xref="Thmthm7.p1.11.11.m11.1.2.1.cmml">∖</mo><msub id="Thmthm7.p1.11.11.m11.1.2.3" xref="Thmthm7.p1.11.11.m11.1.2.3.cmml"><mi id="Thmthm7.p1.11.11.m11.1.2.3.2" xref="Thmthm7.p1.11.11.m11.1.2.3.2.cmml">W</mi><mi id="Thmthm7.p1.11.11.m11.1.2.3.3" mathvariant="normal" xref="Thmthm7.p1.11.11.m11.1.2.3.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.11.11.m11.1b"><apply id="Thmthm7.p1.11.11.m11.1.2.cmml" xref="Thmthm7.p1.11.11.m11.1.2"><setdiff id="Thmthm7.p1.11.11.m11.1.2.1.cmml" xref="Thmthm7.p1.11.11.m11.1.2.1"></setdiff><apply id="Thmthm7.p1.11.11.m11.1.2.2.cmml" xref="Thmthm7.p1.11.11.m11.1.2.2"><times id="Thmthm7.p1.11.11.m11.1.2.2.1.cmml" xref="Thmthm7.p1.11.11.m11.1.2.2.1"></times><ci id="Thmthm7.p1.11.11.m11.1.2.2.2.cmml" xref="Thmthm7.p1.11.11.m11.1.2.2.2">𝑉</ci><ci id="Thmthm7.p1.11.11.m11.1.1.cmml" xref="Thmthm7.p1.11.11.m11.1.1">𝐺</ci></apply><apply id="Thmthm7.p1.11.11.m11.1.2.3.cmml" xref="Thmthm7.p1.11.11.m11.1.2.3"><csymbol cd="ambiguous" id="Thmthm7.p1.11.11.m11.1.2.3.1.cmml" xref="Thmthm7.p1.11.11.m11.1.2.3">subscript</csymbol><ci id="Thmthm7.p1.11.11.m11.1.2.3.2.cmml" xref="Thmthm7.p1.11.11.m11.1.2.3.2">𝑊</ci><ci id="Thmthm7.p1.11.11.m11.1.2.3.3.cmml" xref="Thmthm7.p1.11.11.m11.1.2.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.11.11.m11.1c">V(G)\setminus W_{\ell}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.11.11.m11.1d">italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="W" class="ltx_Math" display="inline" id="Thmthm7.p1.12.12.m12.1"><semantics id="Thmthm7.p1.12.12.m12.1a"><mi id="Thmthm7.p1.12.12.m12.1.1" xref="Thmthm7.p1.12.12.m12.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.12.12.m12.1b"><ci id="Thmthm7.p1.12.12.m12.1.1.cmml" xref="Thmthm7.p1.12.12.m12.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.12.12.m12.1c">W</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.12.12.m12.1d">italic_W</annotation></semantics></math> in <math alttext="G" class="ltx_Math" display="inline" id="Thmthm7.p1.13.13.m13.1"><semantics id="Thmthm7.p1.13.13.m13.1a"><mi id="Thmthm7.p1.13.13.m13.1.1" xref="Thmthm7.p1.13.13.m13.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.13.13.m13.1b"><ci id="Thmthm7.p1.13.13.m13.1.1.cmml" xref="Thmthm7.p1.13.13.m13.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.13.13.m13.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.13.13.m13.1d">italic_G</annotation></semantics></math>. Then every <math alttext="(V(G)\setminus W_{\ell},Z,W)" class="ltx_Math" display="inline" id="Thmthm7.p1.14.14.m14.4"><semantics id="Thmthm7.p1.14.14.m14.4a"><mrow id="Thmthm7.p1.14.14.m14.4.4.1" xref="Thmthm7.p1.14.14.m14.4.4.2.cmml"><mo id="Thmthm7.p1.14.14.m14.4.4.1.2" stretchy="false" xref="Thmthm7.p1.14.14.m14.4.4.2.cmml">(</mo><mrow id="Thmthm7.p1.14.14.m14.4.4.1.1" xref="Thmthm7.p1.14.14.m14.4.4.1.1.cmml"><mrow id="Thmthm7.p1.14.14.m14.4.4.1.1.2" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2.cmml"><mi id="Thmthm7.p1.14.14.m14.4.4.1.1.2.2" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2.2.cmml">V</mi><mo id="Thmthm7.p1.14.14.m14.4.4.1.1.2.1" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2.1.cmml">⁢</mo><mrow id="Thmthm7.p1.14.14.m14.4.4.1.1.2.3.2" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2.cmml"><mo id="Thmthm7.p1.14.14.m14.4.4.1.1.2.3.2.1" stretchy="false" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2.cmml">(</mo><mi id="Thmthm7.p1.14.14.m14.1.1" xref="Thmthm7.p1.14.14.m14.1.1.cmml">G</mi><mo id="Thmthm7.p1.14.14.m14.4.4.1.1.2.3.2.2" stretchy="false" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmthm7.p1.14.14.m14.4.4.1.1.1" xref="Thmthm7.p1.14.14.m14.4.4.1.1.1.cmml">∖</mo><msub id="Thmthm7.p1.14.14.m14.4.4.1.1.3" xref="Thmthm7.p1.14.14.m14.4.4.1.1.3.cmml"><mi id="Thmthm7.p1.14.14.m14.4.4.1.1.3.2" xref="Thmthm7.p1.14.14.m14.4.4.1.1.3.2.cmml">W</mi><mi id="Thmthm7.p1.14.14.m14.4.4.1.1.3.3" mathvariant="normal" xref="Thmthm7.p1.14.14.m14.4.4.1.1.3.3.cmml">ℓ</mi></msub></mrow><mo id="Thmthm7.p1.14.14.m14.4.4.1.3" xref="Thmthm7.p1.14.14.m14.4.4.2.cmml">,</mo><mi id="Thmthm7.p1.14.14.m14.2.2" xref="Thmthm7.p1.14.14.m14.2.2.cmml">Z</mi><mo id="Thmthm7.p1.14.14.m14.4.4.1.4" xref="Thmthm7.p1.14.14.m14.4.4.2.cmml">,</mo><mi id="Thmthm7.p1.14.14.m14.3.3" xref="Thmthm7.p1.14.14.m14.3.3.cmml">W</mi><mo id="Thmthm7.p1.14.14.m14.4.4.1.5" stretchy="false" xref="Thmthm7.p1.14.14.m14.4.4.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.14.14.m14.4b"><vector id="Thmthm7.p1.14.14.m14.4.4.2.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1"><apply id="Thmthm7.p1.14.14.m14.4.4.1.1.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1"><setdiff id="Thmthm7.p1.14.14.m14.4.4.1.1.1.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1.1"></setdiff><apply id="Thmthm7.p1.14.14.m14.4.4.1.1.2.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2"><times id="Thmthm7.p1.14.14.m14.4.4.1.1.2.1.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2.1"></times><ci id="Thmthm7.p1.14.14.m14.4.4.1.1.2.2.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1.2.2">𝑉</ci><ci id="Thmthm7.p1.14.14.m14.1.1.cmml" xref="Thmthm7.p1.14.14.m14.1.1">𝐺</ci></apply><apply id="Thmthm7.p1.14.14.m14.4.4.1.1.3.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1.3"><csymbol cd="ambiguous" id="Thmthm7.p1.14.14.m14.4.4.1.1.3.1.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1.3">subscript</csymbol><ci id="Thmthm7.p1.14.14.m14.4.4.1.1.3.2.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1.3.2">𝑊</ci><ci id="Thmthm7.p1.14.14.m14.4.4.1.1.3.3.cmml" xref="Thmthm7.p1.14.14.m14.4.4.1.1.3.3">ℓ</ci></apply></apply><ci id="Thmthm7.p1.14.14.m14.2.2.cmml" xref="Thmthm7.p1.14.14.m14.2.2">𝑍</ci><ci id="Thmthm7.p1.14.14.m14.3.3.cmml" xref="Thmthm7.p1.14.14.m14.3.3">𝑊</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.14.14.m14.4c">(V(G)\setminus W_{\ell},Z,W)</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.14.14.m14.4d">( italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_Z , italic_W )</annotation></semantics></math>-separation <math alttext="(X,Y)" class="ltx_Math" display="inline" id="Thmthm7.p1.15.15.m15.2"><semantics id="Thmthm7.p1.15.15.m15.2a"><mrow id="Thmthm7.p1.15.15.m15.2.3.2" xref="Thmthm7.p1.15.15.m15.2.3.1.cmml"><mo id="Thmthm7.p1.15.15.m15.2.3.2.1" stretchy="false" xref="Thmthm7.p1.15.15.m15.2.3.1.cmml">(</mo><mi id="Thmthm7.p1.15.15.m15.1.1" xref="Thmthm7.p1.15.15.m15.1.1.cmml">X</mi><mo id="Thmthm7.p1.15.15.m15.2.3.2.2" xref="Thmthm7.p1.15.15.m15.2.3.1.cmml">,</mo><mi id="Thmthm7.p1.15.15.m15.2.2" xref="Thmthm7.p1.15.15.m15.2.2.cmml">Y</mi><mo id="Thmthm7.p1.15.15.m15.2.3.2.3" stretchy="false" xref="Thmthm7.p1.15.15.m15.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.15.15.m15.2b"><interval closure="open" id="Thmthm7.p1.15.15.m15.2.3.1.cmml" xref="Thmthm7.p1.15.15.m15.2.3.2"><ci id="Thmthm7.p1.15.15.m15.1.1.cmml" xref="Thmthm7.p1.15.15.m15.1.1">𝑋</ci><ci id="Thmthm7.p1.15.15.m15.2.2.cmml" xref="Thmthm7.p1.15.15.m15.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.15.15.m15.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.15.15.m15.2d">( italic_X , italic_Y )</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="Thmthm7.p1.16.16.m16.1"><semantics id="Thmthm7.p1.16.16.m16.1a"><mi id="Thmthm7.p1.16.16.m16.1.1" xref="Thmthm7.p1.16.16.m16.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.16.16.m16.1b"><ci id="Thmthm7.p1.16.16.m16.1.1.cmml" xref="Thmthm7.p1.16.16.m16.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.16.16.m16.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.16.16.m16.1d">italic_G</annotation></semantics></math> has the property that, for any separation <math alttext="(A,B)" class="ltx_Math" display="inline" id="Thmthm7.p1.17.17.m17.2"><semantics id="Thmthm7.p1.17.17.m17.2a"><mrow id="Thmthm7.p1.17.17.m17.2.3.2" xref="Thmthm7.p1.17.17.m17.2.3.1.cmml"><mo id="Thmthm7.p1.17.17.m17.2.3.2.1" stretchy="false" xref="Thmthm7.p1.17.17.m17.2.3.1.cmml">(</mo><mi id="Thmthm7.p1.17.17.m17.1.1" xref="Thmthm7.p1.17.17.m17.1.1.cmml">A</mi><mo id="Thmthm7.p1.17.17.m17.2.3.2.2" xref="Thmthm7.p1.17.17.m17.2.3.1.cmml">,</mo><mi id="Thmthm7.p1.17.17.m17.2.2" xref="Thmthm7.p1.17.17.m17.2.2.cmml">B</mi><mo id="Thmthm7.p1.17.17.m17.2.3.2.3" stretchy="false" xref="Thmthm7.p1.17.17.m17.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.17.17.m17.2b"><interval closure="open" id="Thmthm7.p1.17.17.m17.2.3.1.cmml" xref="Thmthm7.p1.17.17.m17.2.3.2"><ci id="Thmthm7.p1.17.17.m17.1.1.cmml" xref="Thmthm7.p1.17.17.m17.1.1">𝐴</ci><ci id="Thmthm7.p1.17.17.m17.2.2.cmml" xref="Thmthm7.p1.17.17.m17.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.17.17.m17.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.17.17.m17.2d">( italic_A , italic_B )</annotation></semantics></math> of <math alttext="G[W_{\ell+1}]" class="ltx_Math" display="inline" id="Thmthm7.p1.18.18.m18.1"><semantics id="Thmthm7.p1.18.18.m18.1a"><mrow id="Thmthm7.p1.18.18.m18.1.1" xref="Thmthm7.p1.18.18.m18.1.1.cmml"><mi id="Thmthm7.p1.18.18.m18.1.1.3" xref="Thmthm7.p1.18.18.m18.1.1.3.cmml">G</mi><mo id="Thmthm7.p1.18.18.m18.1.1.2" xref="Thmthm7.p1.18.18.m18.1.1.2.cmml">⁢</mo><mrow id="Thmthm7.p1.18.18.m18.1.1.1.1" xref="Thmthm7.p1.18.18.m18.1.1.1.2.cmml"><mo id="Thmthm7.p1.18.18.m18.1.1.1.1.2" stretchy="false" xref="Thmthm7.p1.18.18.m18.1.1.1.2.1.cmml">[</mo><msub id="Thmthm7.p1.18.18.m18.1.1.1.1.1" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.cmml"><mi id="Thmthm7.p1.18.18.m18.1.1.1.1.1.2" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.2.cmml">W</mi><mrow id="Thmthm7.p1.18.18.m18.1.1.1.1.1.3" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.cmml"><mi id="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.2" mathvariant="normal" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.2.cmml">ℓ</mi><mo id="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.1" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.1.cmml">+</mo><mn id="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.3" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmthm7.p1.18.18.m18.1.1.1.1.3" stretchy="false" xref="Thmthm7.p1.18.18.m18.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.18.18.m18.1b"><apply id="Thmthm7.p1.18.18.m18.1.1.cmml" xref="Thmthm7.p1.18.18.m18.1.1"><times id="Thmthm7.p1.18.18.m18.1.1.2.cmml" xref="Thmthm7.p1.18.18.m18.1.1.2"></times><ci id="Thmthm7.p1.18.18.m18.1.1.3.cmml" xref="Thmthm7.p1.18.18.m18.1.1.3">𝐺</ci><apply id="Thmthm7.p1.18.18.m18.1.1.1.2.cmml" xref="Thmthm7.p1.18.18.m18.1.1.1.1"><csymbol cd="latexml" id="Thmthm7.p1.18.18.m18.1.1.1.2.1.cmml" xref="Thmthm7.p1.18.18.m18.1.1.1.1.2">delimited-[]</csymbol><apply id="Thmthm7.p1.18.18.m18.1.1.1.1.1.cmml" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm7.p1.18.18.m18.1.1.1.1.1.1.cmml" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1">subscript</csymbol><ci id="Thmthm7.p1.18.18.m18.1.1.1.1.1.2.cmml" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.2">𝑊</ci><apply id="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.cmml" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.3"><plus id="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.1.cmml" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.1"></plus><ci id="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.2.cmml" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.2">ℓ</ci><cn id="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.3.cmml" type="integer" xref="Thmthm7.p1.18.18.m18.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.18.18.m18.1c">G[W_{\ell+1}]</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.18.18.m18.1d">italic_G [ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|W\setminus B|+|Z\setminus B|\leq\frac{(2+\tfrac{1}{6})\,|A\setminus B|}{\ell+% 2}+3\,|A\cap B|\enspace." class="ltx_Math" display="block" id="S3.Ex1.m1.3"><semantics id="S3.Ex1.m1.3a"><mrow id="S3.Ex1.m1.3.3.1" 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id="S3.Ex1.m1.3.3.1.1.3.1.1.1.1.1.cmml" xref="S3.Ex1.m1.3.3.1.1.3.1.1.1.1.1"></intersect><ci id="S3.Ex1.m1.3.3.1.1.3.1.1.1.1.2.cmml" xref="S3.Ex1.m1.3.3.1.1.3.1.1.1.1.2">𝐴</ci><ci id="S3.Ex1.m1.3.3.1.1.3.1.1.1.1.3.cmml" xref="S3.Ex1.m1.3.3.1.1.3.1.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.3c">|W\setminus B|+|Z\setminus B|\leq\frac{(2+\tfrac{1}{6})\,|A\setminus B|}{\ell+% 2}+3\,|A\cap B|\enspace.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.3d">| italic_W ∖ italic_B | + | italic_Z ∖ italic_B | ≤ divide start_ARG ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG + 3 | italic_A ∩ italic_B | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="Thmthm8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm8.1.1.1">Remark 8</span></span><span class="ltx_text ltx_font_bold" id="Thmthm8.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm8.p1"> <p class="ltx_p" id="Thmthm8.p1.1">The proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm7" title="Lemma 7. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">7</span></a> requires some extra effort to obtain <math alttext="\ell+2" class="ltx_Math" display="inline" id="Thmthm8.p1.1.m1.1"><semantics id="Thmthm8.p1.1.m1.1a"><mrow id="Thmthm8.p1.1.m1.1.1" xref="Thmthm8.p1.1.m1.1.1.cmml"><mi id="Thmthm8.p1.1.m1.1.1.2" mathvariant="normal" xref="Thmthm8.p1.1.m1.1.1.2.cmml">ℓ</mi><mo id="Thmthm8.p1.1.m1.1.1.1" xref="Thmthm8.p1.1.m1.1.1.1.cmml">+</mo><mn id="Thmthm8.p1.1.m1.1.1.3" xref="Thmthm8.p1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.1.m1.1b"><apply id="Thmthm8.p1.1.m1.1.1.cmml" xref="Thmthm8.p1.1.m1.1.1"><plus id="Thmthm8.p1.1.m1.1.1.1.cmml" xref="Thmthm8.p1.1.m1.1.1.1"></plus><ci id="Thmthm8.p1.1.m1.1.1.2.cmml" xref="Thmthm8.p1.1.m1.1.1.2">ℓ</ci><cn id="Thmthm8.p1.1.m1.1.1.3.cmml" type="integer" xref="Thmthm8.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.1.m1.1c">\ell+2</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.1.m1.1d">roman_ℓ + 2</annotation></semantics></math> in the denominator. The reader who is not interested in precise constants can already stop at <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E5" title="In Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">5</span></a> in the proof, from which the bound</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|W\setminus B|+|Z\setminus B|\leq\frac{2|A\setminus B|}{\ell+1}+3|A\cap B|" class="ltx_Math" display="block" id="S3.Ex2.m1.4"><semantics id="S3.Ex2.m1.4a"><mrow id="S3.Ex2.m1.4.4" xref="S3.Ex2.m1.4.4.cmml"><mrow id="S3.Ex2.m1.3.3.2" xref="S3.Ex2.m1.3.3.2.cmml"><mrow id="S3.Ex2.m1.2.2.1.1.1" xref="S3.Ex2.m1.2.2.1.1.2.cmml"><mo id="S3.Ex2.m1.2.2.1.1.1.2" stretchy="false" xref="S3.Ex2.m1.2.2.1.1.2.1.cmml">|</mo><mrow id="S3.Ex2.m1.2.2.1.1.1.1" xref="S3.Ex2.m1.2.2.1.1.1.1.cmml"><mi id="S3.Ex2.m1.2.2.1.1.1.1.2" xref="S3.Ex2.m1.2.2.1.1.1.1.2.cmml">W</mi><mo id="S3.Ex2.m1.2.2.1.1.1.1.1" xref="S3.Ex2.m1.2.2.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex2.m1.2.2.1.1.1.1.3" xref="S3.Ex2.m1.2.2.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex2.m1.2.2.1.1.1.3" stretchy="false" xref="S3.Ex2.m1.2.2.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex2.m1.3.3.2.3" xref="S3.Ex2.m1.3.3.2.3.cmml">+</mo><mrow id="S3.Ex2.m1.3.3.2.2.1" xref="S3.Ex2.m1.3.3.2.2.2.cmml"><mo id="S3.Ex2.m1.3.3.2.2.1.2" stretchy="false" xref="S3.Ex2.m1.3.3.2.2.2.1.cmml">|</mo><mrow id="S3.Ex2.m1.3.3.2.2.1.1" xref="S3.Ex2.m1.3.3.2.2.1.1.cmml"><mi id="S3.Ex2.m1.3.3.2.2.1.1.2" xref="S3.Ex2.m1.3.3.2.2.1.1.2.cmml">Z</mi><mo id="S3.Ex2.m1.3.3.2.2.1.1.1" xref="S3.Ex2.m1.3.3.2.2.1.1.1.cmml">∖</mo><mi id="S3.Ex2.m1.3.3.2.2.1.1.3" xref="S3.Ex2.m1.3.3.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex2.m1.3.3.2.2.1.3" stretchy="false" xref="S3.Ex2.m1.3.3.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S3.Ex2.m1.4.4.4" xref="S3.Ex2.m1.4.4.4.cmml">≤</mo><mrow id="S3.Ex2.m1.4.4.3" xref="S3.Ex2.m1.4.4.3.cmml"><mfrac id="S3.Ex2.m1.1.1" xref="S3.Ex2.m1.1.1.cmml"><mrow id="S3.Ex2.m1.1.1.1" xref="S3.Ex2.m1.1.1.1.cmml"><mn id="S3.Ex2.m1.1.1.1.3" xref="S3.Ex2.m1.1.1.1.3.cmml">2</mn><mo id="S3.Ex2.m1.1.1.1.2" xref="S3.Ex2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex2.m1.1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.2.cmml"><mo id="S3.Ex2.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex2.m1.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex2.m1.1.1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.1.1.cmml"><mi id="S3.Ex2.m1.1.1.1.1.1.1.2" xref="S3.Ex2.m1.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex2.m1.1.1.1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex2.m1.1.1.1.1.1.1.3" xref="S3.Ex2.m1.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex2.m1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex2.m1.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mrow id="S3.Ex2.m1.1.1.3" xref="S3.Ex2.m1.1.1.3.cmml"><mi id="S3.Ex2.m1.1.1.3.2" mathvariant="normal" xref="S3.Ex2.m1.1.1.3.2.cmml">ℓ</mi><mo id="S3.Ex2.m1.1.1.3.1" xref="S3.Ex2.m1.1.1.3.1.cmml">+</mo><mn id="S3.Ex2.m1.1.1.3.3" xref="S3.Ex2.m1.1.1.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.Ex2.m1.4.4.3.2" xref="S3.Ex2.m1.4.4.3.2.cmml">+</mo><mrow id="S3.Ex2.m1.4.4.3.1" xref="S3.Ex2.m1.4.4.3.1.cmml"><mn id="S3.Ex2.m1.4.4.3.1.3" xref="S3.Ex2.m1.4.4.3.1.3.cmml">3</mn><mo id="S3.Ex2.m1.4.4.3.1.2" xref="S3.Ex2.m1.4.4.3.1.2.cmml">⁢</mo><mrow id="S3.Ex2.m1.4.4.3.1.1.1" xref="S3.Ex2.m1.4.4.3.1.1.2.cmml"><mo id="S3.Ex2.m1.4.4.3.1.1.1.2" stretchy="false" xref="S3.Ex2.m1.4.4.3.1.1.2.1.cmml">|</mo><mrow id="S3.Ex2.m1.4.4.3.1.1.1.1" xref="S3.Ex2.m1.4.4.3.1.1.1.1.cmml"><mi id="S3.Ex2.m1.4.4.3.1.1.1.1.2" xref="S3.Ex2.m1.4.4.3.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex2.m1.4.4.3.1.1.1.1.1" xref="S3.Ex2.m1.4.4.3.1.1.1.1.1.cmml">∩</mo><mi id="S3.Ex2.m1.4.4.3.1.1.1.1.3" xref="S3.Ex2.m1.4.4.3.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex2.m1.4.4.3.1.1.1.3" stretchy="false" xref="S3.Ex2.m1.4.4.3.1.1.2.1.cmml">|</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex2.m1.4b"><apply id="S3.Ex2.m1.4.4.cmml" xref="S3.Ex2.m1.4.4"><leq id="S3.Ex2.m1.4.4.4.cmml" xref="S3.Ex2.m1.4.4.4"></leq><apply id="S3.Ex2.m1.3.3.2.cmml" xref="S3.Ex2.m1.3.3.2"><plus id="S3.Ex2.m1.3.3.2.3.cmml" xref="S3.Ex2.m1.3.3.2.3"></plus><apply id="S3.Ex2.m1.2.2.1.1.2.cmml" xref="S3.Ex2.m1.2.2.1.1.1"><abs id="S3.Ex2.m1.2.2.1.1.2.1.cmml" xref="S3.Ex2.m1.2.2.1.1.1.2"></abs><apply id="S3.Ex2.m1.2.2.1.1.1.1.cmml" xref="S3.Ex2.m1.2.2.1.1.1.1"><setdiff id="S3.Ex2.m1.2.2.1.1.1.1.1.cmml" xref="S3.Ex2.m1.2.2.1.1.1.1.1"></setdiff><ci id="S3.Ex2.m1.2.2.1.1.1.1.2.cmml" xref="S3.Ex2.m1.2.2.1.1.1.1.2">𝑊</ci><ci id="S3.Ex2.m1.2.2.1.1.1.1.3.cmml" xref="S3.Ex2.m1.2.2.1.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.Ex2.m1.3.3.2.2.2.cmml" xref="S3.Ex2.m1.3.3.2.2.1"><abs id="S3.Ex2.m1.3.3.2.2.2.1.cmml" xref="S3.Ex2.m1.3.3.2.2.1.2"></abs><apply id="S3.Ex2.m1.3.3.2.2.1.1.cmml" xref="S3.Ex2.m1.3.3.2.2.1.1"><setdiff id="S3.Ex2.m1.3.3.2.2.1.1.1.cmml" xref="S3.Ex2.m1.3.3.2.2.1.1.1"></setdiff><ci id="S3.Ex2.m1.3.3.2.2.1.1.2.cmml" xref="S3.Ex2.m1.3.3.2.2.1.1.2">𝑍</ci><ci id="S3.Ex2.m1.3.3.2.2.1.1.3.cmml" xref="S3.Ex2.m1.3.3.2.2.1.1.3">𝐵</ci></apply></apply></apply><apply id="S3.Ex2.m1.4.4.3.cmml" xref="S3.Ex2.m1.4.4.3"><plus id="S3.Ex2.m1.4.4.3.2.cmml" xref="S3.Ex2.m1.4.4.3.2"></plus><apply id="S3.Ex2.m1.1.1.cmml" xref="S3.Ex2.m1.1.1"><divide id="S3.Ex2.m1.1.1.2.cmml" xref="S3.Ex2.m1.1.1"></divide><apply id="S3.Ex2.m1.1.1.1.cmml" xref="S3.Ex2.m1.1.1.1"><times id="S3.Ex2.m1.1.1.1.2.cmml" xref="S3.Ex2.m1.1.1.1.2"></times><cn id="S3.Ex2.m1.1.1.1.3.cmml" type="integer" xref="S3.Ex2.m1.1.1.1.3">2</cn><apply id="S3.Ex2.m1.1.1.1.1.2.cmml" xref="S3.Ex2.m1.1.1.1.1.1"><abs id="S3.Ex2.m1.1.1.1.1.2.1.cmml" xref="S3.Ex2.m1.1.1.1.1.1.2"></abs><apply id="S3.Ex2.m1.1.1.1.1.1.1.cmml" 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xref="S3.Ex2.m1.4.4.3.1.1.1.1"><intersect id="S3.Ex2.m1.4.4.3.1.1.1.1.1.cmml" xref="S3.Ex2.m1.4.4.3.1.1.1.1.1"></intersect><ci id="S3.Ex2.m1.4.4.3.1.1.1.1.2.cmml" xref="S3.Ex2.m1.4.4.3.1.1.1.1.2">𝐴</ci><ci id="S3.Ex2.m1.4.4.3.1.1.1.1.3.cmml" xref="S3.Ex2.m1.4.4.3.1.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex2.m1.4c">|W\setminus B|+|Z\setminus B|\leq\frac{2|A\setminus B|}{\ell+1}+3|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex2.m1.4d">| italic_W ∖ italic_B | + | italic_Z ∖ italic_B | ≤ divide start_ARG 2 | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + 3 | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmthm8.p1.2">follows immediately. This weaker bound is still sufficient to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm1" title="Theorem 1. ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1</span></a> with the constant <math alttext="c<69" class="ltx_Math" display="inline" id="Thmthm8.p1.2.m1.1"><semantics id="Thmthm8.p1.2.m1.1a"><mrow id="Thmthm8.p1.2.m1.1.1" xref="Thmthm8.p1.2.m1.1.1.cmml"><mi id="Thmthm8.p1.2.m1.1.1.2" xref="Thmthm8.p1.2.m1.1.1.2.cmml">c</mi><mo id="Thmthm8.p1.2.m1.1.1.1" xref="Thmthm8.p1.2.m1.1.1.1.cmml"><</mo><mn id="Thmthm8.p1.2.m1.1.1.3" xref="Thmthm8.p1.2.m1.1.1.3.cmml">69</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.2.m1.1b"><apply id="Thmthm8.p1.2.m1.1.1.cmml" xref="Thmthm8.p1.2.m1.1.1"><lt id="Thmthm8.p1.2.m1.1.1.1.cmml" xref="Thmthm8.p1.2.m1.1.1.1"></lt><ci id="Thmthm8.p1.2.m1.1.1.2.cmml" xref="Thmthm8.p1.2.m1.1.1.2">𝑐</ci><cn id="Thmthm8.p1.2.m1.1.1.3.cmml" type="integer" xref="Thmthm8.p1.2.m1.1.1.3">69</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.2.m1.1c">c<69</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.2.m1.1d">italic_c < 69</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S3.9"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.5.p1"> <p class="ltx_p" id="S3.5.p1.11">Let <math alttext="W_{-1}:=\emptyset" class="ltx_Math" display="inline" id="S3.5.p1.1.m1.1"><semantics id="S3.5.p1.1.m1.1a"><mrow id="S3.5.p1.1.m1.1.1" xref="S3.5.p1.1.m1.1.1.cmml"><msub id="S3.5.p1.1.m1.1.1.2" xref="S3.5.p1.1.m1.1.1.2.cmml"><mi id="S3.5.p1.1.m1.1.1.2.2" xref="S3.5.p1.1.m1.1.1.2.2.cmml">W</mi><mrow id="S3.5.p1.1.m1.1.1.2.3" xref="S3.5.p1.1.m1.1.1.2.3.cmml"><mo id="S3.5.p1.1.m1.1.1.2.3a" xref="S3.5.p1.1.m1.1.1.2.3.cmml">−</mo><mn id="S3.5.p1.1.m1.1.1.2.3.2" xref="S3.5.p1.1.m1.1.1.2.3.2.cmml">1</mn></mrow></msub><mo id="S3.5.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.5.p1.1.m1.1.1.1.cmml">:=</mo><mi id="S3.5.p1.1.m1.1.1.3" mathvariant="normal" xref="S3.5.p1.1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.1.m1.1b"><apply id="S3.5.p1.1.m1.1.1.cmml" xref="S3.5.p1.1.m1.1.1"><csymbol cd="latexml" id="S3.5.p1.1.m1.1.1.1.cmml" xref="S3.5.p1.1.m1.1.1.1">assign</csymbol><apply id="S3.5.p1.1.m1.1.1.2.cmml" xref="S3.5.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.5.p1.1.m1.1.1.2.1.cmml" xref="S3.5.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.5.p1.1.m1.1.1.2.2.cmml" xref="S3.5.p1.1.m1.1.1.2.2">𝑊</ci><apply id="S3.5.p1.1.m1.1.1.2.3.cmml" xref="S3.5.p1.1.m1.1.1.2.3"><minus id="S3.5.p1.1.m1.1.1.2.3.1.cmml" xref="S3.5.p1.1.m1.1.1.2.3"></minus><cn id="S3.5.p1.1.m1.1.1.2.3.2.cmml" type="integer" xref="S3.5.p1.1.m1.1.1.2.3.2">1</cn></apply></apply><emptyset id="S3.5.p1.1.m1.1.1.3.cmml" xref="S3.5.p1.1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.1.m1.1c">W_{-1}:=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.1.m1.1d">italic_W start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT := ∅</annotation></semantics></math> and, for each <math alttext="i\in\{0,\ldots,\ell+1\}" class="ltx_Math" display="inline" id="S3.5.p1.2.m2.3"><semantics id="S3.5.p1.2.m2.3a"><mrow id="S3.5.p1.2.m2.3.3" xref="S3.5.p1.2.m2.3.3.cmml"><mi id="S3.5.p1.2.m2.3.3.3" xref="S3.5.p1.2.m2.3.3.3.cmml">i</mi><mo id="S3.5.p1.2.m2.3.3.2" xref="S3.5.p1.2.m2.3.3.2.cmml">∈</mo><mrow id="S3.5.p1.2.m2.3.3.1.1" xref="S3.5.p1.2.m2.3.3.1.2.cmml"><mo id="S3.5.p1.2.m2.3.3.1.1.2" stretchy="false" xref="S3.5.p1.2.m2.3.3.1.2.cmml">{</mo><mn id="S3.5.p1.2.m2.1.1" xref="S3.5.p1.2.m2.1.1.cmml">0</mn><mo id="S3.5.p1.2.m2.3.3.1.1.3" xref="S3.5.p1.2.m2.3.3.1.2.cmml">,</mo><mi id="S3.5.p1.2.m2.2.2" mathvariant="normal" xref="S3.5.p1.2.m2.2.2.cmml">…</mi><mo id="S3.5.p1.2.m2.3.3.1.1.4" xref="S3.5.p1.2.m2.3.3.1.2.cmml">,</mo><mrow id="S3.5.p1.2.m2.3.3.1.1.1" xref="S3.5.p1.2.m2.3.3.1.1.1.cmml"><mi id="S3.5.p1.2.m2.3.3.1.1.1.2" mathvariant="normal" xref="S3.5.p1.2.m2.3.3.1.1.1.2.cmml">ℓ</mi><mo id="S3.5.p1.2.m2.3.3.1.1.1.1" xref="S3.5.p1.2.m2.3.3.1.1.1.1.cmml">+</mo><mn id="S3.5.p1.2.m2.3.3.1.1.1.3" xref="S3.5.p1.2.m2.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S3.5.p1.2.m2.3.3.1.1.5" stretchy="false" xref="S3.5.p1.2.m2.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.2.m2.3b"><apply id="S3.5.p1.2.m2.3.3.cmml" xref="S3.5.p1.2.m2.3.3"><in id="S3.5.p1.2.m2.3.3.2.cmml" xref="S3.5.p1.2.m2.3.3.2"></in><ci id="S3.5.p1.2.m2.3.3.3.cmml" xref="S3.5.p1.2.m2.3.3.3">𝑖</ci><set id="S3.5.p1.2.m2.3.3.1.2.cmml" xref="S3.5.p1.2.m2.3.3.1.1"><cn id="S3.5.p1.2.m2.1.1.cmml" type="integer" xref="S3.5.p1.2.m2.1.1">0</cn><ci id="S3.5.p1.2.m2.2.2.cmml" xref="S3.5.p1.2.m2.2.2">…</ci><apply id="S3.5.p1.2.m2.3.3.1.1.1.cmml" xref="S3.5.p1.2.m2.3.3.1.1.1"><plus id="S3.5.p1.2.m2.3.3.1.1.1.1.cmml" xref="S3.5.p1.2.m2.3.3.1.1.1.1"></plus><ci id="S3.5.p1.2.m2.3.3.1.1.1.2.cmml" xref="S3.5.p1.2.m2.3.3.1.1.1.2">ℓ</ci><cn id="S3.5.p1.2.m2.3.3.1.1.1.3.cmml" type="integer" xref="S3.5.p1.2.m2.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.2.m2.3c">i\in\{0,\ldots,\ell+1\}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.2.m2.3d">italic_i ∈ { 0 , … , roman_ℓ + 1 }</annotation></semantics></math>, let <math alttext="\Delta_{i}:=W_{i}\setminus W_{i-1}" class="ltx_Math" display="inline" id="S3.5.p1.3.m3.1"><semantics id="S3.5.p1.3.m3.1a"><mrow id="S3.5.p1.3.m3.1.1" xref="S3.5.p1.3.m3.1.1.cmml"><msub id="S3.5.p1.3.m3.1.1.2" xref="S3.5.p1.3.m3.1.1.2.cmml"><mi id="S3.5.p1.3.m3.1.1.2.2" mathvariant="normal" xref="S3.5.p1.3.m3.1.1.2.2.cmml">Δ</mi><mi id="S3.5.p1.3.m3.1.1.2.3" xref="S3.5.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S3.5.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.5.p1.3.m3.1.1.1.cmml">:=</mo><mrow id="S3.5.p1.3.m3.1.1.3" xref="S3.5.p1.3.m3.1.1.3.cmml"><msub id="S3.5.p1.3.m3.1.1.3.2" xref="S3.5.p1.3.m3.1.1.3.2.cmml"><mi id="S3.5.p1.3.m3.1.1.3.2.2" xref="S3.5.p1.3.m3.1.1.3.2.2.cmml">W</mi><mi id="S3.5.p1.3.m3.1.1.3.2.3" xref="S3.5.p1.3.m3.1.1.3.2.3.cmml">i</mi></msub><mo id="S3.5.p1.3.m3.1.1.3.1" xref="S3.5.p1.3.m3.1.1.3.1.cmml">∖</mo><msub id="S3.5.p1.3.m3.1.1.3.3" xref="S3.5.p1.3.m3.1.1.3.3.cmml"><mi id="S3.5.p1.3.m3.1.1.3.3.2" xref="S3.5.p1.3.m3.1.1.3.3.2.cmml">W</mi><mrow id="S3.5.p1.3.m3.1.1.3.3.3" xref="S3.5.p1.3.m3.1.1.3.3.3.cmml"><mi id="S3.5.p1.3.m3.1.1.3.3.3.2" xref="S3.5.p1.3.m3.1.1.3.3.3.2.cmml">i</mi><mo id="S3.5.p1.3.m3.1.1.3.3.3.1" xref="S3.5.p1.3.m3.1.1.3.3.3.1.cmml">−</mo><mn id="S3.5.p1.3.m3.1.1.3.3.3.3" xref="S3.5.p1.3.m3.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.3.m3.1b"><apply id="S3.5.p1.3.m3.1.1.cmml" xref="S3.5.p1.3.m3.1.1"><csymbol cd="latexml" id="S3.5.p1.3.m3.1.1.1.cmml" xref="S3.5.p1.3.m3.1.1.1">assign</csymbol><apply id="S3.5.p1.3.m3.1.1.2.cmml" xref="S3.5.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.5.p1.3.m3.1.1.2.1.cmml" xref="S3.5.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.5.p1.3.m3.1.1.2.2.cmml" xref="S3.5.p1.3.m3.1.1.2.2">Δ</ci><ci id="S3.5.p1.3.m3.1.1.2.3.cmml" xref="S3.5.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S3.5.p1.3.m3.1.1.3.cmml" xref="S3.5.p1.3.m3.1.1.3"><setdiff id="S3.5.p1.3.m3.1.1.3.1.cmml" xref="S3.5.p1.3.m3.1.1.3.1"></setdiff><apply id="S3.5.p1.3.m3.1.1.3.2.cmml" xref="S3.5.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S3.5.p1.3.m3.1.1.3.2.1.cmml" xref="S3.5.p1.3.m3.1.1.3.2">subscript</csymbol><ci id="S3.5.p1.3.m3.1.1.3.2.2.cmml" xref="S3.5.p1.3.m3.1.1.3.2.2">𝑊</ci><ci id="S3.5.p1.3.m3.1.1.3.2.3.cmml" xref="S3.5.p1.3.m3.1.1.3.2.3">𝑖</ci></apply><apply id="S3.5.p1.3.m3.1.1.3.3.cmml" xref="S3.5.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.5.p1.3.m3.1.1.3.3.1.cmml" xref="S3.5.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S3.5.p1.3.m3.1.1.3.3.2.cmml" xref="S3.5.p1.3.m3.1.1.3.3.2">𝑊</ci><apply id="S3.5.p1.3.m3.1.1.3.3.3.cmml" xref="S3.5.p1.3.m3.1.1.3.3.3"><minus id="S3.5.p1.3.m3.1.1.3.3.3.1.cmml" xref="S3.5.p1.3.m3.1.1.3.3.3.1"></minus><ci id="S3.5.p1.3.m3.1.1.3.3.3.2.cmml" xref="S3.5.p1.3.m3.1.1.3.3.3.2">𝑖</ci><cn id="S3.5.p1.3.m3.1.1.3.3.3.3.cmml" type="integer" xref="S3.5.p1.3.m3.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.3.m3.1c">\Delta_{i}:=W_{i}\setminus W_{i-1}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.3.m3.1d">roman_Δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ italic_W start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT</annotation></semantics></math> (as in the definition of <math alttext="W" class="ltx_Math" display="inline" id="S3.5.p1.4.m4.1"><semantics id="S3.5.p1.4.m4.1a"><mi id="S3.5.p1.4.m4.1.1" xref="S3.5.p1.4.m4.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.5.p1.4.m4.1b"><ci id="S3.5.p1.4.m4.1.1.cmml" xref="S3.5.p1.4.m4.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.4.m4.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.4.m4.1d">italic_W</annotation></semantics></math>-sequence). By the definition of <math alttext="W" class="ltx_Math" display="inline" id="S3.5.p1.5.m5.1"><semantics id="S3.5.p1.5.m5.1a"><mi id="S3.5.p1.5.m5.1.1" xref="S3.5.p1.5.m5.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.5.p1.5.m5.1b"><ci id="S3.5.p1.5.m5.1.1.cmml" xref="S3.5.p1.5.m5.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.5.m5.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.5.m5.1d">italic_W</annotation></semantics></math>-sequence, <math alttext="G[W_{i}]" class="ltx_Math" display="inline" id="S3.5.p1.6.m6.1"><semantics id="S3.5.p1.6.m6.1a"><mrow id="S3.5.p1.6.m6.1.1" xref="S3.5.p1.6.m6.1.1.cmml"><mi id="S3.5.p1.6.m6.1.1.3" xref="S3.5.p1.6.m6.1.1.3.cmml">G</mi><mo id="S3.5.p1.6.m6.1.1.2" xref="S3.5.p1.6.m6.1.1.2.cmml">⁢</mo><mrow id="S3.5.p1.6.m6.1.1.1.1" xref="S3.5.p1.6.m6.1.1.1.2.cmml"><mo id="S3.5.p1.6.m6.1.1.1.1.2" stretchy="false" xref="S3.5.p1.6.m6.1.1.1.2.1.cmml">[</mo><msub id="S3.5.p1.6.m6.1.1.1.1.1" xref="S3.5.p1.6.m6.1.1.1.1.1.cmml"><mi id="S3.5.p1.6.m6.1.1.1.1.1.2" xref="S3.5.p1.6.m6.1.1.1.1.1.2.cmml">W</mi><mi id="S3.5.p1.6.m6.1.1.1.1.1.3" xref="S3.5.p1.6.m6.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.5.p1.6.m6.1.1.1.1.3" stretchy="false" xref="S3.5.p1.6.m6.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.6.m6.1b"><apply id="S3.5.p1.6.m6.1.1.cmml" xref="S3.5.p1.6.m6.1.1"><times id="S3.5.p1.6.m6.1.1.2.cmml" xref="S3.5.p1.6.m6.1.1.2"></times><ci id="S3.5.p1.6.m6.1.1.3.cmml" xref="S3.5.p1.6.m6.1.1.3">𝐺</ci><apply id="S3.5.p1.6.m6.1.1.1.2.cmml" xref="S3.5.p1.6.m6.1.1.1.1"><csymbol cd="latexml" id="S3.5.p1.6.m6.1.1.1.2.1.cmml" xref="S3.5.p1.6.m6.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.5.p1.6.m6.1.1.1.1.1.cmml" xref="S3.5.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.5.p1.6.m6.1.1.1.1.1.1.cmml" xref="S3.5.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S3.5.p1.6.m6.1.1.1.1.1.2.cmml" xref="S3.5.p1.6.m6.1.1.1.1.1.2">𝑊</ci><ci id="S3.5.p1.6.m6.1.1.1.1.1.3.cmml" xref="S3.5.p1.6.m6.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.6.m6.1c">G[W_{i}]</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.6.m6.1d">italic_G [ italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ]</annotation></semantics></math> contains a set <math alttext="\mathcal{P}_{i}" class="ltx_Math" display="inline" id="S3.5.p1.7.m7.1"><semantics id="S3.5.p1.7.m7.1a"><msub id="S3.5.p1.7.m7.1.1" xref="S3.5.p1.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.5.p1.7.m7.1.1.2" xref="S3.5.p1.7.m7.1.1.2.cmml">𝒫</mi><mi id="S3.5.p1.7.m7.1.1.3" xref="S3.5.p1.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.5.p1.7.m7.1b"><apply id="S3.5.p1.7.m7.1.1.cmml" xref="S3.5.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.5.p1.7.m7.1.1.1.cmml" xref="S3.5.p1.7.m7.1.1">subscript</csymbol><ci id="S3.5.p1.7.m7.1.1.2.cmml" xref="S3.5.p1.7.m7.1.1.2">𝒫</ci><ci id="S3.5.p1.7.m7.1.1.3.cmml" xref="S3.5.p1.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.7.m7.1c">\mathcal{P}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.7.m7.1d">caligraphic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="|W\setminus B|" class="ltx_Math" display="inline" id="S3.5.p1.8.m8.1"><semantics id="S3.5.p1.8.m8.1a"><mrow id="S3.5.p1.8.m8.1.1.1" xref="S3.5.p1.8.m8.1.1.2.cmml"><mo id="S3.5.p1.8.m8.1.1.1.2" stretchy="false" xref="S3.5.p1.8.m8.1.1.2.1.cmml">|</mo><mrow id="S3.5.p1.8.m8.1.1.1.1" xref="S3.5.p1.8.m8.1.1.1.1.cmml"><mi id="S3.5.p1.8.m8.1.1.1.1.2" xref="S3.5.p1.8.m8.1.1.1.1.2.cmml">W</mi><mo id="S3.5.p1.8.m8.1.1.1.1.1" xref="S3.5.p1.8.m8.1.1.1.1.1.cmml">∖</mo><mi id="S3.5.p1.8.m8.1.1.1.1.3" xref="S3.5.p1.8.m8.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.5.p1.8.m8.1.1.1.3" stretchy="false" xref="S3.5.p1.8.m8.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.8.m8.1b"><apply id="S3.5.p1.8.m8.1.1.2.cmml" xref="S3.5.p1.8.m8.1.1.1"><abs id="S3.5.p1.8.m8.1.1.2.1.cmml" xref="S3.5.p1.8.m8.1.1.1.2"></abs><apply id="S3.5.p1.8.m8.1.1.1.1.cmml" xref="S3.5.p1.8.m8.1.1.1.1"><setdiff id="S3.5.p1.8.m8.1.1.1.1.1.cmml" xref="S3.5.p1.8.m8.1.1.1.1.1"></setdiff><ci id="S3.5.p1.8.m8.1.1.1.1.2.cmml" xref="S3.5.p1.8.m8.1.1.1.1.2">𝑊</ci><ci id="S3.5.p1.8.m8.1.1.1.1.3.cmml" xref="S3.5.p1.8.m8.1.1.1.1.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.8.m8.1c">|W\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.8.m8.1d">| italic_W ∖ italic_B |</annotation></semantics></math> pairwise vertex-disjoint <math alttext="\Delta_{i}" class="ltx_Math" display="inline" id="S3.5.p1.9.m9.1"><semantics id="S3.5.p1.9.m9.1a"><msub id="S3.5.p1.9.m9.1.1" xref="S3.5.p1.9.m9.1.1.cmml"><mi id="S3.5.p1.9.m9.1.1.2" mathvariant="normal" xref="S3.5.p1.9.m9.1.1.2.cmml">Δ</mi><mi id="S3.5.p1.9.m9.1.1.3" xref="S3.5.p1.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.5.p1.9.m9.1b"><apply id="S3.5.p1.9.m9.1.1.cmml" xref="S3.5.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.5.p1.9.m9.1.1.1.cmml" xref="S3.5.p1.9.m9.1.1">subscript</csymbol><ci id="S3.5.p1.9.m9.1.1.2.cmml" xref="S3.5.p1.9.m9.1.1.2">Δ</ci><ci id="S3.5.p1.9.m9.1.1.3.cmml" xref="S3.5.p1.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.9.m9.1c">\Delta_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.9.m9.1d">roman_Δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>-<math alttext="(W\setminus B)" class="ltx_Math" display="inline" id="S3.5.p1.10.m10.1"><semantics id="S3.5.p1.10.m10.1a"><mrow id="S3.5.p1.10.m10.1.1.1" xref="S3.5.p1.10.m10.1.1.1.1.cmml"><mo id="S3.5.p1.10.m10.1.1.1.2" stretchy="false" xref="S3.5.p1.10.m10.1.1.1.1.cmml">(</mo><mrow id="S3.5.p1.10.m10.1.1.1.1" xref="S3.5.p1.10.m10.1.1.1.1.cmml"><mi id="S3.5.p1.10.m10.1.1.1.1.2" xref="S3.5.p1.10.m10.1.1.1.1.2.cmml">W</mi><mo id="S3.5.p1.10.m10.1.1.1.1.1" xref="S3.5.p1.10.m10.1.1.1.1.1.cmml">∖</mo><mi id="S3.5.p1.10.m10.1.1.1.1.3" xref="S3.5.p1.10.m10.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.5.p1.10.m10.1.1.1.3" stretchy="false" xref="S3.5.p1.10.m10.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.10.m10.1b"><apply id="S3.5.p1.10.m10.1.1.1.1.cmml" xref="S3.5.p1.10.m10.1.1.1"><setdiff id="S3.5.p1.10.m10.1.1.1.1.1.cmml" xref="S3.5.p1.10.m10.1.1.1.1.1"></setdiff><ci id="S3.5.p1.10.m10.1.1.1.1.2.cmml" xref="S3.5.p1.10.m10.1.1.1.1.2">𝑊</ci><ci id="S3.5.p1.10.m10.1.1.1.1.3.cmml" xref="S3.5.p1.10.m10.1.1.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.10.m10.1c">(W\setminus B)</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.10.m10.1d">( italic_W ∖ italic_B )</annotation></semantics></math> paths, for each <math alttext="i\in\{0,\ldots,\ell\}" class="ltx_Math" display="inline" id="S3.5.p1.11.m11.3"><semantics id="S3.5.p1.11.m11.3a"><mrow id="S3.5.p1.11.m11.3.4" xref="S3.5.p1.11.m11.3.4.cmml"><mi id="S3.5.p1.11.m11.3.4.2" xref="S3.5.p1.11.m11.3.4.2.cmml">i</mi><mo id="S3.5.p1.11.m11.3.4.1" xref="S3.5.p1.11.m11.3.4.1.cmml">∈</mo><mrow id="S3.5.p1.11.m11.3.4.3.2" xref="S3.5.p1.11.m11.3.4.3.1.cmml"><mo id="S3.5.p1.11.m11.3.4.3.2.1" stretchy="false" xref="S3.5.p1.11.m11.3.4.3.1.cmml">{</mo><mn id="S3.5.p1.11.m11.1.1" xref="S3.5.p1.11.m11.1.1.cmml">0</mn><mo id="S3.5.p1.11.m11.3.4.3.2.2" xref="S3.5.p1.11.m11.3.4.3.1.cmml">,</mo><mi id="S3.5.p1.11.m11.2.2" mathvariant="normal" xref="S3.5.p1.11.m11.2.2.cmml">…</mi><mo id="S3.5.p1.11.m11.3.4.3.2.3" xref="S3.5.p1.11.m11.3.4.3.1.cmml">,</mo><mi id="S3.5.p1.11.m11.3.3" mathvariant="normal" xref="S3.5.p1.11.m11.3.3.cmml">ℓ</mi><mo id="S3.5.p1.11.m11.3.4.3.2.4" stretchy="false" xref="S3.5.p1.11.m11.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.11.m11.3b"><apply id="S3.5.p1.11.m11.3.4.cmml" xref="S3.5.p1.11.m11.3.4"><in id="S3.5.p1.11.m11.3.4.1.cmml" xref="S3.5.p1.11.m11.3.4.1"></in><ci id="S3.5.p1.11.m11.3.4.2.cmml" xref="S3.5.p1.11.m11.3.4.2">𝑖</ci><set id="S3.5.p1.11.m11.3.4.3.1.cmml" xref="S3.5.p1.11.m11.3.4.3.2"><cn id="S3.5.p1.11.m11.1.1.cmml" type="integer" xref="S3.5.p1.11.m11.1.1">0</cn><ci id="S3.5.p1.11.m11.2.2.cmml" xref="S3.5.p1.11.m11.2.2">…</ci><ci id="S3.5.p1.11.m11.3.3.cmml" xref="S3.5.p1.11.m11.3.3">ℓ</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.11.m11.3c">i\in\{0,\ldots,\ell\}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.11.m11.3d">italic_i ∈ { 0 , … , roman_ℓ }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.6.p2"> <p class="ltx_p" id="S3.6.p2.21">We begin by bounding <math alttext="|W\setminus B|" class="ltx_Math" display="inline" id="S3.6.p2.1.m1.1"><semantics id="S3.6.p2.1.m1.1a"><mrow id="S3.6.p2.1.m1.1.1.1" xref="S3.6.p2.1.m1.1.1.2.cmml"><mo id="S3.6.p2.1.m1.1.1.1.2" stretchy="false" xref="S3.6.p2.1.m1.1.1.2.1.cmml">|</mo><mrow id="S3.6.p2.1.m1.1.1.1.1" xref="S3.6.p2.1.m1.1.1.1.1.cmml"><mi id="S3.6.p2.1.m1.1.1.1.1.2" xref="S3.6.p2.1.m1.1.1.1.1.2.cmml">W</mi><mo id="S3.6.p2.1.m1.1.1.1.1.1" xref="S3.6.p2.1.m1.1.1.1.1.1.cmml">∖</mo><mi id="S3.6.p2.1.m1.1.1.1.1.3" xref="S3.6.p2.1.m1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.6.p2.1.m1.1.1.1.3" stretchy="false" xref="S3.6.p2.1.m1.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.1.m1.1b"><apply id="S3.6.p2.1.m1.1.1.2.cmml" xref="S3.6.p2.1.m1.1.1.1"><abs id="S3.6.p2.1.m1.1.1.2.1.cmml" xref="S3.6.p2.1.m1.1.1.1.2"></abs><apply id="S3.6.p2.1.m1.1.1.1.1.cmml" xref="S3.6.p2.1.m1.1.1.1.1"><setdiff id="S3.6.p2.1.m1.1.1.1.1.1.cmml" xref="S3.6.p2.1.m1.1.1.1.1.1"></setdiff><ci id="S3.6.p2.1.m1.1.1.1.1.2.cmml" xref="S3.6.p2.1.m1.1.1.1.1.2">𝑊</ci><ci id="S3.6.p2.1.m1.1.1.1.1.3.cmml" xref="S3.6.p2.1.m1.1.1.1.1.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.1.m1.1c">|W\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.1.m1.1d">| italic_W ∖ italic_B |</annotation></semantics></math> using the path sets <math alttext="\mathcal{P}_{0},\ldots,\mathcal{P}_{\ell}" class="ltx_Math" display="inline" id="S3.6.p2.2.m2.3"><semantics id="S3.6.p2.2.m2.3a"><mrow id="S3.6.p2.2.m2.3.3.2" xref="S3.6.p2.2.m2.3.3.3.cmml"><msub id="S3.6.p2.2.m2.2.2.1.1" xref="S3.6.p2.2.m2.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.2.m2.2.2.1.1.2" xref="S3.6.p2.2.m2.2.2.1.1.2.cmml">𝒫</mi><mn id="S3.6.p2.2.m2.2.2.1.1.3" xref="S3.6.p2.2.m2.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.6.p2.2.m2.3.3.2.3" xref="S3.6.p2.2.m2.3.3.3.cmml">,</mo><mi id="S3.6.p2.2.m2.1.1" mathvariant="normal" xref="S3.6.p2.2.m2.1.1.cmml">…</mi><mo id="S3.6.p2.2.m2.3.3.2.4" xref="S3.6.p2.2.m2.3.3.3.cmml">,</mo><msub id="S3.6.p2.2.m2.3.3.2.2" xref="S3.6.p2.2.m2.3.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.2.m2.3.3.2.2.2" xref="S3.6.p2.2.m2.3.3.2.2.2.cmml">𝒫</mi><mi id="S3.6.p2.2.m2.3.3.2.2.3" mathvariant="normal" xref="S3.6.p2.2.m2.3.3.2.2.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.2.m2.3b"><list id="S3.6.p2.2.m2.3.3.3.cmml" xref="S3.6.p2.2.m2.3.3.2"><apply id="S3.6.p2.2.m2.2.2.1.1.cmml" xref="S3.6.p2.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3.6.p2.2.m2.2.2.1.1.1.cmml" xref="S3.6.p2.2.m2.2.2.1.1">subscript</csymbol><ci id="S3.6.p2.2.m2.2.2.1.1.2.cmml" xref="S3.6.p2.2.m2.2.2.1.1.2">𝒫</ci><cn id="S3.6.p2.2.m2.2.2.1.1.3.cmml" type="integer" xref="S3.6.p2.2.m2.2.2.1.1.3">0</cn></apply><ci id="S3.6.p2.2.m2.1.1.cmml" xref="S3.6.p2.2.m2.1.1">…</ci><apply id="S3.6.p2.2.m2.3.3.2.2.cmml" xref="S3.6.p2.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S3.6.p2.2.m2.3.3.2.2.1.cmml" xref="S3.6.p2.2.m2.3.3.2.2">subscript</csymbol><ci id="S3.6.p2.2.m2.3.3.2.2.2.cmml" xref="S3.6.p2.2.m2.3.3.2.2.2">𝒫</ci><ci id="S3.6.p2.2.m2.3.3.2.2.3.cmml" xref="S3.6.p2.2.m2.3.3.2.2.3">ℓ</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.2.m2.3c">\mathcal{P}_{0},\ldots,\mathcal{P}_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.2.m2.3d">caligraphic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , caligraphic_P start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. For each <math alttext="i\in\{0,\ldots,\ell\}" class="ltx_Math" display="inline" id="S3.6.p2.3.m3.3"><semantics id="S3.6.p2.3.m3.3a"><mrow id="S3.6.p2.3.m3.3.4" xref="S3.6.p2.3.m3.3.4.cmml"><mi id="S3.6.p2.3.m3.3.4.2" xref="S3.6.p2.3.m3.3.4.2.cmml">i</mi><mo id="S3.6.p2.3.m3.3.4.1" xref="S3.6.p2.3.m3.3.4.1.cmml">∈</mo><mrow id="S3.6.p2.3.m3.3.4.3.2" xref="S3.6.p2.3.m3.3.4.3.1.cmml"><mo id="S3.6.p2.3.m3.3.4.3.2.1" stretchy="false" xref="S3.6.p2.3.m3.3.4.3.1.cmml">{</mo><mn id="S3.6.p2.3.m3.1.1" xref="S3.6.p2.3.m3.1.1.cmml">0</mn><mo id="S3.6.p2.3.m3.3.4.3.2.2" xref="S3.6.p2.3.m3.3.4.3.1.cmml">,</mo><mi id="S3.6.p2.3.m3.2.2" mathvariant="normal" xref="S3.6.p2.3.m3.2.2.cmml">…</mi><mo id="S3.6.p2.3.m3.3.4.3.2.3" xref="S3.6.p2.3.m3.3.4.3.1.cmml">,</mo><mi id="S3.6.p2.3.m3.3.3" mathvariant="normal" xref="S3.6.p2.3.m3.3.3.cmml">ℓ</mi><mo id="S3.6.p2.3.m3.3.4.3.2.4" stretchy="false" xref="S3.6.p2.3.m3.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.3.m3.3b"><apply id="S3.6.p2.3.m3.3.4.cmml" xref="S3.6.p2.3.m3.3.4"><in id="S3.6.p2.3.m3.3.4.1.cmml" xref="S3.6.p2.3.m3.3.4.1"></in><ci id="S3.6.p2.3.m3.3.4.2.cmml" xref="S3.6.p2.3.m3.3.4.2">𝑖</ci><set id="S3.6.p2.3.m3.3.4.3.1.cmml" xref="S3.6.p2.3.m3.3.4.3.2"><cn id="S3.6.p2.3.m3.1.1.cmml" type="integer" xref="S3.6.p2.3.m3.1.1">0</cn><ci id="S3.6.p2.3.m3.2.2.cmml" xref="S3.6.p2.3.m3.2.2">…</ci><ci id="S3.6.p2.3.m3.3.3.cmml" xref="S3.6.p2.3.m3.3.3">ℓ</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.3.m3.3c">i\in\{0,\ldots,\ell\}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.3.m3.3d">italic_i ∈ { 0 , … , roman_ℓ }</annotation></semantics></math>, let <math alttext="\mathcal{Q}_{i}\subseteq\mathcal{P}_{i}" class="ltx_Math" display="inline" id="S3.6.p2.4.m4.1"><semantics id="S3.6.p2.4.m4.1a"><mrow id="S3.6.p2.4.m4.1.1" xref="S3.6.p2.4.m4.1.1.cmml"><msub id="S3.6.p2.4.m4.1.1.2" xref="S3.6.p2.4.m4.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.4.m4.1.1.2.2" xref="S3.6.p2.4.m4.1.1.2.2.cmml">𝒬</mi><mi id="S3.6.p2.4.m4.1.1.2.3" xref="S3.6.p2.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S3.6.p2.4.m4.1.1.1" xref="S3.6.p2.4.m4.1.1.1.cmml">⊆</mo><msub id="S3.6.p2.4.m4.1.1.3" xref="S3.6.p2.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.4.m4.1.1.3.2" xref="S3.6.p2.4.m4.1.1.3.2.cmml">𝒫</mi><mi id="S3.6.p2.4.m4.1.1.3.3" xref="S3.6.p2.4.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.4.m4.1b"><apply id="S3.6.p2.4.m4.1.1.cmml" xref="S3.6.p2.4.m4.1.1"><subset id="S3.6.p2.4.m4.1.1.1.cmml" xref="S3.6.p2.4.m4.1.1.1"></subset><apply id="S3.6.p2.4.m4.1.1.2.cmml" xref="S3.6.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.6.p2.4.m4.1.1.2.1.cmml" xref="S3.6.p2.4.m4.1.1.2">subscript</csymbol><ci id="S3.6.p2.4.m4.1.1.2.2.cmml" xref="S3.6.p2.4.m4.1.1.2.2">𝒬</ci><ci id="S3.6.p2.4.m4.1.1.2.3.cmml" xref="S3.6.p2.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S3.6.p2.4.m4.1.1.3.cmml" xref="S3.6.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.6.p2.4.m4.1.1.3.1.cmml" xref="S3.6.p2.4.m4.1.1.3">subscript</csymbol><ci id="S3.6.p2.4.m4.1.1.3.2.cmml" xref="S3.6.p2.4.m4.1.1.3.2">𝒫</ci><ci id="S3.6.p2.4.m4.1.1.3.3.cmml" xref="S3.6.p2.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.4.m4.1c">\mathcal{Q}_{i}\subseteq\mathcal{P}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.4.m4.1d">caligraphic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊆ caligraphic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> contain the paths in <math alttext="\mathcal{P}_{i}" class="ltx_Math" display="inline" id="S3.6.p2.5.m5.1"><semantics id="S3.6.p2.5.m5.1a"><msub id="S3.6.p2.5.m5.1.1" xref="S3.6.p2.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.5.m5.1.1.2" xref="S3.6.p2.5.m5.1.1.2.cmml">𝒫</mi><mi id="S3.6.p2.5.m5.1.1.3" xref="S3.6.p2.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p2.5.m5.1b"><apply id="S3.6.p2.5.m5.1.1.cmml" xref="S3.6.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.6.p2.5.m5.1.1.1.cmml" xref="S3.6.p2.5.m5.1.1">subscript</csymbol><ci id="S3.6.p2.5.m5.1.1.2.cmml" xref="S3.6.p2.5.m5.1.1.2">𝒫</ci><ci id="S3.6.p2.5.m5.1.1.3.cmml" xref="S3.6.p2.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.5.m5.1c">\mathcal{P}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.5.m5.1d">caligraphic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> that begin at a vertex in <math alttext="A\setminus B" class="ltx_Math" display="inline" id="S3.6.p2.6.m6.1"><semantics id="S3.6.p2.6.m6.1a"><mrow id="S3.6.p2.6.m6.1.1" xref="S3.6.p2.6.m6.1.1.cmml"><mi id="S3.6.p2.6.m6.1.1.2" xref="S3.6.p2.6.m6.1.1.2.cmml">A</mi><mo id="S3.6.p2.6.m6.1.1.1" xref="S3.6.p2.6.m6.1.1.1.cmml">∖</mo><mi id="S3.6.p2.6.m6.1.1.3" xref="S3.6.p2.6.m6.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.6.m6.1b"><apply id="S3.6.p2.6.m6.1.1.cmml" xref="S3.6.p2.6.m6.1.1"><setdiff id="S3.6.p2.6.m6.1.1.1.cmml" xref="S3.6.p2.6.m6.1.1.1"></setdiff><ci id="S3.6.p2.6.m6.1.1.2.cmml" xref="S3.6.p2.6.m6.1.1.2">𝐴</ci><ci id="S3.6.p2.6.m6.1.1.3.cmml" xref="S3.6.p2.6.m6.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.6.m6.1c">A\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.6.m6.1d">italic_A ∖ italic_B</annotation></semantics></math> and let <math alttext="\overline{\mathcal{Q}}_{i}:=\mathcal{P}_{i}\setminus\mathcal{Q}_{i}" class="ltx_Math" display="inline" id="S3.6.p2.7.m7.1"><semantics id="S3.6.p2.7.m7.1a"><mrow id="S3.6.p2.7.m7.1.1" xref="S3.6.p2.7.m7.1.1.cmml"><msub id="S3.6.p2.7.m7.1.1.2" xref="S3.6.p2.7.m7.1.1.2.cmml"><mover accent="true" id="S3.6.p2.7.m7.1.1.2.2" xref="S3.6.p2.7.m7.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.7.m7.1.1.2.2.2" xref="S3.6.p2.7.m7.1.1.2.2.2.cmml">𝒬</mi><mo id="S3.6.p2.7.m7.1.1.2.2.1" xref="S3.6.p2.7.m7.1.1.2.2.1.cmml">¯</mo></mover><mi id="S3.6.p2.7.m7.1.1.2.3" xref="S3.6.p2.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="S3.6.p2.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.6.p2.7.m7.1.1.1.cmml">:=</mo><mrow id="S3.6.p2.7.m7.1.1.3" xref="S3.6.p2.7.m7.1.1.3.cmml"><msub id="S3.6.p2.7.m7.1.1.3.2" xref="S3.6.p2.7.m7.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.7.m7.1.1.3.2.2" xref="S3.6.p2.7.m7.1.1.3.2.2.cmml">𝒫</mi><mi id="S3.6.p2.7.m7.1.1.3.2.3" xref="S3.6.p2.7.m7.1.1.3.2.3.cmml">i</mi></msub><mo id="S3.6.p2.7.m7.1.1.3.1" xref="S3.6.p2.7.m7.1.1.3.1.cmml">∖</mo><msub id="S3.6.p2.7.m7.1.1.3.3" xref="S3.6.p2.7.m7.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.7.m7.1.1.3.3.2" xref="S3.6.p2.7.m7.1.1.3.3.2.cmml">𝒬</mi><mi id="S3.6.p2.7.m7.1.1.3.3.3" xref="S3.6.p2.7.m7.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.7.m7.1b"><apply id="S3.6.p2.7.m7.1.1.cmml" xref="S3.6.p2.7.m7.1.1"><csymbol cd="latexml" id="S3.6.p2.7.m7.1.1.1.cmml" xref="S3.6.p2.7.m7.1.1.1">assign</csymbol><apply id="S3.6.p2.7.m7.1.1.2.cmml" xref="S3.6.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.6.p2.7.m7.1.1.2.1.cmml" xref="S3.6.p2.7.m7.1.1.2">subscript</csymbol><apply id="S3.6.p2.7.m7.1.1.2.2.cmml" xref="S3.6.p2.7.m7.1.1.2.2"><ci id="S3.6.p2.7.m7.1.1.2.2.1.cmml" xref="S3.6.p2.7.m7.1.1.2.2.1">¯</ci><ci id="S3.6.p2.7.m7.1.1.2.2.2.cmml" xref="S3.6.p2.7.m7.1.1.2.2.2">𝒬</ci></apply><ci id="S3.6.p2.7.m7.1.1.2.3.cmml" xref="S3.6.p2.7.m7.1.1.2.3">𝑖</ci></apply><apply id="S3.6.p2.7.m7.1.1.3.cmml" xref="S3.6.p2.7.m7.1.1.3"><setdiff id="S3.6.p2.7.m7.1.1.3.1.cmml" xref="S3.6.p2.7.m7.1.1.3.1"></setdiff><apply id="S3.6.p2.7.m7.1.1.3.2.cmml" xref="S3.6.p2.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.6.p2.7.m7.1.1.3.2.1.cmml" xref="S3.6.p2.7.m7.1.1.3.2">subscript</csymbol><ci id="S3.6.p2.7.m7.1.1.3.2.2.cmml" xref="S3.6.p2.7.m7.1.1.3.2.2">𝒫</ci><ci id="S3.6.p2.7.m7.1.1.3.2.3.cmml" xref="S3.6.p2.7.m7.1.1.3.2.3">𝑖</ci></apply><apply id="S3.6.p2.7.m7.1.1.3.3.cmml" xref="S3.6.p2.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.6.p2.7.m7.1.1.3.3.1.cmml" xref="S3.6.p2.7.m7.1.1.3.3">subscript</csymbol><ci id="S3.6.p2.7.m7.1.1.3.3.2.cmml" xref="S3.6.p2.7.m7.1.1.3.3.2">𝒬</ci><ci id="S3.6.p2.7.m7.1.1.3.3.3.cmml" xref="S3.6.p2.7.m7.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.7.m7.1c">\overline{\mathcal{Q}}_{i}:=\mathcal{P}_{i}\setminus\mathcal{Q}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.7.m7.1d">over¯ start_ARG caligraphic_Q end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := caligraphic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ caligraphic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> contain the paths in <math alttext="\mathcal{P}_{i}" class="ltx_Math" display="inline" id="S3.6.p2.8.m8.1"><semantics id="S3.6.p2.8.m8.1a"><msub id="S3.6.p2.8.m8.1.1" xref="S3.6.p2.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.8.m8.1.1.2" xref="S3.6.p2.8.m8.1.1.2.cmml">𝒫</mi><mi id="S3.6.p2.8.m8.1.1.3" xref="S3.6.p2.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p2.8.m8.1b"><apply id="S3.6.p2.8.m8.1.1.cmml" xref="S3.6.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S3.6.p2.8.m8.1.1.1.cmml" xref="S3.6.p2.8.m8.1.1">subscript</csymbol><ci id="S3.6.p2.8.m8.1.1.2.cmml" xref="S3.6.p2.8.m8.1.1.2">𝒫</ci><ci id="S3.6.p2.8.m8.1.1.3.cmml" xref="S3.6.p2.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.8.m8.1c">\mathcal{P}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.8.m8.1d">caligraphic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> that begin at a vertex in <math alttext="B" class="ltx_Math" display="inline" id="S3.6.p2.9.m9.1"><semantics id="S3.6.p2.9.m9.1a"><mi id="S3.6.p2.9.m9.1.1" xref="S3.6.p2.9.m9.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.6.p2.9.m9.1b"><ci id="S3.6.p2.9.m9.1.1.cmml" xref="S3.6.p2.9.m9.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.9.m9.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.9.m9.1d">italic_B</annotation></semantics></math>. Since each path in <math alttext="\mathcal{Q}_{i}" class="ltx_Math" display="inline" id="S3.6.p2.10.m10.1"><semantics id="S3.6.p2.10.m10.1a"><msub id="S3.6.p2.10.m10.1.1" xref="S3.6.p2.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.10.m10.1.1.2" xref="S3.6.p2.10.m10.1.1.2.cmml">𝒬</mi><mi id="S3.6.p2.10.m10.1.1.3" xref="S3.6.p2.10.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p2.10.m10.1b"><apply id="S3.6.p2.10.m10.1.1.cmml" xref="S3.6.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S3.6.p2.10.m10.1.1.1.cmml" xref="S3.6.p2.10.m10.1.1">subscript</csymbol><ci id="S3.6.p2.10.m10.1.1.2.cmml" xref="S3.6.p2.10.m10.1.1.2">𝒬</ci><ci id="S3.6.p2.10.m10.1.1.3.cmml" xref="S3.6.p2.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.10.m10.1c">\mathcal{Q}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.10.m10.1d">caligraphic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> begins at a distinct vertex in <math alttext="\Delta_{i}\setminus B" class="ltx_Math" display="inline" id="S3.6.p2.11.m11.1"><semantics id="S3.6.p2.11.m11.1a"><mrow id="S3.6.p2.11.m11.1.1" xref="S3.6.p2.11.m11.1.1.cmml"><msub id="S3.6.p2.11.m11.1.1.2" xref="S3.6.p2.11.m11.1.1.2.cmml"><mi id="S3.6.p2.11.m11.1.1.2.2" mathvariant="normal" xref="S3.6.p2.11.m11.1.1.2.2.cmml">Δ</mi><mi id="S3.6.p2.11.m11.1.1.2.3" xref="S3.6.p2.11.m11.1.1.2.3.cmml">i</mi></msub><mo id="S3.6.p2.11.m11.1.1.1" xref="S3.6.p2.11.m11.1.1.1.cmml">∖</mo><mi id="S3.6.p2.11.m11.1.1.3" xref="S3.6.p2.11.m11.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.11.m11.1b"><apply id="S3.6.p2.11.m11.1.1.cmml" xref="S3.6.p2.11.m11.1.1"><setdiff id="S3.6.p2.11.m11.1.1.1.cmml" xref="S3.6.p2.11.m11.1.1.1"></setdiff><apply id="S3.6.p2.11.m11.1.1.2.cmml" xref="S3.6.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.6.p2.11.m11.1.1.2.1.cmml" xref="S3.6.p2.11.m11.1.1.2">subscript</csymbol><ci id="S3.6.p2.11.m11.1.1.2.2.cmml" xref="S3.6.p2.11.m11.1.1.2.2">Δ</ci><ci id="S3.6.p2.11.m11.1.1.2.3.cmml" xref="S3.6.p2.11.m11.1.1.2.3">𝑖</ci></apply><ci id="S3.6.p2.11.m11.1.1.3.cmml" xref="S3.6.p2.11.m11.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.11.m11.1c">\Delta_{i}\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.11.m11.1d">roman_Δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ italic_B</annotation></semantics></math>, <math alttext="|\mathcal{Q}_{i}|\leq|\Delta_{i}\setminus B|" class="ltx_Math" display="inline" id="S3.6.p2.12.m12.2"><semantics id="S3.6.p2.12.m12.2a"><mrow id="S3.6.p2.12.m12.2.2" xref="S3.6.p2.12.m12.2.2.cmml"><mrow id="S3.6.p2.12.m12.1.1.1.1" xref="S3.6.p2.12.m12.1.1.1.2.cmml"><mo id="S3.6.p2.12.m12.1.1.1.1.2" stretchy="false" xref="S3.6.p2.12.m12.1.1.1.2.1.cmml">|</mo><msub id="S3.6.p2.12.m12.1.1.1.1.1" xref="S3.6.p2.12.m12.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.12.m12.1.1.1.1.1.2" xref="S3.6.p2.12.m12.1.1.1.1.1.2.cmml">𝒬</mi><mi id="S3.6.p2.12.m12.1.1.1.1.1.3" xref="S3.6.p2.12.m12.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.6.p2.12.m12.1.1.1.1.3" stretchy="false" xref="S3.6.p2.12.m12.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.6.p2.12.m12.2.2.3" xref="S3.6.p2.12.m12.2.2.3.cmml">≤</mo><mrow id="S3.6.p2.12.m12.2.2.2.1" xref="S3.6.p2.12.m12.2.2.2.2.cmml"><mo id="S3.6.p2.12.m12.2.2.2.1.2" stretchy="false" xref="S3.6.p2.12.m12.2.2.2.2.1.cmml">|</mo><mrow id="S3.6.p2.12.m12.2.2.2.1.1" xref="S3.6.p2.12.m12.2.2.2.1.1.cmml"><msub id="S3.6.p2.12.m12.2.2.2.1.1.2" xref="S3.6.p2.12.m12.2.2.2.1.1.2.cmml"><mi id="S3.6.p2.12.m12.2.2.2.1.1.2.2" mathvariant="normal" xref="S3.6.p2.12.m12.2.2.2.1.1.2.2.cmml">Δ</mi><mi id="S3.6.p2.12.m12.2.2.2.1.1.2.3" xref="S3.6.p2.12.m12.2.2.2.1.1.2.3.cmml">i</mi></msub><mo id="S3.6.p2.12.m12.2.2.2.1.1.1" xref="S3.6.p2.12.m12.2.2.2.1.1.1.cmml">∖</mo><mi id="S3.6.p2.12.m12.2.2.2.1.1.3" xref="S3.6.p2.12.m12.2.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.6.p2.12.m12.2.2.2.1.3" stretchy="false" xref="S3.6.p2.12.m12.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.12.m12.2b"><apply id="S3.6.p2.12.m12.2.2.cmml" xref="S3.6.p2.12.m12.2.2"><leq id="S3.6.p2.12.m12.2.2.3.cmml" xref="S3.6.p2.12.m12.2.2.3"></leq><apply id="S3.6.p2.12.m12.1.1.1.2.cmml" xref="S3.6.p2.12.m12.1.1.1.1"><abs id="S3.6.p2.12.m12.1.1.1.2.1.cmml" xref="S3.6.p2.12.m12.1.1.1.1.2"></abs><apply id="S3.6.p2.12.m12.1.1.1.1.1.cmml" xref="S3.6.p2.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.6.p2.12.m12.1.1.1.1.1.1.cmml" xref="S3.6.p2.12.m12.1.1.1.1.1">subscript</csymbol><ci id="S3.6.p2.12.m12.1.1.1.1.1.2.cmml" xref="S3.6.p2.12.m12.1.1.1.1.1.2">𝒬</ci><ci id="S3.6.p2.12.m12.1.1.1.1.1.3.cmml" xref="S3.6.p2.12.m12.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S3.6.p2.12.m12.2.2.2.2.cmml" xref="S3.6.p2.12.m12.2.2.2.1"><abs id="S3.6.p2.12.m12.2.2.2.2.1.cmml" xref="S3.6.p2.12.m12.2.2.2.1.2"></abs><apply id="S3.6.p2.12.m12.2.2.2.1.1.cmml" xref="S3.6.p2.12.m12.2.2.2.1.1"><setdiff id="S3.6.p2.12.m12.2.2.2.1.1.1.cmml" xref="S3.6.p2.12.m12.2.2.2.1.1.1"></setdiff><apply id="S3.6.p2.12.m12.2.2.2.1.1.2.cmml" xref="S3.6.p2.12.m12.2.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.6.p2.12.m12.2.2.2.1.1.2.1.cmml" xref="S3.6.p2.12.m12.2.2.2.1.1.2">subscript</csymbol><ci id="S3.6.p2.12.m12.2.2.2.1.1.2.2.cmml" xref="S3.6.p2.12.m12.2.2.2.1.1.2.2">Δ</ci><ci id="S3.6.p2.12.m12.2.2.2.1.1.2.3.cmml" xref="S3.6.p2.12.m12.2.2.2.1.1.2.3">𝑖</ci></apply><ci id="S3.6.p2.12.m12.2.2.2.1.1.3.cmml" xref="S3.6.p2.12.m12.2.2.2.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.12.m12.2c">|\mathcal{Q}_{i}|\leq|\Delta_{i}\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.12.m12.2d">| caligraphic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ≤ | roman_Δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ italic_B |</annotation></semantics></math>. Each path in <math alttext="\overline{\mathcal{Q}}_{i}" class="ltx_Math" display="inline" id="S3.6.p2.13.m13.1"><semantics id="S3.6.p2.13.m13.1a"><msub id="S3.6.p2.13.m13.1.1" xref="S3.6.p2.13.m13.1.1.cmml"><mover accent="true" id="S3.6.p2.13.m13.1.1.2" xref="S3.6.p2.13.m13.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.13.m13.1.1.2.2" xref="S3.6.p2.13.m13.1.1.2.2.cmml">𝒬</mi><mo id="S3.6.p2.13.m13.1.1.2.1" xref="S3.6.p2.13.m13.1.1.2.1.cmml">¯</mo></mover><mi id="S3.6.p2.13.m13.1.1.3" xref="S3.6.p2.13.m13.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p2.13.m13.1b"><apply id="S3.6.p2.13.m13.1.1.cmml" xref="S3.6.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S3.6.p2.13.m13.1.1.1.cmml" xref="S3.6.p2.13.m13.1.1">subscript</csymbol><apply id="S3.6.p2.13.m13.1.1.2.cmml" xref="S3.6.p2.13.m13.1.1.2"><ci id="S3.6.p2.13.m13.1.1.2.1.cmml" xref="S3.6.p2.13.m13.1.1.2.1">¯</ci><ci id="S3.6.p2.13.m13.1.1.2.2.cmml" xref="S3.6.p2.13.m13.1.1.2.2">𝒬</ci></apply><ci id="S3.6.p2.13.m13.1.1.3.cmml" xref="S3.6.p2.13.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.13.m13.1c">\overline{\mathcal{Q}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.13.m13.1d">over¯ start_ARG caligraphic_Q end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> begins at a vertex in <math alttext="B" class="ltx_Math" display="inline" id="S3.6.p2.14.m14.1"><semantics id="S3.6.p2.14.m14.1a"><mi id="S3.6.p2.14.m14.1.1" xref="S3.6.p2.14.m14.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.6.p2.14.m14.1b"><ci id="S3.6.p2.14.m14.1.1.cmml" xref="S3.6.p2.14.m14.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.14.m14.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.14.m14.1d">italic_B</annotation></semantics></math> and ends at a vertex in <math alttext="W\setminus B" class="ltx_Math" display="inline" id="S3.6.p2.15.m15.1"><semantics id="S3.6.p2.15.m15.1a"><mrow id="S3.6.p2.15.m15.1.1" xref="S3.6.p2.15.m15.1.1.cmml"><mi id="S3.6.p2.15.m15.1.1.2" xref="S3.6.p2.15.m15.1.1.2.cmml">W</mi><mo id="S3.6.p2.15.m15.1.1.1" xref="S3.6.p2.15.m15.1.1.1.cmml">∖</mo><mi id="S3.6.p2.15.m15.1.1.3" xref="S3.6.p2.15.m15.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.15.m15.1b"><apply id="S3.6.p2.15.m15.1.1.cmml" xref="S3.6.p2.15.m15.1.1"><setdiff id="S3.6.p2.15.m15.1.1.1.cmml" xref="S3.6.p2.15.m15.1.1.1"></setdiff><ci id="S3.6.p2.15.m15.1.1.2.cmml" xref="S3.6.p2.15.m15.1.1.2">𝑊</ci><ci id="S3.6.p2.15.m15.1.1.3.cmml" xref="S3.6.p2.15.m15.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.15.m15.1c">W\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.15.m15.1d">italic_W ∖ italic_B</annotation></semantics></math>. Since <math alttext="(A,B)" class="ltx_Math" display="inline" id="S3.6.p2.16.m16.2"><semantics id="S3.6.p2.16.m16.2a"><mrow id="S3.6.p2.16.m16.2.3.2" xref="S3.6.p2.16.m16.2.3.1.cmml"><mo id="S3.6.p2.16.m16.2.3.2.1" stretchy="false" xref="S3.6.p2.16.m16.2.3.1.cmml">(</mo><mi id="S3.6.p2.16.m16.1.1" xref="S3.6.p2.16.m16.1.1.cmml">A</mi><mo id="S3.6.p2.16.m16.2.3.2.2" xref="S3.6.p2.16.m16.2.3.1.cmml">,</mo><mi id="S3.6.p2.16.m16.2.2" xref="S3.6.p2.16.m16.2.2.cmml">B</mi><mo id="S3.6.p2.16.m16.2.3.2.3" stretchy="false" xref="S3.6.p2.16.m16.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.16.m16.2b"><interval closure="open" id="S3.6.p2.16.m16.2.3.1.cmml" xref="S3.6.p2.16.m16.2.3.2"><ci id="S3.6.p2.16.m16.1.1.cmml" xref="S3.6.p2.16.m16.1.1">𝐴</ci><ci id="S3.6.p2.16.m16.2.2.cmml" xref="S3.6.p2.16.m16.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.16.m16.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.16.m16.2d">( italic_A , italic_B )</annotation></semantics></math> is a separation of <math alttext="G[W_{\ell+1}]" class="ltx_Math" display="inline" id="S3.6.p2.17.m17.1"><semantics id="S3.6.p2.17.m17.1a"><mrow id="S3.6.p2.17.m17.1.1" xref="S3.6.p2.17.m17.1.1.cmml"><mi id="S3.6.p2.17.m17.1.1.3" xref="S3.6.p2.17.m17.1.1.3.cmml">G</mi><mo id="S3.6.p2.17.m17.1.1.2" xref="S3.6.p2.17.m17.1.1.2.cmml">⁢</mo><mrow id="S3.6.p2.17.m17.1.1.1.1" xref="S3.6.p2.17.m17.1.1.1.2.cmml"><mo id="S3.6.p2.17.m17.1.1.1.1.2" stretchy="false" xref="S3.6.p2.17.m17.1.1.1.2.1.cmml">[</mo><msub id="S3.6.p2.17.m17.1.1.1.1.1" xref="S3.6.p2.17.m17.1.1.1.1.1.cmml"><mi id="S3.6.p2.17.m17.1.1.1.1.1.2" xref="S3.6.p2.17.m17.1.1.1.1.1.2.cmml">W</mi><mrow id="S3.6.p2.17.m17.1.1.1.1.1.3" xref="S3.6.p2.17.m17.1.1.1.1.1.3.cmml"><mi id="S3.6.p2.17.m17.1.1.1.1.1.3.2" mathvariant="normal" xref="S3.6.p2.17.m17.1.1.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.6.p2.17.m17.1.1.1.1.1.3.1" xref="S3.6.p2.17.m17.1.1.1.1.1.3.1.cmml">+</mo><mn id="S3.6.p2.17.m17.1.1.1.1.1.3.3" xref="S3.6.p2.17.m17.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.6.p2.17.m17.1.1.1.1.3" stretchy="false" xref="S3.6.p2.17.m17.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.17.m17.1b"><apply id="S3.6.p2.17.m17.1.1.cmml" xref="S3.6.p2.17.m17.1.1"><times id="S3.6.p2.17.m17.1.1.2.cmml" xref="S3.6.p2.17.m17.1.1.2"></times><ci id="S3.6.p2.17.m17.1.1.3.cmml" xref="S3.6.p2.17.m17.1.1.3">𝐺</ci><apply id="S3.6.p2.17.m17.1.1.1.2.cmml" xref="S3.6.p2.17.m17.1.1.1.1"><csymbol cd="latexml" id="S3.6.p2.17.m17.1.1.1.2.1.cmml" xref="S3.6.p2.17.m17.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.6.p2.17.m17.1.1.1.1.1.cmml" xref="S3.6.p2.17.m17.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.6.p2.17.m17.1.1.1.1.1.1.cmml" xref="S3.6.p2.17.m17.1.1.1.1.1">subscript</csymbol><ci id="S3.6.p2.17.m17.1.1.1.1.1.2.cmml" xref="S3.6.p2.17.m17.1.1.1.1.1.2">𝑊</ci><apply id="S3.6.p2.17.m17.1.1.1.1.1.3.cmml" xref="S3.6.p2.17.m17.1.1.1.1.1.3"><plus id="S3.6.p2.17.m17.1.1.1.1.1.3.1.cmml" xref="S3.6.p2.17.m17.1.1.1.1.1.3.1"></plus><ci id="S3.6.p2.17.m17.1.1.1.1.1.3.2.cmml" xref="S3.6.p2.17.m17.1.1.1.1.1.3.2">ℓ</ci><cn id="S3.6.p2.17.m17.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.6.p2.17.m17.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.17.m17.1c">G[W_{\ell+1}]</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.17.m17.1d">italic_G [ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math>, this implies that each path in <math alttext="\overline{\mathcal{Q}}_{i}" class="ltx_Math" display="inline" id="S3.6.p2.18.m18.1"><semantics id="S3.6.p2.18.m18.1a"><msub id="S3.6.p2.18.m18.1.1" xref="S3.6.p2.18.m18.1.1.cmml"><mover accent="true" id="S3.6.p2.18.m18.1.1.2" xref="S3.6.p2.18.m18.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.18.m18.1.1.2.2" xref="S3.6.p2.18.m18.1.1.2.2.cmml">𝒬</mi><mo id="S3.6.p2.18.m18.1.1.2.1" xref="S3.6.p2.18.m18.1.1.2.1.cmml">¯</mo></mover><mi id="S3.6.p2.18.m18.1.1.3" xref="S3.6.p2.18.m18.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p2.18.m18.1b"><apply id="S3.6.p2.18.m18.1.1.cmml" xref="S3.6.p2.18.m18.1.1"><csymbol cd="ambiguous" id="S3.6.p2.18.m18.1.1.1.cmml" xref="S3.6.p2.18.m18.1.1">subscript</csymbol><apply id="S3.6.p2.18.m18.1.1.2.cmml" xref="S3.6.p2.18.m18.1.1.2"><ci id="S3.6.p2.18.m18.1.1.2.1.cmml" xref="S3.6.p2.18.m18.1.1.2.1">¯</ci><ci id="S3.6.p2.18.m18.1.1.2.2.cmml" xref="S3.6.p2.18.m18.1.1.2.2">𝒬</ci></apply><ci id="S3.6.p2.18.m18.1.1.3.cmml" xref="S3.6.p2.18.m18.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.18.m18.1c">\overline{\mathcal{Q}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.18.m18.1d">over¯ start_ARG caligraphic_Q end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> contains a vertex in <math alttext="A\cap B" class="ltx_Math" display="inline" id="S3.6.p2.19.m19.1"><semantics id="S3.6.p2.19.m19.1a"><mrow id="S3.6.p2.19.m19.1.1" xref="S3.6.p2.19.m19.1.1.cmml"><mi id="S3.6.p2.19.m19.1.1.2" xref="S3.6.p2.19.m19.1.1.2.cmml">A</mi><mo id="S3.6.p2.19.m19.1.1.1" xref="S3.6.p2.19.m19.1.1.1.cmml">∩</mo><mi id="S3.6.p2.19.m19.1.1.3" xref="S3.6.p2.19.m19.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.19.m19.1b"><apply id="S3.6.p2.19.m19.1.1.cmml" xref="S3.6.p2.19.m19.1.1"><intersect id="S3.6.p2.19.m19.1.1.1.cmml" xref="S3.6.p2.19.m19.1.1.1"></intersect><ci id="S3.6.p2.19.m19.1.1.2.cmml" xref="S3.6.p2.19.m19.1.1.2">𝐴</ci><ci id="S3.6.p2.19.m19.1.1.3.cmml" xref="S3.6.p2.19.m19.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.19.m19.1c">A\cap B</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.19.m19.1d">italic_A ∩ italic_B</annotation></semantics></math>. Since the paths in <math alttext="\overline{\mathcal{Q}}_{i}" class="ltx_Math" display="inline" id="S3.6.p2.20.m20.1"><semantics id="S3.6.p2.20.m20.1a"><msub id="S3.6.p2.20.m20.1.1" xref="S3.6.p2.20.m20.1.1.cmml"><mover accent="true" id="S3.6.p2.20.m20.1.1.2" xref="S3.6.p2.20.m20.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.20.m20.1.1.2.2" xref="S3.6.p2.20.m20.1.1.2.2.cmml">𝒬</mi><mo id="S3.6.p2.20.m20.1.1.2.1" xref="S3.6.p2.20.m20.1.1.2.1.cmml">¯</mo></mover><mi id="S3.6.p2.20.m20.1.1.3" xref="S3.6.p2.20.m20.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p2.20.m20.1b"><apply id="S3.6.p2.20.m20.1.1.cmml" xref="S3.6.p2.20.m20.1.1"><csymbol cd="ambiguous" id="S3.6.p2.20.m20.1.1.1.cmml" xref="S3.6.p2.20.m20.1.1">subscript</csymbol><apply id="S3.6.p2.20.m20.1.1.2.cmml" xref="S3.6.p2.20.m20.1.1.2"><ci id="S3.6.p2.20.m20.1.1.2.1.cmml" xref="S3.6.p2.20.m20.1.1.2.1">¯</ci><ci id="S3.6.p2.20.m20.1.1.2.2.cmml" xref="S3.6.p2.20.m20.1.1.2.2">𝒬</ci></apply><ci id="S3.6.p2.20.m20.1.1.3.cmml" xref="S3.6.p2.20.m20.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.20.m20.1c">\overline{\mathcal{Q}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.20.m20.1d">over¯ start_ARG caligraphic_Q end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are pairwise vertex-disjoint, <math alttext="|\overline{\mathcal{Q}}_{i}|\leq|A\cap B|" class="ltx_Math" display="inline" id="S3.6.p2.21.m21.2"><semantics id="S3.6.p2.21.m21.2a"><mrow id="S3.6.p2.21.m21.2.2" xref="S3.6.p2.21.m21.2.2.cmml"><mrow id="S3.6.p2.21.m21.1.1.1.1" xref="S3.6.p2.21.m21.1.1.1.2.cmml"><mo id="S3.6.p2.21.m21.1.1.1.1.2" stretchy="false" xref="S3.6.p2.21.m21.1.1.1.2.1.cmml">|</mo><msub id="S3.6.p2.21.m21.1.1.1.1.1" xref="S3.6.p2.21.m21.1.1.1.1.1.cmml"><mover accent="true" id="S3.6.p2.21.m21.1.1.1.1.1.2" xref="S3.6.p2.21.m21.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.6.p2.21.m21.1.1.1.1.1.2.2" xref="S3.6.p2.21.m21.1.1.1.1.1.2.2.cmml">𝒬</mi><mo id="S3.6.p2.21.m21.1.1.1.1.1.2.1" xref="S3.6.p2.21.m21.1.1.1.1.1.2.1.cmml">¯</mo></mover><mi id="S3.6.p2.21.m21.1.1.1.1.1.3" xref="S3.6.p2.21.m21.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.6.p2.21.m21.1.1.1.1.3" stretchy="false" xref="S3.6.p2.21.m21.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.6.p2.21.m21.2.2.3" xref="S3.6.p2.21.m21.2.2.3.cmml">≤</mo><mrow id="S3.6.p2.21.m21.2.2.2.1" xref="S3.6.p2.21.m21.2.2.2.2.cmml"><mo id="S3.6.p2.21.m21.2.2.2.1.2" stretchy="false" xref="S3.6.p2.21.m21.2.2.2.2.1.cmml">|</mo><mrow id="S3.6.p2.21.m21.2.2.2.1.1" xref="S3.6.p2.21.m21.2.2.2.1.1.cmml"><mi id="S3.6.p2.21.m21.2.2.2.1.1.2" xref="S3.6.p2.21.m21.2.2.2.1.1.2.cmml">A</mi><mo id="S3.6.p2.21.m21.2.2.2.1.1.1" xref="S3.6.p2.21.m21.2.2.2.1.1.1.cmml">∩</mo><mi id="S3.6.p2.21.m21.2.2.2.1.1.3" xref="S3.6.p2.21.m21.2.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.6.p2.21.m21.2.2.2.1.3" stretchy="false" xref="S3.6.p2.21.m21.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p2.21.m21.2b"><apply id="S3.6.p2.21.m21.2.2.cmml" xref="S3.6.p2.21.m21.2.2"><leq id="S3.6.p2.21.m21.2.2.3.cmml" xref="S3.6.p2.21.m21.2.2.3"></leq><apply id="S3.6.p2.21.m21.1.1.1.2.cmml" xref="S3.6.p2.21.m21.1.1.1.1"><abs id="S3.6.p2.21.m21.1.1.1.2.1.cmml" xref="S3.6.p2.21.m21.1.1.1.1.2"></abs><apply id="S3.6.p2.21.m21.1.1.1.1.1.cmml" xref="S3.6.p2.21.m21.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.6.p2.21.m21.1.1.1.1.1.1.cmml" xref="S3.6.p2.21.m21.1.1.1.1.1">subscript</csymbol><apply id="S3.6.p2.21.m21.1.1.1.1.1.2.cmml" xref="S3.6.p2.21.m21.1.1.1.1.1.2"><ci id="S3.6.p2.21.m21.1.1.1.1.1.2.1.cmml" xref="S3.6.p2.21.m21.1.1.1.1.1.2.1">¯</ci><ci id="S3.6.p2.21.m21.1.1.1.1.1.2.2.cmml" xref="S3.6.p2.21.m21.1.1.1.1.1.2.2">𝒬</ci></apply><ci id="S3.6.p2.21.m21.1.1.1.1.1.3.cmml" xref="S3.6.p2.21.m21.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S3.6.p2.21.m21.2.2.2.2.cmml" xref="S3.6.p2.21.m21.2.2.2.1"><abs id="S3.6.p2.21.m21.2.2.2.2.1.cmml" xref="S3.6.p2.21.m21.2.2.2.1.2"></abs><apply id="S3.6.p2.21.m21.2.2.2.1.1.cmml" xref="S3.6.p2.21.m21.2.2.2.1.1"><intersect id="S3.6.p2.21.m21.2.2.2.1.1.1.cmml" xref="S3.6.p2.21.m21.2.2.2.1.1.1"></intersect><ci id="S3.6.p2.21.m21.2.2.2.1.1.2.cmml" xref="S3.6.p2.21.m21.2.2.2.1.1.2">𝐴</ci><ci id="S3.6.p2.21.m21.2.2.2.1.1.3.cmml" xref="S3.6.p2.21.m21.2.2.2.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p2.21.m21.2c">|\overline{\mathcal{Q}}_{i}|\leq|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.6.p2.21.m21.2d">| over¯ start_ARG caligraphic_Q end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ≤ | italic_A ∩ italic_B |</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.7.p3"> <p class="ltx_p" id="S3.7.p3.8">Each vertex <math alttext="w\in W\setminus B" class="ltx_Math" display="inline" id="S3.7.p3.1.m1.1"><semantics id="S3.7.p3.1.m1.1a"><mrow id="S3.7.p3.1.m1.1.1" xref="S3.7.p3.1.m1.1.1.cmml"><mi id="S3.7.p3.1.m1.1.1.2" xref="S3.7.p3.1.m1.1.1.2.cmml">w</mi><mo id="S3.7.p3.1.m1.1.1.1" xref="S3.7.p3.1.m1.1.1.1.cmml">∈</mo><mrow id="S3.7.p3.1.m1.1.1.3" xref="S3.7.p3.1.m1.1.1.3.cmml"><mi id="S3.7.p3.1.m1.1.1.3.2" xref="S3.7.p3.1.m1.1.1.3.2.cmml">W</mi><mo id="S3.7.p3.1.m1.1.1.3.1" xref="S3.7.p3.1.m1.1.1.3.1.cmml">∖</mo><mi id="S3.7.p3.1.m1.1.1.3.3" xref="S3.7.p3.1.m1.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.7.p3.1.m1.1b"><apply id="S3.7.p3.1.m1.1.1.cmml" xref="S3.7.p3.1.m1.1.1"><in id="S3.7.p3.1.m1.1.1.1.cmml" xref="S3.7.p3.1.m1.1.1.1"></in><ci id="S3.7.p3.1.m1.1.1.2.cmml" xref="S3.7.p3.1.m1.1.1.2">𝑤</ci><apply id="S3.7.p3.1.m1.1.1.3.cmml" xref="S3.7.p3.1.m1.1.1.3"><setdiff id="S3.7.p3.1.m1.1.1.3.1.cmml" xref="S3.7.p3.1.m1.1.1.3.1"></setdiff><ci id="S3.7.p3.1.m1.1.1.3.2.cmml" xref="S3.7.p3.1.m1.1.1.3.2">𝑊</ci><ci id="S3.7.p3.1.m1.1.1.3.3.cmml" xref="S3.7.p3.1.m1.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p3.1.m1.1c">w\in W\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.7.p3.1.m1.1d">italic_w ∈ italic_W ∖ italic_B</annotation></semantics></math> is the last vertex of exactly one path in <math alttext="\mathcal{P}_{i}" class="ltx_Math" display="inline" id="S3.7.p3.2.m2.1"><semantics id="S3.7.p3.2.m2.1a"><msub id="S3.7.p3.2.m2.1.1" xref="S3.7.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.7.p3.2.m2.1.1.2" xref="S3.7.p3.2.m2.1.1.2.cmml">𝒫</mi><mi id="S3.7.p3.2.m2.1.1.3" xref="S3.7.p3.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.7.p3.2.m2.1b"><apply id="S3.7.p3.2.m2.1.1.cmml" xref="S3.7.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.7.p3.2.m2.1.1.1.cmml" xref="S3.7.p3.2.m2.1.1">subscript</csymbol><ci id="S3.7.p3.2.m2.1.1.2.cmml" xref="S3.7.p3.2.m2.1.1.2">𝒫</ci><ci id="S3.7.p3.2.m2.1.1.3.cmml" xref="S3.7.p3.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p3.2.m2.1c">\mathcal{P}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.7.p3.2.m2.1d">caligraphic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, for each <math alttext="i\in\{0,\ldots,\ell\}" class="ltx_Math" display="inline" id="S3.7.p3.3.m3.3"><semantics id="S3.7.p3.3.m3.3a"><mrow id="S3.7.p3.3.m3.3.4" xref="S3.7.p3.3.m3.3.4.cmml"><mi id="S3.7.p3.3.m3.3.4.2" xref="S3.7.p3.3.m3.3.4.2.cmml">i</mi><mo id="S3.7.p3.3.m3.3.4.1" xref="S3.7.p3.3.m3.3.4.1.cmml">∈</mo><mrow id="S3.7.p3.3.m3.3.4.3.2" xref="S3.7.p3.3.m3.3.4.3.1.cmml"><mo id="S3.7.p3.3.m3.3.4.3.2.1" stretchy="false" xref="S3.7.p3.3.m3.3.4.3.1.cmml">{</mo><mn id="S3.7.p3.3.m3.1.1" xref="S3.7.p3.3.m3.1.1.cmml">0</mn><mo id="S3.7.p3.3.m3.3.4.3.2.2" xref="S3.7.p3.3.m3.3.4.3.1.cmml">,</mo><mi id="S3.7.p3.3.m3.2.2" mathvariant="normal" xref="S3.7.p3.3.m3.2.2.cmml">…</mi><mo id="S3.7.p3.3.m3.3.4.3.2.3" xref="S3.7.p3.3.m3.3.4.3.1.cmml">,</mo><mi id="S3.7.p3.3.m3.3.3" mathvariant="normal" xref="S3.7.p3.3.m3.3.3.cmml">ℓ</mi><mo id="S3.7.p3.3.m3.3.4.3.2.4" stretchy="false" xref="S3.7.p3.3.m3.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.7.p3.3.m3.3b"><apply id="S3.7.p3.3.m3.3.4.cmml" xref="S3.7.p3.3.m3.3.4"><in id="S3.7.p3.3.m3.3.4.1.cmml" xref="S3.7.p3.3.m3.3.4.1"></in><ci id="S3.7.p3.3.m3.3.4.2.cmml" xref="S3.7.p3.3.m3.3.4.2">𝑖</ci><set id="S3.7.p3.3.m3.3.4.3.1.cmml" xref="S3.7.p3.3.m3.3.4.3.2"><cn id="S3.7.p3.3.m3.1.1.cmml" type="integer" xref="S3.7.p3.3.m3.1.1">0</cn><ci id="S3.7.p3.3.m3.2.2.cmml" xref="S3.7.p3.3.m3.2.2">…</ci><ci id="S3.7.p3.3.m3.3.3.cmml" xref="S3.7.p3.3.m3.3.3">ℓ</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p3.3.m3.3c">i\in\{0,\ldots,\ell\}</annotation><annotation encoding="application/x-llamapun" id="S3.7.p3.3.m3.3d">italic_i ∈ { 0 , … , roman_ℓ }</annotation></semantics></math>. Thus, each vertex <math alttext="w\in W\setminus B" class="ltx_Math" display="inline" id="S3.7.p3.4.m4.1"><semantics id="S3.7.p3.4.m4.1a"><mrow id="S3.7.p3.4.m4.1.1" xref="S3.7.p3.4.m4.1.1.cmml"><mi id="S3.7.p3.4.m4.1.1.2" xref="S3.7.p3.4.m4.1.1.2.cmml">w</mi><mo id="S3.7.p3.4.m4.1.1.1" xref="S3.7.p3.4.m4.1.1.1.cmml">∈</mo><mrow id="S3.7.p3.4.m4.1.1.3" xref="S3.7.p3.4.m4.1.1.3.cmml"><mi id="S3.7.p3.4.m4.1.1.3.2" xref="S3.7.p3.4.m4.1.1.3.2.cmml">W</mi><mo id="S3.7.p3.4.m4.1.1.3.1" xref="S3.7.p3.4.m4.1.1.3.1.cmml">∖</mo><mi id="S3.7.p3.4.m4.1.1.3.3" xref="S3.7.p3.4.m4.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.7.p3.4.m4.1b"><apply id="S3.7.p3.4.m4.1.1.cmml" xref="S3.7.p3.4.m4.1.1"><in id="S3.7.p3.4.m4.1.1.1.cmml" xref="S3.7.p3.4.m4.1.1.1"></in><ci id="S3.7.p3.4.m4.1.1.2.cmml" xref="S3.7.p3.4.m4.1.1.2">𝑤</ci><apply id="S3.7.p3.4.m4.1.1.3.cmml" xref="S3.7.p3.4.m4.1.1.3"><setdiff id="S3.7.p3.4.m4.1.1.3.1.cmml" xref="S3.7.p3.4.m4.1.1.3.1"></setdiff><ci id="S3.7.p3.4.m4.1.1.3.2.cmml" xref="S3.7.p3.4.m4.1.1.3.2">𝑊</ci><ci id="S3.7.p3.4.m4.1.1.3.3.cmml" xref="S3.7.p3.4.m4.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p3.4.m4.1c">w\in W\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.7.p3.4.m4.1d">italic_w ∈ italic_W ∖ italic_B</annotation></semantics></math> is the endpoint of exactly <math alttext="\ell+1" class="ltx_Math" display="inline" id="S3.7.p3.5.m5.1"><semantics id="S3.7.p3.5.m5.1a"><mrow id="S3.7.p3.5.m5.1.1" xref="S3.7.p3.5.m5.1.1.cmml"><mi id="S3.7.p3.5.m5.1.1.2" mathvariant="normal" xref="S3.7.p3.5.m5.1.1.2.cmml">ℓ</mi><mo id="S3.7.p3.5.m5.1.1.1" xref="S3.7.p3.5.m5.1.1.1.cmml">+</mo><mn id="S3.7.p3.5.m5.1.1.3" xref="S3.7.p3.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.7.p3.5.m5.1b"><apply id="S3.7.p3.5.m5.1.1.cmml" xref="S3.7.p3.5.m5.1.1"><plus id="S3.7.p3.5.m5.1.1.1.cmml" xref="S3.7.p3.5.m5.1.1.1"></plus><ci id="S3.7.p3.5.m5.1.1.2.cmml" xref="S3.7.p3.5.m5.1.1.2">ℓ</ci><cn id="S3.7.p3.5.m5.1.1.3.cmml" type="integer" xref="S3.7.p3.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p3.5.m5.1c">\ell+1</annotation><annotation encoding="application/x-llamapun" id="S3.7.p3.5.m5.1d">roman_ℓ + 1</annotation></semantics></math> paths in <math alttext="\bigcup_{i=0}^{\ell}\mathcal{P}_{i}" class="ltx_Math" display="inline" id="S3.7.p3.6.m6.1"><semantics id="S3.7.p3.6.m6.1a"><mrow id="S3.7.p3.6.m6.1.1" xref="S3.7.p3.6.m6.1.1.cmml"><msubsup id="S3.7.p3.6.m6.1.1.1" xref="S3.7.p3.6.m6.1.1.1.cmml"><mo id="S3.7.p3.6.m6.1.1.1.2.2" xref="S3.7.p3.6.m6.1.1.1.2.2.cmml">⋃</mo><mrow id="S3.7.p3.6.m6.1.1.1.2.3" xref="S3.7.p3.6.m6.1.1.1.2.3.cmml"><mi id="S3.7.p3.6.m6.1.1.1.2.3.2" xref="S3.7.p3.6.m6.1.1.1.2.3.2.cmml">i</mi><mo id="S3.7.p3.6.m6.1.1.1.2.3.1" xref="S3.7.p3.6.m6.1.1.1.2.3.1.cmml">=</mo><mn id="S3.7.p3.6.m6.1.1.1.2.3.3" xref="S3.7.p3.6.m6.1.1.1.2.3.3.cmml">0</mn></mrow><mi id="S3.7.p3.6.m6.1.1.1.3" mathvariant="normal" xref="S3.7.p3.6.m6.1.1.1.3.cmml">ℓ</mi></msubsup><msub id="S3.7.p3.6.m6.1.1.2" xref="S3.7.p3.6.m6.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.7.p3.6.m6.1.1.2.2" xref="S3.7.p3.6.m6.1.1.2.2.cmml">𝒫</mi><mi id="S3.7.p3.6.m6.1.1.2.3" xref="S3.7.p3.6.m6.1.1.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.7.p3.6.m6.1b"><apply id="S3.7.p3.6.m6.1.1.cmml" xref="S3.7.p3.6.m6.1.1"><apply id="S3.7.p3.6.m6.1.1.1.cmml" xref="S3.7.p3.6.m6.1.1.1"><csymbol cd="ambiguous" id="S3.7.p3.6.m6.1.1.1.1.cmml" xref="S3.7.p3.6.m6.1.1.1">superscript</csymbol><apply id="S3.7.p3.6.m6.1.1.1.2.cmml" xref="S3.7.p3.6.m6.1.1.1"><csymbol cd="ambiguous" id="S3.7.p3.6.m6.1.1.1.2.1.cmml" xref="S3.7.p3.6.m6.1.1.1">subscript</csymbol><union id="S3.7.p3.6.m6.1.1.1.2.2.cmml" xref="S3.7.p3.6.m6.1.1.1.2.2"></union><apply id="S3.7.p3.6.m6.1.1.1.2.3.cmml" xref="S3.7.p3.6.m6.1.1.1.2.3"><eq id="S3.7.p3.6.m6.1.1.1.2.3.1.cmml" xref="S3.7.p3.6.m6.1.1.1.2.3.1"></eq><ci id="S3.7.p3.6.m6.1.1.1.2.3.2.cmml" xref="S3.7.p3.6.m6.1.1.1.2.3.2">𝑖</ci><cn id="S3.7.p3.6.m6.1.1.1.2.3.3.cmml" type="integer" xref="S3.7.p3.6.m6.1.1.1.2.3.3">0</cn></apply></apply><ci id="S3.7.p3.6.m6.1.1.1.3.cmml" xref="S3.7.p3.6.m6.1.1.1.3">ℓ</ci></apply><apply id="S3.7.p3.6.m6.1.1.2.cmml" xref="S3.7.p3.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.7.p3.6.m6.1.1.2.1.cmml" xref="S3.7.p3.6.m6.1.1.2">subscript</csymbol><ci id="S3.7.p3.6.m6.1.1.2.2.cmml" xref="S3.7.p3.6.m6.1.1.2.2">𝒫</ci><ci id="S3.7.p3.6.m6.1.1.2.3.cmml" xref="S3.7.p3.6.m6.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p3.6.m6.1c">\bigcup_{i=0}^{\ell}\mathcal{P}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.7.p3.6.m6.1d">⋃ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT caligraphic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="\{\Delta_{0},\ldots,\Delta_{\ell+1}\}" class="ltx_Math" display="inline" id="S3.7.p3.7.m7.3"><semantics id="S3.7.p3.7.m7.3a"><mrow id="S3.7.p3.7.m7.3.3.2" xref="S3.7.p3.7.m7.3.3.3.cmml"><mo id="S3.7.p3.7.m7.3.3.2.3" stretchy="false" xref="S3.7.p3.7.m7.3.3.3.cmml">{</mo><msub id="S3.7.p3.7.m7.2.2.1.1" xref="S3.7.p3.7.m7.2.2.1.1.cmml"><mi id="S3.7.p3.7.m7.2.2.1.1.2" mathvariant="normal" xref="S3.7.p3.7.m7.2.2.1.1.2.cmml">Δ</mi><mn id="S3.7.p3.7.m7.2.2.1.1.3" xref="S3.7.p3.7.m7.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.7.p3.7.m7.3.3.2.4" xref="S3.7.p3.7.m7.3.3.3.cmml">,</mo><mi id="S3.7.p3.7.m7.1.1" mathvariant="normal" xref="S3.7.p3.7.m7.1.1.cmml">…</mi><mo id="S3.7.p3.7.m7.3.3.2.5" xref="S3.7.p3.7.m7.3.3.3.cmml">,</mo><msub id="S3.7.p3.7.m7.3.3.2.2" xref="S3.7.p3.7.m7.3.3.2.2.cmml"><mi id="S3.7.p3.7.m7.3.3.2.2.2" mathvariant="normal" xref="S3.7.p3.7.m7.3.3.2.2.2.cmml">Δ</mi><mrow id="S3.7.p3.7.m7.3.3.2.2.3" xref="S3.7.p3.7.m7.3.3.2.2.3.cmml"><mi id="S3.7.p3.7.m7.3.3.2.2.3.2" mathvariant="normal" xref="S3.7.p3.7.m7.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.7.p3.7.m7.3.3.2.2.3.1" xref="S3.7.p3.7.m7.3.3.2.2.3.1.cmml">+</mo><mn id="S3.7.p3.7.m7.3.3.2.2.3.3" xref="S3.7.p3.7.m7.3.3.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.7.p3.7.m7.3.3.2.6" stretchy="false" xref="S3.7.p3.7.m7.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.7.p3.7.m7.3b"><set id="S3.7.p3.7.m7.3.3.3.cmml" xref="S3.7.p3.7.m7.3.3.2"><apply id="S3.7.p3.7.m7.2.2.1.1.cmml" xref="S3.7.p3.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S3.7.p3.7.m7.2.2.1.1.1.cmml" xref="S3.7.p3.7.m7.2.2.1.1">subscript</csymbol><ci id="S3.7.p3.7.m7.2.2.1.1.2.cmml" xref="S3.7.p3.7.m7.2.2.1.1.2">Δ</ci><cn id="S3.7.p3.7.m7.2.2.1.1.3.cmml" type="integer" xref="S3.7.p3.7.m7.2.2.1.1.3">0</cn></apply><ci id="S3.7.p3.7.m7.1.1.cmml" xref="S3.7.p3.7.m7.1.1">…</ci><apply id="S3.7.p3.7.m7.3.3.2.2.cmml" xref="S3.7.p3.7.m7.3.3.2.2"><csymbol cd="ambiguous" id="S3.7.p3.7.m7.3.3.2.2.1.cmml" xref="S3.7.p3.7.m7.3.3.2.2">subscript</csymbol><ci id="S3.7.p3.7.m7.3.3.2.2.2.cmml" xref="S3.7.p3.7.m7.3.3.2.2.2">Δ</ci><apply id="S3.7.p3.7.m7.3.3.2.2.3.cmml" xref="S3.7.p3.7.m7.3.3.2.2.3"><plus id="S3.7.p3.7.m7.3.3.2.2.3.1.cmml" xref="S3.7.p3.7.m7.3.3.2.2.3.1"></plus><ci id="S3.7.p3.7.m7.3.3.2.2.3.2.cmml" xref="S3.7.p3.7.m7.3.3.2.2.3.2">ℓ</ci><cn id="S3.7.p3.7.m7.3.3.2.2.3.3.cmml" type="integer" xref="S3.7.p3.7.m7.3.3.2.2.3.3">1</cn></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p3.7.m7.3c">\{\Delta_{0},\ldots,\Delta_{\ell+1}\}</annotation><annotation encoding="application/x-llamapun" id="S3.7.p3.7.m7.3d">{ roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT }</annotation></semantics></math> is a partition of <math alttext="W_{\ell+1}" class="ltx_Math" display="inline" id="S3.7.p3.8.m8.1"><semantics id="S3.7.p3.8.m8.1a"><msub id="S3.7.p3.8.m8.1.1" xref="S3.7.p3.8.m8.1.1.cmml"><mi id="S3.7.p3.8.m8.1.1.2" xref="S3.7.p3.8.m8.1.1.2.cmml">W</mi><mrow id="S3.7.p3.8.m8.1.1.3" xref="S3.7.p3.8.m8.1.1.3.cmml"><mi id="S3.7.p3.8.m8.1.1.3.2" mathvariant="normal" xref="S3.7.p3.8.m8.1.1.3.2.cmml">ℓ</mi><mo id="S3.7.p3.8.m8.1.1.3.1" xref="S3.7.p3.8.m8.1.1.3.1.cmml">+</mo><mn id="S3.7.p3.8.m8.1.1.3.3" xref="S3.7.p3.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.7.p3.8.m8.1b"><apply id="S3.7.p3.8.m8.1.1.cmml" xref="S3.7.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S3.7.p3.8.m8.1.1.1.cmml" xref="S3.7.p3.8.m8.1.1">subscript</csymbol><ci id="S3.7.p3.8.m8.1.1.2.cmml" xref="S3.7.p3.8.m8.1.1.2">𝑊</ci><apply id="S3.7.p3.8.m8.1.1.3.cmml" xref="S3.7.p3.8.m8.1.1.3"><plus id="S3.7.p3.8.m8.1.1.3.1.cmml" xref="S3.7.p3.8.m8.1.1.3.1"></plus><ci id="S3.7.p3.8.m8.1.1.3.2.cmml" xref="S3.7.p3.8.m8.1.1.3.2">ℓ</ci><cn id="S3.7.p3.8.m8.1.1.3.3.cmml" type="integer" xref="S3.7.p3.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p3.8.m8.1c">W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.7.p3.8.m8.1d">italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, we have:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx1"> <tbody id="S3.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell" colspan="2"><math alttext="\displaystyle\begin{split}|W\setminus B|&=\frac{1}{\ell+1}\cdot\sum_{i=0}^{% \ell}|\mathcal{P}_{i}|\\ &=\frac{1}{\ell+1}\cdot\sum_{i=0}^{\ell}\left(|\mathcal{Q}_{i}|+|\overline{% \mathcal{Q}_{i}}|\right)\\ &\leq\frac{1}{\ell+1}\cdot\sum_{i=0}^{\ell}\left(|\Delta_{i}\setminus B|+|A% \cap B|\right)\\ &=\frac{|W_{\ell}\setminus B|}{\ell+1}+|A\cap B|\\ &=\frac{|A\setminus B|-|\Delta_{\ell+1}\setminus B|}{\ell+1}+|A\cap B|\enspace% .\end{split}" class="ltx_Math" display="inline" id="S3.E1.m1.75"><semantics id="S3.E1.m1.75a"><mtable columnspacing="0pt" id="S3.E1.m1.75.75.7" rowspacing="0pt"><mtr id="S3.E1.m1.75.75.7a"><mtd class="ltx_align_right" columnalign="right" id="S3.E1.m1.75.75.7b"><mrow id="S3.E1.m1.70.70.2.69.16.6.6"><mo id="S3.E1.m1.1.1.1.1.1.1" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo><mrow id="S3.E1.m1.70.70.2.69.16.6.6.1"><mi id="S3.E1.m1.2.2.2.2.2.2" xref="S3.E1.m1.2.2.2.2.2.2.cmml">W</mi><mo id="S3.E1.m1.3.3.3.3.3.3" xref="S3.E1.m1.3.3.3.3.3.3.cmml">∖</mo><mi id="S3.E1.m1.4.4.4.4.4.4" xref="S3.E1.m1.4.4.4.4.4.4.cmml">B</mi></mrow><mo id="S3.E1.m1.5.5.5.5.5.5" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S3.E1.m1.75.75.7c"><mrow id="S3.E1.m1.71.71.3.70.17.11"><mi id="S3.E1.m1.71.71.3.70.17.11.12" xref="S3.E1.m1.69.69.1.1.1.cmml"></mi><mo id="S3.E1.m1.6.6.6.6.1.1" xref="S3.E1.m1.6.6.6.6.1.1.cmml">=</mo><mrow id="S3.E1.m1.71.71.3.70.17.11.11"><mstyle displaystyle="true" id="S3.E1.m1.7.7.7.7.2.2" xref="S3.E1.m1.7.7.7.7.2.2.cmml"><mfrac id="S3.E1.m1.7.7.7.7.2.2a" xref="S3.E1.m1.7.7.7.7.2.2.cmml"><mn id="S3.E1.m1.7.7.7.7.2.2.2" xref="S3.E1.m1.7.7.7.7.2.2.2.cmml">1</mn><mrow id="S3.E1.m1.7.7.7.7.2.2.3" xref="S3.E1.m1.7.7.7.7.2.2.3.cmml"><mi id="S3.E1.m1.7.7.7.7.2.2.3.2" mathvariant="normal" xref="S3.E1.m1.7.7.7.7.2.2.3.2.cmml">ℓ</mi><mo id="S3.E1.m1.7.7.7.7.2.2.3.1" xref="S3.E1.m1.7.7.7.7.2.2.3.1.cmml">+</mo><mn id="S3.E1.m1.7.7.7.7.2.2.3.3" xref="S3.E1.m1.7.7.7.7.2.2.3.3.cmml">1</mn></mrow></mfrac></mstyle><mo id="S3.E1.m1.8.8.8.8.3.3" lspace="0.222em" rspace="0.222em" xref="S3.E1.m1.8.8.8.8.3.3.cmml">⋅</mo><mrow id="S3.E1.m1.71.71.3.70.17.11.11.1"><mstyle displaystyle="true" id="S3.E1.m1.71.71.3.70.17.11.11.1.2"><munderover id="S3.E1.m1.71.71.3.70.17.11.11.1.2a"><mo id="S3.E1.m1.9.9.9.9.4.4" movablelimits="false" xref="S3.E1.m1.9.9.9.9.4.4.cmml">∑</mo><mrow id="S3.E1.m1.10.10.10.10.5.5.1" xref="S3.E1.m1.10.10.10.10.5.5.1.cmml"><mi id="S3.E1.m1.10.10.10.10.5.5.1.2" xref="S3.E1.m1.10.10.10.10.5.5.1.2.cmml">i</mi><mo id="S3.E1.m1.10.10.10.10.5.5.1.1" xref="S3.E1.m1.10.10.10.10.5.5.1.1.cmml">=</mo><mn id="S3.E1.m1.10.10.10.10.5.5.1.3" xref="S3.E1.m1.10.10.10.10.5.5.1.3.cmml">0</mn></mrow><mi id="S3.E1.m1.11.11.11.11.6.6.1" mathvariant="normal" xref="S3.E1.m1.11.11.11.11.6.6.1.cmml">ℓ</mi></munderover></mstyle><mrow id="S3.E1.m1.71.71.3.70.17.11.11.1.1.1"><mo id="S3.E1.m1.12.12.12.12.7.7" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo><msub id="S3.E1.m1.71.71.3.70.17.11.11.1.1.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.13.13.13.13.8.8" xref="S3.E1.m1.13.13.13.13.8.8.cmml">𝒫</mi><mi id="S3.E1.m1.14.14.14.14.9.9.1" xref="S3.E1.m1.14.14.14.14.9.9.1.cmml">i</mi></msub><mo id="S3.E1.m1.15.15.15.15.10.10" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E1.m1.75.75.7d"><mtd id="S3.E1.m1.75.75.7e" xref="S3.E1.m1.69.69.1.1.1.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S3.E1.m1.75.75.7f"><mrow id="S3.E1.m1.72.72.4.71.17.17"><mi id="S3.E1.m1.72.72.4.71.17.17.18" xref="S3.E1.m1.69.69.1.1.1.cmml"></mi><mo id="S3.E1.m1.16.16.16.1.1.1" xref="S3.E1.m1.16.16.16.1.1.1.cmml">=</mo><mrow id="S3.E1.m1.72.72.4.71.17.17.17"><mstyle displaystyle="true" id="S3.E1.m1.17.17.17.2.2.2" xref="S3.E1.m1.17.17.17.2.2.2.cmml"><mfrac id="S3.E1.m1.17.17.17.2.2.2a" xref="S3.E1.m1.17.17.17.2.2.2.cmml"><mn id="S3.E1.m1.17.17.17.2.2.2.2" xref="S3.E1.m1.17.17.17.2.2.2.2.cmml">1</mn><mrow id="S3.E1.m1.17.17.17.2.2.2.3" xref="S3.E1.m1.17.17.17.2.2.2.3.cmml"><mi id="S3.E1.m1.17.17.17.2.2.2.3.2" mathvariant="normal" xref="S3.E1.m1.17.17.17.2.2.2.3.2.cmml">ℓ</mi><mo id="S3.E1.m1.17.17.17.2.2.2.3.1" xref="S3.E1.m1.17.17.17.2.2.2.3.1.cmml">+</mo><mn id="S3.E1.m1.17.17.17.2.2.2.3.3" xref="S3.E1.m1.17.17.17.2.2.2.3.3.cmml">1</mn></mrow></mfrac></mstyle><mo id="S3.E1.m1.18.18.18.3.3.3" lspace="0.222em" rspace="0.222em" xref="S3.E1.m1.18.18.18.3.3.3.cmml">⋅</mo><mrow id="S3.E1.m1.72.72.4.71.17.17.17.1"><mstyle displaystyle="true" id="S3.E1.m1.72.72.4.71.17.17.17.1.2"><munderover id="S3.E1.m1.72.72.4.71.17.17.17.1.2a"><mo id="S3.E1.m1.19.19.19.4.4.4" movablelimits="false" xref="S3.E1.m1.19.19.19.4.4.4.cmml">∑</mo><mrow id="S3.E1.m1.20.20.20.5.5.5.1" xref="S3.E1.m1.20.20.20.5.5.5.1.cmml"><mi id="S3.E1.m1.20.20.20.5.5.5.1.2" xref="S3.E1.m1.20.20.20.5.5.5.1.2.cmml">i</mi><mo id="S3.E1.m1.20.20.20.5.5.5.1.1" xref="S3.E1.m1.20.20.20.5.5.5.1.1.cmml">=</mo><mn id="S3.E1.m1.20.20.20.5.5.5.1.3" xref="S3.E1.m1.20.20.20.5.5.5.1.3.cmml">0</mn></mrow><mi id="S3.E1.m1.21.21.21.6.6.6.1" mathvariant="normal" xref="S3.E1.m1.21.21.21.6.6.6.1.cmml">ℓ</mi></munderover></mstyle><mrow id="S3.E1.m1.72.72.4.71.17.17.17.1.1.1"><mo id="S3.E1.m1.22.22.22.7.7.7" xref="S3.E1.m1.69.69.1.1.1.cmml">(</mo><mrow id="S3.E1.m1.72.72.4.71.17.17.17.1.1.1.1"><mrow id="S3.E1.m1.72.72.4.71.17.17.17.1.1.1.1.1.1"><mo id="S3.E1.m1.23.23.23.8.8.8" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo><msub id="S3.E1.m1.72.72.4.71.17.17.17.1.1.1.1.1.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.24.24.24.9.9.9" xref="S3.E1.m1.24.24.24.9.9.9.cmml">𝒬</mi><mi id="S3.E1.m1.25.25.25.10.10.10.1" xref="S3.E1.m1.25.25.25.10.10.10.1.cmml">i</mi></msub><mo id="S3.E1.m1.26.26.26.11.11.11" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo></mrow><mo id="S3.E1.m1.27.27.27.12.12.12" xref="S3.E1.m1.27.27.27.12.12.12.cmml">+</mo><mrow id="S3.E1.m1.72.72.4.71.17.17.17.1.1.1.1.2"><mo id="S3.E1.m1.28.28.28.13.13.13" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo><mover accent="true" id="S3.E1.m1.29.29.29.14.14.14" xref="S3.E1.m1.29.29.29.14.14.14.cmml"><msub id="S3.E1.m1.29.29.29.14.14.14.2" xref="S3.E1.m1.29.29.29.14.14.14.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.29.29.29.14.14.14.2.2" xref="S3.E1.m1.29.29.29.14.14.14.2.2.cmml">𝒬</mi><mi id="S3.E1.m1.29.29.29.14.14.14.2.3" xref="S3.E1.m1.29.29.29.14.14.14.2.3.cmml">i</mi></msub><mo id="S3.E1.m1.29.29.29.14.14.14.1" xref="S3.E1.m1.29.29.29.14.14.14.1.cmml">¯</mo></mover><mo id="S3.E1.m1.30.30.30.15.15.15" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo></mrow></mrow><mo id="S3.E1.m1.31.31.31.16.16.16" xref="S3.E1.m1.69.69.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E1.m1.75.75.7g"><mtd id="S3.E1.m1.75.75.7h" xref="S3.E1.m1.69.69.1.1.1.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S3.E1.m1.75.75.7i"><mrow id="S3.E1.m1.73.73.5.72.21.21"><mi id="S3.E1.m1.73.73.5.72.21.21.22" xref="S3.E1.m1.69.69.1.1.1.cmml"></mi><mo id="S3.E1.m1.32.32.32.1.1.1" xref="S3.E1.m1.32.32.32.1.1.1.cmml">≤</mo><mrow id="S3.E1.m1.73.73.5.72.21.21.21"><mstyle displaystyle="true" id="S3.E1.m1.33.33.33.2.2.2" xref="S3.E1.m1.33.33.33.2.2.2.cmml"><mfrac id="S3.E1.m1.33.33.33.2.2.2a" xref="S3.E1.m1.33.33.33.2.2.2.cmml"><mn id="S3.E1.m1.33.33.33.2.2.2.2" xref="S3.E1.m1.33.33.33.2.2.2.2.cmml">1</mn><mrow id="S3.E1.m1.33.33.33.2.2.2.3" xref="S3.E1.m1.33.33.33.2.2.2.3.cmml"><mi id="S3.E1.m1.33.33.33.2.2.2.3.2" mathvariant="normal" xref="S3.E1.m1.33.33.33.2.2.2.3.2.cmml">ℓ</mi><mo id="S3.E1.m1.33.33.33.2.2.2.3.1" xref="S3.E1.m1.33.33.33.2.2.2.3.1.cmml">+</mo><mn id="S3.E1.m1.33.33.33.2.2.2.3.3" xref="S3.E1.m1.33.33.33.2.2.2.3.3.cmml">1</mn></mrow></mfrac></mstyle><mo id="S3.E1.m1.34.34.34.3.3.3" lspace="0.222em" rspace="0.222em" xref="S3.E1.m1.34.34.34.3.3.3.cmml">⋅</mo><mrow id="S3.E1.m1.73.73.5.72.21.21.21.1"><mstyle displaystyle="true" id="S3.E1.m1.73.73.5.72.21.21.21.1.2"><munderover id="S3.E1.m1.73.73.5.72.21.21.21.1.2a"><mo id="S3.E1.m1.35.35.35.4.4.4" movablelimits="false" xref="S3.E1.m1.35.35.35.4.4.4.cmml">∑</mo><mrow id="S3.E1.m1.36.36.36.5.5.5.1" xref="S3.E1.m1.36.36.36.5.5.5.1.cmml"><mi id="S3.E1.m1.36.36.36.5.5.5.1.2" xref="S3.E1.m1.36.36.36.5.5.5.1.2.cmml">i</mi><mo id="S3.E1.m1.36.36.36.5.5.5.1.1" 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xref="S3.E1.m1.43.43.43.12.12.12.cmml">B</mi></mrow><mo id="S3.E1.m1.44.44.44.13.13.13" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo></mrow><mo id="S3.E1.m1.45.45.45.14.14.14" xref="S3.E1.m1.45.45.45.14.14.14.cmml">+</mo><mrow id="S3.E1.m1.73.73.5.72.21.21.21.1.1.1.1.2.1"><mo id="S3.E1.m1.46.46.46.15.15.15" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo><mrow id="S3.E1.m1.73.73.5.72.21.21.21.1.1.1.1.2.1.1"><mi id="S3.E1.m1.47.47.47.16.16.16" xref="S3.E1.m1.47.47.47.16.16.16.cmml">A</mi><mo id="S3.E1.m1.48.48.48.17.17.17" xref="S3.E1.m1.48.48.48.17.17.17.cmml">∩</mo><mi id="S3.E1.m1.49.49.49.18.18.18" xref="S3.E1.m1.49.49.49.18.18.18.cmml">B</mi></mrow><mo id="S3.E1.m1.50.50.50.19.19.19" stretchy="false" xref="S3.E1.m1.69.69.1.1.1.cmml">|</mo></mrow></mrow><mo id="S3.E1.m1.51.51.51.20.20.20" xref="S3.E1.m1.69.69.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E1.m1.75.75.7j"><mtd id="S3.E1.m1.75.75.7k" xref="S3.E1.m1.69.69.1.1.1.cmml"></mtd><mtd 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xref="S3.E1.m1.1.1.1.1.1.1"><intersect id="S3.E1.m1.65.65.65.6.6.6.cmml" xref="S3.E1.m1.65.65.65.6.6.6"></intersect><ci id="S3.E1.m1.64.64.64.5.5.5.cmml" xref="S3.E1.m1.64.64.64.5.5.5">𝐴</ci><ci id="S3.E1.m1.66.66.66.7.7.7.cmml" xref="S3.E1.m1.66.66.66.7.7.7">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1.m1.75c">\displaystyle\begin{split}|W\setminus B|&=\frac{1}{\ell+1}\cdot\sum_{i=0}^{% \ell}|\mathcal{P}_{i}|\\ &=\frac{1}{\ell+1}\cdot\sum_{i=0}^{\ell}\left(|\mathcal{Q}_{i}|+|\overline{% \mathcal{Q}_{i}}|\right)\\ &\leq\frac{1}{\ell+1}\cdot\sum_{i=0}^{\ell}\left(|\Delta_{i}\setminus B|+|A% \cap B|\right)\\ &=\frac{|W_{\ell}\setminus B|}{\ell+1}+|A\cap B|\\ &=\frac{|A\setminus B|-|\Delta_{\ell+1}\setminus B|}{\ell+1}+|A\cap B|\enspace% .\end{split}</annotation><annotation encoding="application/x-llamapun" id="S3.E1.m1.75d">start_ROW start_CELL | italic_W ∖ italic_B | end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG roman_ℓ + 1 end_ARG ⋅ ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT | caligraphic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG roman_ℓ + 1 end_ARG ⋅ ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ( | caligraphic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | + | over¯ start_ARG caligraphic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ divide start_ARG 1 end_ARG start_ARG roman_ℓ + 1 end_ARG ⋅ ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ( | roman_Δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ italic_B | + | italic_A ∩ italic_B | ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG | italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + | italic_A ∩ italic_B | end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG | italic_A ∖ italic_B | - | roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + | italic_A ∩ italic_B | . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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> </div> <div class="ltx_para" id="S3.8.p4"> <p class="ltx_p" id="S3.8.p4.12">Next, we bound <math alttext="|Z\setminus B|" class="ltx_Math" display="inline" id="S3.8.p4.1.m1.1"><semantics id="S3.8.p4.1.m1.1a"><mrow id="S3.8.p4.1.m1.1.1.1" xref="S3.8.p4.1.m1.1.1.2.cmml"><mo id="S3.8.p4.1.m1.1.1.1.2" stretchy="false" xref="S3.8.p4.1.m1.1.1.2.1.cmml">|</mo><mrow id="S3.8.p4.1.m1.1.1.1.1" xref="S3.8.p4.1.m1.1.1.1.1.cmml"><mi id="S3.8.p4.1.m1.1.1.1.1.2" xref="S3.8.p4.1.m1.1.1.1.1.2.cmml">Z</mi><mo id="S3.8.p4.1.m1.1.1.1.1.1" xref="S3.8.p4.1.m1.1.1.1.1.1.cmml">∖</mo><mi id="S3.8.p4.1.m1.1.1.1.1.3" xref="S3.8.p4.1.m1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.8.p4.1.m1.1.1.1.3" stretchy="false" xref="S3.8.p4.1.m1.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p4.1.m1.1b"><apply id="S3.8.p4.1.m1.1.1.2.cmml" xref="S3.8.p4.1.m1.1.1.1"><abs id="S3.8.p4.1.m1.1.1.2.1.cmml" xref="S3.8.p4.1.m1.1.1.1.2"></abs><apply id="S3.8.p4.1.m1.1.1.1.1.cmml" xref="S3.8.p4.1.m1.1.1.1.1"><setdiff id="S3.8.p4.1.m1.1.1.1.1.1.cmml" xref="S3.8.p4.1.m1.1.1.1.1.1"></setdiff><ci id="S3.8.p4.1.m1.1.1.1.1.2.cmml" xref="S3.8.p4.1.m1.1.1.1.1.2">𝑍</ci><ci id="S3.8.p4.1.m1.1.1.1.1.3.cmml" xref="S3.8.p4.1.m1.1.1.1.1.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.1.m1.1c">|Z\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.1.m1.1d">| italic_Z ∖ italic_B |</annotation></semantics></math>. By the definition of <math alttext="W" class="ltx_Math" display="inline" id="S3.8.p4.2.m2.1"><semantics id="S3.8.p4.2.m2.1a"><mi id="S3.8.p4.2.m2.1.1" xref="S3.8.p4.2.m2.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.8.p4.2.m2.1b"><ci id="S3.8.p4.2.m2.1.1.cmml" xref="S3.8.p4.2.m2.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.2.m2.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.2.m2.1d">italic_W</annotation></semantics></math>-sequence, <math alttext="G[W_{\ell+1}]" class="ltx_Math" display="inline" id="S3.8.p4.3.m3.1"><semantics id="S3.8.p4.3.m3.1a"><mrow id="S3.8.p4.3.m3.1.1" xref="S3.8.p4.3.m3.1.1.cmml"><mi id="S3.8.p4.3.m3.1.1.3" xref="S3.8.p4.3.m3.1.1.3.cmml">G</mi><mo id="S3.8.p4.3.m3.1.1.2" xref="S3.8.p4.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.8.p4.3.m3.1.1.1.1" xref="S3.8.p4.3.m3.1.1.1.2.cmml"><mo id="S3.8.p4.3.m3.1.1.1.1.2" stretchy="false" xref="S3.8.p4.3.m3.1.1.1.2.1.cmml">[</mo><msub 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xref="S3.8.p4.3.m3.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.8.p4.3.m3.1.1.1.1.1.cmml" xref="S3.8.p4.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.8.p4.3.m3.1.1.1.1.1.1.cmml" xref="S3.8.p4.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S3.8.p4.3.m3.1.1.1.1.1.2.cmml" xref="S3.8.p4.3.m3.1.1.1.1.1.2">𝑊</ci><apply id="S3.8.p4.3.m3.1.1.1.1.1.3.cmml" xref="S3.8.p4.3.m3.1.1.1.1.1.3"><plus id="S3.8.p4.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.8.p4.3.m3.1.1.1.1.1.3.1"></plus><ci id="S3.8.p4.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.8.p4.3.m3.1.1.1.1.1.3.2">ℓ</ci><cn id="S3.8.p4.3.m3.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.8.p4.3.m3.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.3.m3.1c">G[W_{\ell+1}]</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.3.m3.1d">italic_G [ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math> has a set <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S3.8.p4.4.m4.1"><semantics id="S3.8.p4.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S3.8.p4.4.m4.1.1" xref="S3.8.p4.4.m4.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S3.8.p4.4.m4.1b"><ci id="S3.8.p4.4.m4.1.1.cmml" xref="S3.8.p4.4.m4.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.4.m4.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.4.m4.1d">caligraphic_R</annotation></semantics></math> of <math alttext="|\Delta_{\ell+1}|=|Z|" class="ltx_Math" display="inline" id="S3.8.p4.5.m5.2"><semantics id="S3.8.p4.5.m5.2a"><mrow id="S3.8.p4.5.m5.2.2" xref="S3.8.p4.5.m5.2.2.cmml"><mrow id="S3.8.p4.5.m5.2.2.1.1" xref="S3.8.p4.5.m5.2.2.1.2.cmml"><mo id="S3.8.p4.5.m5.2.2.1.1.2" stretchy="false" xref="S3.8.p4.5.m5.2.2.1.2.1.cmml">|</mo><msub id="S3.8.p4.5.m5.2.2.1.1.1" xref="S3.8.p4.5.m5.2.2.1.1.1.cmml"><mi id="S3.8.p4.5.m5.2.2.1.1.1.2" mathvariant="normal" xref="S3.8.p4.5.m5.2.2.1.1.1.2.cmml">Δ</mi><mrow id="S3.8.p4.5.m5.2.2.1.1.1.3" xref="S3.8.p4.5.m5.2.2.1.1.1.3.cmml"><mi id="S3.8.p4.5.m5.2.2.1.1.1.3.2" mathvariant="normal" xref="S3.8.p4.5.m5.2.2.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.8.p4.5.m5.2.2.1.1.1.3.1" xref="S3.8.p4.5.m5.2.2.1.1.1.3.1.cmml">+</mo><mn id="S3.8.p4.5.m5.2.2.1.1.1.3.3" xref="S3.8.p4.5.m5.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.8.p4.5.m5.2.2.1.1.3" stretchy="false" xref="S3.8.p4.5.m5.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.8.p4.5.m5.2.2.2" xref="S3.8.p4.5.m5.2.2.2.cmml">=</mo><mrow id="S3.8.p4.5.m5.2.2.3.2" xref="S3.8.p4.5.m5.2.2.3.1.cmml"><mo id="S3.8.p4.5.m5.2.2.3.2.1" stretchy="false" xref="S3.8.p4.5.m5.2.2.3.1.1.cmml">|</mo><mi id="S3.8.p4.5.m5.1.1" xref="S3.8.p4.5.m5.1.1.cmml">Z</mi><mo id="S3.8.p4.5.m5.2.2.3.2.2" stretchy="false" xref="S3.8.p4.5.m5.2.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p4.5.m5.2b"><apply id="S3.8.p4.5.m5.2.2.cmml" xref="S3.8.p4.5.m5.2.2"><eq id="S3.8.p4.5.m5.2.2.2.cmml" xref="S3.8.p4.5.m5.2.2.2"></eq><apply id="S3.8.p4.5.m5.2.2.1.2.cmml" xref="S3.8.p4.5.m5.2.2.1.1"><abs id="S3.8.p4.5.m5.2.2.1.2.1.cmml" xref="S3.8.p4.5.m5.2.2.1.1.2"></abs><apply id="S3.8.p4.5.m5.2.2.1.1.1.cmml" xref="S3.8.p4.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.8.p4.5.m5.2.2.1.1.1.1.cmml" xref="S3.8.p4.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S3.8.p4.5.m5.2.2.1.1.1.2.cmml" xref="S3.8.p4.5.m5.2.2.1.1.1.2">Δ</ci><apply id="S3.8.p4.5.m5.2.2.1.1.1.3.cmml" xref="S3.8.p4.5.m5.2.2.1.1.1.3"><plus id="S3.8.p4.5.m5.2.2.1.1.1.3.1.cmml" xref="S3.8.p4.5.m5.2.2.1.1.1.3.1"></plus><ci id="S3.8.p4.5.m5.2.2.1.1.1.3.2.cmml" xref="S3.8.p4.5.m5.2.2.1.1.1.3.2">ℓ</ci><cn id="S3.8.p4.5.m5.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.8.p4.5.m5.2.2.1.1.1.3.3">1</cn></apply></apply></apply><apply id="S3.8.p4.5.m5.2.2.3.1.cmml" xref="S3.8.p4.5.m5.2.2.3.2"><abs id="S3.8.p4.5.m5.2.2.3.1.1.cmml" xref="S3.8.p4.5.m5.2.2.3.2.1"></abs><ci id="S3.8.p4.5.m5.1.1.cmml" xref="S3.8.p4.5.m5.1.1">𝑍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.5.m5.2c">|\Delta_{\ell+1}|=|Z|</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.5.m5.2d">| roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT | = | italic_Z |</annotation></semantics></math> pairwise vertex-disjoint <math alttext="\Delta_{\ell+1}" class="ltx_Math" display="inline" id="S3.8.p4.6.m6.1"><semantics id="S3.8.p4.6.m6.1a"><msub id="S3.8.p4.6.m6.1.1" xref="S3.8.p4.6.m6.1.1.cmml"><mi id="S3.8.p4.6.m6.1.1.2" mathvariant="normal" xref="S3.8.p4.6.m6.1.1.2.cmml">Δ</mi><mrow id="S3.8.p4.6.m6.1.1.3" xref="S3.8.p4.6.m6.1.1.3.cmml"><mi id="S3.8.p4.6.m6.1.1.3.2" mathvariant="normal" xref="S3.8.p4.6.m6.1.1.3.2.cmml">ℓ</mi><mo id="S3.8.p4.6.m6.1.1.3.1" xref="S3.8.p4.6.m6.1.1.3.1.cmml">+</mo><mn id="S3.8.p4.6.m6.1.1.3.3" xref="S3.8.p4.6.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.8.p4.6.m6.1b"><apply id="S3.8.p4.6.m6.1.1.cmml" xref="S3.8.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S3.8.p4.6.m6.1.1.1.cmml" xref="S3.8.p4.6.m6.1.1">subscript</csymbol><ci id="S3.8.p4.6.m6.1.1.2.cmml" xref="S3.8.p4.6.m6.1.1.2">Δ</ci><apply id="S3.8.p4.6.m6.1.1.3.cmml" xref="S3.8.p4.6.m6.1.1.3"><plus id="S3.8.p4.6.m6.1.1.3.1.cmml" xref="S3.8.p4.6.m6.1.1.3.1"></plus><ci id="S3.8.p4.6.m6.1.1.3.2.cmml" xref="S3.8.p4.6.m6.1.1.3.2">ℓ</ci><cn id="S3.8.p4.6.m6.1.1.3.3.cmml" type="integer" xref="S3.8.p4.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.6.m6.1c">\Delta_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.6.m6.1d">roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math>-<math alttext="W" class="ltx_Math" display="inline" id="S3.8.p4.7.m7.1"><semantics id="S3.8.p4.7.m7.1a"><mi id="S3.8.p4.7.m7.1.1" xref="S3.8.p4.7.m7.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.8.p4.7.m7.1b"><ci id="S3.8.p4.7.m7.1.1.cmml" xref="S3.8.p4.7.m7.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.7.m7.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.7.m7.1d">italic_W</annotation></semantics></math> paths. Let <math alttext="\mathcal{P}^{\star}:=\{P\in\mathcal{R}:V(P)\cap(Z\setminus B)\neq\emptyset\}" class="ltx_Math" display="inline" id="S3.8.p4.8.m8.3"><semantics id="S3.8.p4.8.m8.3a"><mrow id="S3.8.p4.8.m8.3.3" xref="S3.8.p4.8.m8.3.3.cmml"><msup id="S3.8.p4.8.m8.3.3.4" xref="S3.8.p4.8.m8.3.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.8.p4.8.m8.3.3.4.2" xref="S3.8.p4.8.m8.3.3.4.2.cmml">𝒫</mi><mo id="S3.8.p4.8.m8.3.3.4.3" xref="S3.8.p4.8.m8.3.3.4.3.cmml">⋆</mo></msup><mo id="S3.8.p4.8.m8.3.3.3" lspace="0.278em" rspace="0.278em" xref="S3.8.p4.8.m8.3.3.3.cmml">:=</mo><mrow id="S3.8.p4.8.m8.3.3.2.2" xref="S3.8.p4.8.m8.3.3.2.3.cmml"><mo id="S3.8.p4.8.m8.3.3.2.2.3" stretchy="false" xref="S3.8.p4.8.m8.3.3.2.3.1.cmml">{</mo><mrow id="S3.8.p4.8.m8.2.2.1.1.1" xref="S3.8.p4.8.m8.2.2.1.1.1.cmml"><mi id="S3.8.p4.8.m8.2.2.1.1.1.2" xref="S3.8.p4.8.m8.2.2.1.1.1.2.cmml">P</mi><mo id="S3.8.p4.8.m8.2.2.1.1.1.1" xref="S3.8.p4.8.m8.2.2.1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.8.p4.8.m8.2.2.1.1.1.3" xref="S3.8.p4.8.m8.2.2.1.1.1.3.cmml">ℛ</mi></mrow><mo id="S3.8.p4.8.m8.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.8.p4.8.m8.3.3.2.3.1.cmml">:</mo><mrow id="S3.8.p4.8.m8.3.3.2.2.2" xref="S3.8.p4.8.m8.3.3.2.2.2.cmml"><mrow id="S3.8.p4.8.m8.3.3.2.2.2.1" xref="S3.8.p4.8.m8.3.3.2.2.2.1.cmml"><mrow id="S3.8.p4.8.m8.3.3.2.2.2.1.3" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3.cmml"><mi id="S3.8.p4.8.m8.3.3.2.2.2.1.3.2" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3.2.cmml">V</mi><mo id="S3.8.p4.8.m8.3.3.2.2.2.1.3.1" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3.1.cmml">⁢</mo><mrow id="S3.8.p4.8.m8.3.3.2.2.2.1.3.3.2" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3.cmml"><mo id="S3.8.p4.8.m8.3.3.2.2.2.1.3.3.2.1" stretchy="false" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3.cmml">(</mo><mi id="S3.8.p4.8.m8.1.1" xref="S3.8.p4.8.m8.1.1.cmml">P</mi><mo id="S3.8.p4.8.m8.3.3.2.2.2.1.3.3.2.2" stretchy="false" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3.cmml">)</mo></mrow></mrow><mo id="S3.8.p4.8.m8.3.3.2.2.2.1.2" xref="S3.8.p4.8.m8.3.3.2.2.2.1.2.cmml">∩</mo><mrow id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.cmml"><mo id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.2" stretchy="false" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.cmml"><mi id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.2" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.2.cmml">Z</mi><mo id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.1" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.1.cmml">∖</mo><mi id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.3" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.3" stretchy="false" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.8.p4.8.m8.3.3.2.2.2.2" xref="S3.8.p4.8.m8.3.3.2.2.2.2.cmml">≠</mo><mi id="S3.8.p4.8.m8.3.3.2.2.2.3" mathvariant="normal" xref="S3.8.p4.8.m8.3.3.2.2.2.3.cmml">∅</mi></mrow><mo id="S3.8.p4.8.m8.3.3.2.2.5" stretchy="false" xref="S3.8.p4.8.m8.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p4.8.m8.3b"><apply id="S3.8.p4.8.m8.3.3.cmml" xref="S3.8.p4.8.m8.3.3"><csymbol cd="latexml" id="S3.8.p4.8.m8.3.3.3.cmml" xref="S3.8.p4.8.m8.3.3.3">assign</csymbol><apply id="S3.8.p4.8.m8.3.3.4.cmml" xref="S3.8.p4.8.m8.3.3.4"><csymbol cd="ambiguous" id="S3.8.p4.8.m8.3.3.4.1.cmml" xref="S3.8.p4.8.m8.3.3.4">superscript</csymbol><ci id="S3.8.p4.8.m8.3.3.4.2.cmml" xref="S3.8.p4.8.m8.3.3.4.2">𝒫</ci><ci id="S3.8.p4.8.m8.3.3.4.3.cmml" xref="S3.8.p4.8.m8.3.3.4.3">⋆</ci></apply><apply id="S3.8.p4.8.m8.3.3.2.3.cmml" xref="S3.8.p4.8.m8.3.3.2.2"><csymbol cd="latexml" id="S3.8.p4.8.m8.3.3.2.3.1.cmml" xref="S3.8.p4.8.m8.3.3.2.2.3">conditional-set</csymbol><apply id="S3.8.p4.8.m8.2.2.1.1.1.cmml" xref="S3.8.p4.8.m8.2.2.1.1.1"><in id="S3.8.p4.8.m8.2.2.1.1.1.1.cmml" xref="S3.8.p4.8.m8.2.2.1.1.1.1"></in><ci id="S3.8.p4.8.m8.2.2.1.1.1.2.cmml" xref="S3.8.p4.8.m8.2.2.1.1.1.2">𝑃</ci><ci id="S3.8.p4.8.m8.2.2.1.1.1.3.cmml" xref="S3.8.p4.8.m8.2.2.1.1.1.3">ℛ</ci></apply><apply id="S3.8.p4.8.m8.3.3.2.2.2.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2"><neq id="S3.8.p4.8.m8.3.3.2.2.2.2.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.2"></neq><apply id="S3.8.p4.8.m8.3.3.2.2.2.1.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1"><intersect id="S3.8.p4.8.m8.3.3.2.2.2.1.2.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1.2"></intersect><apply id="S3.8.p4.8.m8.3.3.2.2.2.1.3.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3"><times id="S3.8.p4.8.m8.3.3.2.2.2.1.3.1.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3.1"></times><ci id="S3.8.p4.8.m8.3.3.2.2.2.1.3.2.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1.3.2">𝑉</ci><ci id="S3.8.p4.8.m8.1.1.cmml" xref="S3.8.p4.8.m8.1.1">𝑃</ci></apply><apply id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1"><setdiff id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.1.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.1"></setdiff><ci id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.2.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.2">𝑍</ci><ci id="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.3.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.1.1.1.1.3">𝐵</ci></apply></apply><emptyset id="S3.8.p4.8.m8.3.3.2.2.2.3.cmml" xref="S3.8.p4.8.m8.3.3.2.2.2.3"></emptyset></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.8.m8.3c">\mathcal{P}^{\star}:=\{P\in\mathcal{R}:V(P)\cap(Z\setminus B)\neq\emptyset\}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.8.m8.3d">caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT := { italic_P ∈ caligraphic_R : italic_V ( italic_P ) ∩ ( italic_Z ∖ italic_B ) ≠ ∅ }</annotation></semantics></math>. Since each of the paths in <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S3.8.p4.9.m9.1"><semantics id="S3.8.p4.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S3.8.p4.9.m9.1.1" xref="S3.8.p4.9.m9.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S3.8.p4.9.m9.1b"><ci id="S3.8.p4.9.m9.1.1.cmml" xref="S3.8.p4.9.m9.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.9.m9.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.9.m9.1d">caligraphic_R</annotation></semantics></math> contains a distinct vertex in <math alttext="Z" class="ltx_Math" display="inline" id="S3.8.p4.10.m10.1"><semantics id="S3.8.p4.10.m10.1a"><mi id="S3.8.p4.10.m10.1.1" xref="S3.8.p4.10.m10.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S3.8.p4.10.m10.1b"><ci id="S3.8.p4.10.m10.1.1.cmml" xref="S3.8.p4.10.m10.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.10.m10.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.10.m10.1d">italic_Z</annotation></semantics></math>, <math alttext="|\mathcal{P}^{\star}|=|Z\setminus B|" class="ltx_Math" display="inline" id="S3.8.p4.11.m11.2"><semantics id="S3.8.p4.11.m11.2a"><mrow id="S3.8.p4.11.m11.2.2" xref="S3.8.p4.11.m11.2.2.cmml"><mrow id="S3.8.p4.11.m11.1.1.1.1" xref="S3.8.p4.11.m11.1.1.1.2.cmml"><mo id="S3.8.p4.11.m11.1.1.1.1.2" stretchy="false" xref="S3.8.p4.11.m11.1.1.1.2.1.cmml">|</mo><msup id="S3.8.p4.11.m11.1.1.1.1.1" xref="S3.8.p4.11.m11.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.8.p4.11.m11.1.1.1.1.1.2" xref="S3.8.p4.11.m11.1.1.1.1.1.2.cmml">𝒫</mi><mo id="S3.8.p4.11.m11.1.1.1.1.1.3" xref="S3.8.p4.11.m11.1.1.1.1.1.3.cmml">⋆</mo></msup><mo id="S3.8.p4.11.m11.1.1.1.1.3" stretchy="false" xref="S3.8.p4.11.m11.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.8.p4.11.m11.2.2.3" xref="S3.8.p4.11.m11.2.2.3.cmml">=</mo><mrow id="S3.8.p4.11.m11.2.2.2.1" xref="S3.8.p4.11.m11.2.2.2.2.cmml"><mo id="S3.8.p4.11.m11.2.2.2.1.2" stretchy="false" xref="S3.8.p4.11.m11.2.2.2.2.1.cmml">|</mo><mrow id="S3.8.p4.11.m11.2.2.2.1.1" xref="S3.8.p4.11.m11.2.2.2.1.1.cmml"><mi id="S3.8.p4.11.m11.2.2.2.1.1.2" xref="S3.8.p4.11.m11.2.2.2.1.1.2.cmml">Z</mi><mo id="S3.8.p4.11.m11.2.2.2.1.1.1" xref="S3.8.p4.11.m11.2.2.2.1.1.1.cmml">∖</mo><mi id="S3.8.p4.11.m11.2.2.2.1.1.3" xref="S3.8.p4.11.m11.2.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.8.p4.11.m11.2.2.2.1.3" stretchy="false" xref="S3.8.p4.11.m11.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p4.11.m11.2b"><apply id="S3.8.p4.11.m11.2.2.cmml" xref="S3.8.p4.11.m11.2.2"><eq id="S3.8.p4.11.m11.2.2.3.cmml" xref="S3.8.p4.11.m11.2.2.3"></eq><apply id="S3.8.p4.11.m11.1.1.1.2.cmml" xref="S3.8.p4.11.m11.1.1.1.1"><abs id="S3.8.p4.11.m11.1.1.1.2.1.cmml" xref="S3.8.p4.11.m11.1.1.1.1.2"></abs><apply id="S3.8.p4.11.m11.1.1.1.1.1.cmml" xref="S3.8.p4.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.8.p4.11.m11.1.1.1.1.1.1.cmml" xref="S3.8.p4.11.m11.1.1.1.1.1">superscript</csymbol><ci id="S3.8.p4.11.m11.1.1.1.1.1.2.cmml" xref="S3.8.p4.11.m11.1.1.1.1.1.2">𝒫</ci><ci id="S3.8.p4.11.m11.1.1.1.1.1.3.cmml" xref="S3.8.p4.11.m11.1.1.1.1.1.3">⋆</ci></apply></apply><apply id="S3.8.p4.11.m11.2.2.2.2.cmml" xref="S3.8.p4.11.m11.2.2.2.1"><abs id="S3.8.p4.11.m11.2.2.2.2.1.cmml" xref="S3.8.p4.11.m11.2.2.2.1.2"></abs><apply id="S3.8.p4.11.m11.2.2.2.1.1.cmml" xref="S3.8.p4.11.m11.2.2.2.1.1"><setdiff id="S3.8.p4.11.m11.2.2.2.1.1.1.cmml" xref="S3.8.p4.11.m11.2.2.2.1.1.1"></setdiff><ci id="S3.8.p4.11.m11.2.2.2.1.1.2.cmml" xref="S3.8.p4.11.m11.2.2.2.1.1.2">𝑍</ci><ci id="S3.8.p4.11.m11.2.2.2.1.1.3.cmml" xref="S3.8.p4.11.m11.2.2.2.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.11.m11.2c">|\mathcal{P}^{\star}|=|Z\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.11.m11.2d">| caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | = | italic_Z ∖ italic_B |</annotation></semantics></math>. Partition <math alttext="\mathcal{P}^{\star}" class="ltx_Math" display="inline" id="S3.8.p4.12.m12.1"><semantics id="S3.8.p4.12.m12.1a"><msup id="S3.8.p4.12.m12.1.1" xref="S3.8.p4.12.m12.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.8.p4.12.m12.1.1.2" xref="S3.8.p4.12.m12.1.1.2.cmml">𝒫</mi><mo id="S3.8.p4.12.m12.1.1.3" xref="S3.8.p4.12.m12.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S3.8.p4.12.m12.1b"><apply id="S3.8.p4.12.m12.1.1.cmml" xref="S3.8.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S3.8.p4.12.m12.1.1.1.cmml" xref="S3.8.p4.12.m12.1.1">superscript</csymbol><ci id="S3.8.p4.12.m12.1.1.2.cmml" xref="S3.8.p4.12.m12.1.1.2">𝒫</ci><ci id="S3.8.p4.12.m12.1.1.3.cmml" xref="S3.8.p4.12.m12.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p4.12.m12.1c">\mathcal{P}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p4.12.m12.1d">caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> into three sets:</p> </div> <div class="ltx_para" id="S3.9.p5"> <ol class="ltx_enumerate" id="S3.I2"> <li class="ltx_item" id="S3.I2.i0" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">0.</span> <div class="ltx_para" id="S3.I2.i0.p1"> <p class="ltx_p" id="S3.I2.i0.p1.4"><math alttext="\mathcal{P}^{\star}_{0}" class="ltx_Math" display="inline" id="S3.I2.i0.p1.1.m1.1"><semantics id="S3.I2.i0.p1.1.m1.1a"><msubsup id="S3.I2.i0.p1.1.m1.1.1" xref="S3.I2.i0.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i0.p1.1.m1.1.1.2.2" xref="S3.I2.i0.p1.1.m1.1.1.2.2.cmml">𝒫</mi><mn id="S3.I2.i0.p1.1.m1.1.1.3" xref="S3.I2.i0.p1.1.m1.1.1.3.cmml">0</mn><mo id="S3.I2.i0.p1.1.m1.1.1.2.3" xref="S3.I2.i0.p1.1.m1.1.1.2.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I2.i0.p1.1.m1.1b"><apply id="S3.I2.i0.p1.1.m1.1.1.cmml" xref="S3.I2.i0.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I2.i0.p1.1.m1.1.1.1.cmml" xref="S3.I2.i0.p1.1.m1.1.1">subscript</csymbol><apply id="S3.I2.i0.p1.1.m1.1.1.2.cmml" xref="S3.I2.i0.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I2.i0.p1.1.m1.1.1.2.1.cmml" xref="S3.I2.i0.p1.1.m1.1.1">superscript</csymbol><ci id="S3.I2.i0.p1.1.m1.1.1.2.2.cmml" xref="S3.I2.i0.p1.1.m1.1.1.2.2">𝒫</ci><ci id="S3.I2.i0.p1.1.m1.1.1.2.3.cmml" xref="S3.I2.i0.p1.1.m1.1.1.2.3">⋆</ci></apply><cn id="S3.I2.i0.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.I2.i0.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i0.p1.1.m1.1c">\mathcal{P}^{\star}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i0.p1.1.m1.1d">caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> are the paths in <math alttext="\mathcal{P}^{\star}" class="ltx_Math" display="inline" id="S3.I2.i0.p1.2.m2.1"><semantics id="S3.I2.i0.p1.2.m2.1a"><msup id="S3.I2.i0.p1.2.m2.1.1" xref="S3.I2.i0.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i0.p1.2.m2.1.1.2" xref="S3.I2.i0.p1.2.m2.1.1.2.cmml">𝒫</mi><mo id="S3.I2.i0.p1.2.m2.1.1.3" xref="S3.I2.i0.p1.2.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I2.i0.p1.2.m2.1b"><apply id="S3.I2.i0.p1.2.m2.1.1.cmml" xref="S3.I2.i0.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I2.i0.p1.2.m2.1.1.1.cmml" xref="S3.I2.i0.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I2.i0.p1.2.m2.1.1.2.cmml" xref="S3.I2.i0.p1.2.m2.1.1.2">𝒫</ci><ci id="S3.I2.i0.p1.2.m2.1.1.3.cmml" xref="S3.I2.i0.p1.2.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i0.p1.2.m2.1c">\mathcal{P}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i0.p1.2.m2.1d">caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> that start at a vertex of <math alttext="\Delta_{\ell+1}\setminus B" class="ltx_Math" display="inline" id="S3.I2.i0.p1.3.m3.1"><semantics id="S3.I2.i0.p1.3.m3.1a"><mrow id="S3.I2.i0.p1.3.m3.1.1" xref="S3.I2.i0.p1.3.m3.1.1.cmml"><msub id="S3.I2.i0.p1.3.m3.1.1.2" xref="S3.I2.i0.p1.3.m3.1.1.2.cmml"><mi id="S3.I2.i0.p1.3.m3.1.1.2.2" mathvariant="normal" xref="S3.I2.i0.p1.3.m3.1.1.2.2.cmml">Δ</mi><mrow id="S3.I2.i0.p1.3.m3.1.1.2.3" xref="S3.I2.i0.p1.3.m3.1.1.2.3.cmml"><mi id="S3.I2.i0.p1.3.m3.1.1.2.3.2" mathvariant="normal" xref="S3.I2.i0.p1.3.m3.1.1.2.3.2.cmml">ℓ</mi><mo id="S3.I2.i0.p1.3.m3.1.1.2.3.1" xref="S3.I2.i0.p1.3.m3.1.1.2.3.1.cmml">+</mo><mn id="S3.I2.i0.p1.3.m3.1.1.2.3.3" xref="S3.I2.i0.p1.3.m3.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.I2.i0.p1.3.m3.1.1.1" xref="S3.I2.i0.p1.3.m3.1.1.1.cmml">∖</mo><mi id="S3.I2.i0.p1.3.m3.1.1.3" xref="S3.I2.i0.p1.3.m3.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i0.p1.3.m3.1b"><apply id="S3.I2.i0.p1.3.m3.1.1.cmml" xref="S3.I2.i0.p1.3.m3.1.1"><setdiff id="S3.I2.i0.p1.3.m3.1.1.1.cmml" xref="S3.I2.i0.p1.3.m3.1.1.1"></setdiff><apply id="S3.I2.i0.p1.3.m3.1.1.2.cmml" xref="S3.I2.i0.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i0.p1.3.m3.1.1.2.1.cmml" xref="S3.I2.i0.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.I2.i0.p1.3.m3.1.1.2.2.cmml" xref="S3.I2.i0.p1.3.m3.1.1.2.2">Δ</ci><apply id="S3.I2.i0.p1.3.m3.1.1.2.3.cmml" xref="S3.I2.i0.p1.3.m3.1.1.2.3"><plus id="S3.I2.i0.p1.3.m3.1.1.2.3.1.cmml" xref="S3.I2.i0.p1.3.m3.1.1.2.3.1"></plus><ci id="S3.I2.i0.p1.3.m3.1.1.2.3.2.cmml" xref="S3.I2.i0.p1.3.m3.1.1.2.3.2">ℓ</ci><cn id="S3.I2.i0.p1.3.m3.1.1.2.3.3.cmml" type="integer" xref="S3.I2.i0.p1.3.m3.1.1.2.3.3">1</cn></apply></apply><ci id="S3.I2.i0.p1.3.m3.1.1.3.cmml" xref="S3.I2.i0.p1.3.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i0.p1.3.m3.1c">\Delta_{\ell+1}\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i0.p1.3.m3.1d">roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∖ italic_B</annotation></semantics></math> and end at a vertex in <math alttext="W\setminus B" class="ltx_Math" display="inline" id="S3.I2.i0.p1.4.m4.1"><semantics id="S3.I2.i0.p1.4.m4.1a"><mrow id="S3.I2.i0.p1.4.m4.1.1" xref="S3.I2.i0.p1.4.m4.1.1.cmml"><mi id="S3.I2.i0.p1.4.m4.1.1.2" xref="S3.I2.i0.p1.4.m4.1.1.2.cmml">W</mi><mo id="S3.I2.i0.p1.4.m4.1.1.1" xref="S3.I2.i0.p1.4.m4.1.1.1.cmml">∖</mo><mi id="S3.I2.i0.p1.4.m4.1.1.3" xref="S3.I2.i0.p1.4.m4.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i0.p1.4.m4.1b"><apply id="S3.I2.i0.p1.4.m4.1.1.cmml" xref="S3.I2.i0.p1.4.m4.1.1"><setdiff id="S3.I2.i0.p1.4.m4.1.1.1.cmml" xref="S3.I2.i0.p1.4.m4.1.1.1"></setdiff><ci id="S3.I2.i0.p1.4.m4.1.1.2.cmml" xref="S3.I2.i0.p1.4.m4.1.1.2">𝑊</ci><ci id="S3.I2.i0.p1.4.m4.1.1.3.cmml" xref="S3.I2.i0.p1.4.m4.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i0.p1.4.m4.1c">W\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i0.p1.4.m4.1d">italic_W ∖ italic_B</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.4"><math alttext="\mathcal{P}^{\star}_{1}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.1"><semantics id="S3.I2.i1.p1.1.m1.1a"><msubsup id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i1.p1.1.m1.1.1.2.2" xref="S3.I2.i1.p1.1.m1.1.1.2.2.cmml">𝒫</mi><mn id="S3.I2.i1.p1.1.m1.1.1.3" xref="S3.I2.i1.p1.1.m1.1.1.3.cmml">1</mn><mo id="S3.I2.i1.p1.1.m1.1.1.2.3" xref="S3.I2.i1.p1.1.m1.1.1.2.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.1b"><apply id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1">subscript</csymbol><apply id="S3.I2.i1.p1.1.m1.1.1.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.1.1.2.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1">superscript</csymbol><ci id="S3.I2.i1.p1.1.m1.1.1.2.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2.2">𝒫</ci><ci id="S3.I2.i1.p1.1.m1.1.1.2.3.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2.3">⋆</ci></apply><cn id="S3.I2.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.I2.i1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.1c">\mathcal{P}^{\star}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.1d">caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> are the paths in <math alttext="\mathcal{P}^{\star}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.1"><semantics id="S3.I2.i1.p1.2.m2.1a"><msup id="S3.I2.i1.p1.2.m2.1.1" xref="S3.I2.i1.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i1.p1.2.m2.1.1.2" xref="S3.I2.i1.p1.2.m2.1.1.2.cmml">𝒫</mi><mo id="S3.I2.i1.p1.2.m2.1.1.3" xref="S3.I2.i1.p1.2.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.1b"><apply id="S3.I2.i1.p1.2.m2.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I2.i1.p1.2.m2.1.1.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.2">𝒫</ci><ci id="S3.I2.i1.p1.2.m2.1.1.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.1c">\mathcal{P}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.1d">caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> that start at a vertex in <math alttext="\Delta_{\ell+1}\cap B" class="ltx_Math" display="inline" id="S3.I2.i1.p1.3.m3.1"><semantics id="S3.I2.i1.p1.3.m3.1a"><mrow id="S3.I2.i1.p1.3.m3.1.1" xref="S3.I2.i1.p1.3.m3.1.1.cmml"><msub id="S3.I2.i1.p1.3.m3.1.1.2" xref="S3.I2.i1.p1.3.m3.1.1.2.cmml"><mi id="S3.I2.i1.p1.3.m3.1.1.2.2" mathvariant="normal" xref="S3.I2.i1.p1.3.m3.1.1.2.2.cmml">Δ</mi><mrow id="S3.I2.i1.p1.3.m3.1.1.2.3" xref="S3.I2.i1.p1.3.m3.1.1.2.3.cmml"><mi id="S3.I2.i1.p1.3.m3.1.1.2.3.2" mathvariant="normal" xref="S3.I2.i1.p1.3.m3.1.1.2.3.2.cmml">ℓ</mi><mo id="S3.I2.i1.p1.3.m3.1.1.2.3.1" xref="S3.I2.i1.p1.3.m3.1.1.2.3.1.cmml">+</mo><mn id="S3.I2.i1.p1.3.m3.1.1.2.3.3" xref="S3.I2.i1.p1.3.m3.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.I2.i1.p1.3.m3.1.1.1" xref="S3.I2.i1.p1.3.m3.1.1.1.cmml">∩</mo><mi id="S3.I2.i1.p1.3.m3.1.1.3" xref="S3.I2.i1.p1.3.m3.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.3.m3.1b"><apply id="S3.I2.i1.p1.3.m3.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1"><intersect id="S3.I2.i1.p1.3.m3.1.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.1"></intersect><apply id="S3.I2.i1.p1.3.m3.1.1.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.3.m3.1.1.2.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.I2.i1.p1.3.m3.1.1.2.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2.2">Δ</ci><apply id="S3.I2.i1.p1.3.m3.1.1.2.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2.3"><plus id="S3.I2.i1.p1.3.m3.1.1.2.3.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2.3.1"></plus><ci id="S3.I2.i1.p1.3.m3.1.1.2.3.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2.3.2">ℓ</ci><cn id="S3.I2.i1.p1.3.m3.1.1.2.3.3.cmml" type="integer" xref="S3.I2.i1.p1.3.m3.1.1.2.3.3">1</cn></apply></apply><ci id="S3.I2.i1.p1.3.m3.1.1.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.3.m3.1c">\Delta_{\ell+1}\cap B</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.3.m3.1d">roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∩ italic_B</annotation></semantics></math> and end at a vertex in <math alttext="W\setminus B" class="ltx_Math" display="inline" id="S3.I2.i1.p1.4.m4.1"><semantics id="S3.I2.i1.p1.4.m4.1a"><mrow id="S3.I2.i1.p1.4.m4.1.1" xref="S3.I2.i1.p1.4.m4.1.1.cmml"><mi id="S3.I2.i1.p1.4.m4.1.1.2" xref="S3.I2.i1.p1.4.m4.1.1.2.cmml">W</mi><mo id="S3.I2.i1.p1.4.m4.1.1.1" xref="S3.I2.i1.p1.4.m4.1.1.1.cmml">∖</mo><mi id="S3.I2.i1.p1.4.m4.1.1.3" xref="S3.I2.i1.p1.4.m4.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.4.m4.1b"><apply id="S3.I2.i1.p1.4.m4.1.1.cmml" xref="S3.I2.i1.p1.4.m4.1.1"><setdiff id="S3.I2.i1.p1.4.m4.1.1.1.cmml" xref="S3.I2.i1.p1.4.m4.1.1.1"></setdiff><ci id="S3.I2.i1.p1.4.m4.1.1.2.cmml" xref="S3.I2.i1.p1.4.m4.1.1.2">𝑊</ci><ci id="S3.I2.i1.p1.4.m4.1.1.3.cmml" xref="S3.I2.i1.p1.4.m4.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.4.m4.1c">W\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.4.m4.1d">italic_W ∖ italic_B</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.4"><math alttext="\mathcal{P}^{\star}_{2}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><msubsup id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i2.p1.1.m1.1.1.2.2" xref="S3.I2.i2.p1.1.m1.1.1.2.2.cmml">𝒫</mi><mn id="S3.I2.i2.p1.1.m1.1.1.3" xref="S3.I2.i2.p1.1.m1.1.1.3.cmml">2</mn><mo id="S3.I2.i2.p1.1.m1.1.1.2.3" xref="S3.I2.i2.p1.1.m1.1.1.2.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><apply id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1">subscript</csymbol><apply id="S3.I2.i2.p1.1.m1.1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.1.1.2.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1">superscript</csymbol><ci id="S3.I2.i2.p1.1.m1.1.1.2.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2.2">𝒫</ci><ci id="S3.I2.i2.p1.1.m1.1.1.2.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2.3">⋆</ci></apply><cn id="S3.I2.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.I2.i2.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">\mathcal{P}^{\star}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are the paths in <math alttext="\mathcal{P}^{\star}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><msup id="S3.I2.i2.p1.2.m2.1.1" xref="S3.I2.i2.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i2.p1.2.m2.1.1.2" xref="S3.I2.i2.p1.2.m2.1.1.2.cmml">𝒫</mi><mo id="S3.I2.i2.p1.2.m2.1.1.3" xref="S3.I2.i2.p1.2.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I2.i2.p1.2.m2.1.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.2">𝒫</ci><ci id="S3.I2.i2.p1.2.m2.1.1.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">\mathcal{P}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> that start at a vertex in <math alttext="\Delta_{\ell+1}\setminus B" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.1"><semantics id="S3.I2.i2.p1.3.m3.1a"><mrow id="S3.I2.i2.p1.3.m3.1.1" xref="S3.I2.i2.p1.3.m3.1.1.cmml"><msub id="S3.I2.i2.p1.3.m3.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.2.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.2.2" mathvariant="normal" xref="S3.I2.i2.p1.3.m3.1.1.2.2.cmml">Δ</mi><mrow id="S3.I2.i2.p1.3.m3.1.1.2.3" xref="S3.I2.i2.p1.3.m3.1.1.2.3.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.2.3.2" mathvariant="normal" xref="S3.I2.i2.p1.3.m3.1.1.2.3.2.cmml">ℓ</mi><mo id="S3.I2.i2.p1.3.m3.1.1.2.3.1" xref="S3.I2.i2.p1.3.m3.1.1.2.3.1.cmml">+</mo><mn id="S3.I2.i2.p1.3.m3.1.1.2.3.3" xref="S3.I2.i2.p1.3.m3.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.I2.i2.p1.3.m3.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.cmml">∖</mo><mi id="S3.I2.i2.p1.3.m3.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.1b"><apply id="S3.I2.i2.p1.3.m3.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1"><setdiff id="S3.I2.i2.p1.3.m3.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1"></setdiff><apply id="S3.I2.i2.p1.3.m3.1.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.3.m3.1.1.2.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.I2.i2.p1.3.m3.1.1.2.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2.2">Δ</ci><apply id="S3.I2.i2.p1.3.m3.1.1.2.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2.3"><plus id="S3.I2.i2.p1.3.m3.1.1.2.3.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2.3.1"></plus><ci id="S3.I2.i2.p1.3.m3.1.1.2.3.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2.3.2">ℓ</ci><cn id="S3.I2.i2.p1.3.m3.1.1.2.3.3.cmml" type="integer" xref="S3.I2.i2.p1.3.m3.1.1.2.3.3">1</cn></apply></apply><ci id="S3.I2.i2.p1.3.m3.1.1.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.1c">\Delta_{\ell+1}\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.1d">roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∖ italic_B</annotation></semantics></math> and end at a vertex in <math alttext="W\cap B" class="ltx_Math" display="inline" id="S3.I2.i2.p1.4.m4.1"><semantics id="S3.I2.i2.p1.4.m4.1a"><mrow id="S3.I2.i2.p1.4.m4.1.1" xref="S3.I2.i2.p1.4.m4.1.1.cmml"><mi id="S3.I2.i2.p1.4.m4.1.1.2" xref="S3.I2.i2.p1.4.m4.1.1.2.cmml">W</mi><mo id="S3.I2.i2.p1.4.m4.1.1.1" xref="S3.I2.i2.p1.4.m4.1.1.1.cmml">∩</mo><mi id="S3.I2.i2.p1.4.m4.1.1.3" xref="S3.I2.i2.p1.4.m4.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.4.m4.1b"><apply id="S3.I2.i2.p1.4.m4.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1"><intersect id="S3.I2.i2.p1.4.m4.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1"></intersect><ci id="S3.I2.i2.p1.4.m4.1.1.2.cmml" xref="S3.I2.i2.p1.4.m4.1.1.2">𝑊</ci><ci id="S3.I2.i2.p1.4.m4.1.1.3.cmml" xref="S3.I2.i2.p1.4.m4.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.4.m4.1c">W\cap B</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.4.m4.1d">italic_W ∩ italic_B</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S3.9.p5.12">Since the paths in <math alttext="\mathcal{P}_{0}^{\star}\cup\mathcal{P}_{1}^{\star}" class="ltx_Math" display="inline" id="S3.9.p5.1.m1.1"><semantics id="S3.9.p5.1.m1.1a"><mrow id="S3.9.p5.1.m1.1.1" xref="S3.9.p5.1.m1.1.1.cmml"><msubsup id="S3.9.p5.1.m1.1.1.2" xref="S3.9.p5.1.m1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.1.m1.1.1.2.2.2" xref="S3.9.p5.1.m1.1.1.2.2.2.cmml">𝒫</mi><mn id="S3.9.p5.1.m1.1.1.2.2.3" xref="S3.9.p5.1.m1.1.1.2.2.3.cmml">0</mn><mo id="S3.9.p5.1.m1.1.1.2.3" xref="S3.9.p5.1.m1.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S3.9.p5.1.m1.1.1.1" xref="S3.9.p5.1.m1.1.1.1.cmml">∪</mo><msubsup id="S3.9.p5.1.m1.1.1.3" xref="S3.9.p5.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.1.m1.1.1.3.2.2" xref="S3.9.p5.1.m1.1.1.3.2.2.cmml">𝒫</mi><mn id="S3.9.p5.1.m1.1.1.3.2.3" xref="S3.9.p5.1.m1.1.1.3.2.3.cmml">1</mn><mo id="S3.9.p5.1.m1.1.1.3.3" xref="S3.9.p5.1.m1.1.1.3.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.1.m1.1b"><apply id="S3.9.p5.1.m1.1.1.cmml" xref="S3.9.p5.1.m1.1.1"><union id="S3.9.p5.1.m1.1.1.1.cmml" xref="S3.9.p5.1.m1.1.1.1"></union><apply id="S3.9.p5.1.m1.1.1.2.cmml" xref="S3.9.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.1.m1.1.1.2.1.cmml" xref="S3.9.p5.1.m1.1.1.2">superscript</csymbol><apply id="S3.9.p5.1.m1.1.1.2.2.cmml" xref="S3.9.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.1.m1.1.1.2.2.1.cmml" xref="S3.9.p5.1.m1.1.1.2">subscript</csymbol><ci id="S3.9.p5.1.m1.1.1.2.2.2.cmml" xref="S3.9.p5.1.m1.1.1.2.2.2">𝒫</ci><cn id="S3.9.p5.1.m1.1.1.2.2.3.cmml" type="integer" xref="S3.9.p5.1.m1.1.1.2.2.3">0</cn></apply><ci id="S3.9.p5.1.m1.1.1.2.3.cmml" xref="S3.9.p5.1.m1.1.1.2.3">⋆</ci></apply><apply id="S3.9.p5.1.m1.1.1.3.cmml" xref="S3.9.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.1.m1.1.1.3.1.cmml" xref="S3.9.p5.1.m1.1.1.3">superscript</csymbol><apply id="S3.9.p5.1.m1.1.1.3.2.cmml" xref="S3.9.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.1.m1.1.1.3.2.1.cmml" xref="S3.9.p5.1.m1.1.1.3">subscript</csymbol><ci id="S3.9.p5.1.m1.1.1.3.2.2.cmml" xref="S3.9.p5.1.m1.1.1.3.2.2">𝒫</ci><cn id="S3.9.p5.1.m1.1.1.3.2.3.cmml" type="integer" xref="S3.9.p5.1.m1.1.1.3.2.3">1</cn></apply><ci id="S3.9.p5.1.m1.1.1.3.3.cmml" xref="S3.9.p5.1.m1.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.1.m1.1c">\mathcal{P}_{0}^{\star}\cup\mathcal{P}_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.1.m1.1d">caligraphic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∪ caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> are pairwise vertex-disjoint and each contains a vertex of <math alttext="W\setminus B" class="ltx_Math" display="inline" id="S3.9.p5.2.m2.1"><semantics id="S3.9.p5.2.m2.1a"><mrow id="S3.9.p5.2.m2.1.1" xref="S3.9.p5.2.m2.1.1.cmml"><mi id="S3.9.p5.2.m2.1.1.2" xref="S3.9.p5.2.m2.1.1.2.cmml">W</mi><mo id="S3.9.p5.2.m2.1.1.1" xref="S3.9.p5.2.m2.1.1.1.cmml">∖</mo><mi id="S3.9.p5.2.m2.1.1.3" xref="S3.9.p5.2.m2.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.2.m2.1b"><apply id="S3.9.p5.2.m2.1.1.cmml" xref="S3.9.p5.2.m2.1.1"><setdiff id="S3.9.p5.2.m2.1.1.1.cmml" xref="S3.9.p5.2.m2.1.1.1"></setdiff><ci id="S3.9.p5.2.m2.1.1.2.cmml" xref="S3.9.p5.2.m2.1.1.2">𝑊</ci><ci id="S3.9.p5.2.m2.1.1.3.cmml" xref="S3.9.p5.2.m2.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.2.m2.1c">W\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.2.m2.1d">italic_W ∖ italic_B</annotation></semantics></math>, <math alttext="|\mathcal{P}_{0}^{\star}|+|\mathcal{P}_{1}^{\star}|\leq|W\setminus B|" class="ltx_Math" display="inline" id="S3.9.p5.3.m3.3"><semantics id="S3.9.p5.3.m3.3a"><mrow id="S3.9.p5.3.m3.3.3" xref="S3.9.p5.3.m3.3.3.cmml"><mrow id="S3.9.p5.3.m3.2.2.2" xref="S3.9.p5.3.m3.2.2.2.cmml"><mrow id="S3.9.p5.3.m3.1.1.1.1.1" xref="S3.9.p5.3.m3.1.1.1.1.2.cmml"><mo id="S3.9.p5.3.m3.1.1.1.1.1.2" stretchy="false" xref="S3.9.p5.3.m3.1.1.1.1.2.1.cmml">|</mo><msubsup id="S3.9.p5.3.m3.1.1.1.1.1.1" xref="S3.9.p5.3.m3.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.3.m3.1.1.1.1.1.1.2.2" xref="S3.9.p5.3.m3.1.1.1.1.1.1.2.2.cmml">𝒫</mi><mn id="S3.9.p5.3.m3.1.1.1.1.1.1.2.3" xref="S3.9.p5.3.m3.1.1.1.1.1.1.2.3.cmml">0</mn><mo id="S3.9.p5.3.m3.1.1.1.1.1.1.3" xref="S3.9.p5.3.m3.1.1.1.1.1.1.3.cmml">⋆</mo></msubsup><mo id="S3.9.p5.3.m3.1.1.1.1.1.3" stretchy="false" xref="S3.9.p5.3.m3.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.9.p5.3.m3.2.2.2.3" xref="S3.9.p5.3.m3.2.2.2.3.cmml">+</mo><mrow id="S3.9.p5.3.m3.2.2.2.2.1" xref="S3.9.p5.3.m3.2.2.2.2.2.cmml"><mo id="S3.9.p5.3.m3.2.2.2.2.1.2" stretchy="false" xref="S3.9.p5.3.m3.2.2.2.2.2.1.cmml">|</mo><msubsup id="S3.9.p5.3.m3.2.2.2.2.1.1" xref="S3.9.p5.3.m3.2.2.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.3.m3.2.2.2.2.1.1.2.2" xref="S3.9.p5.3.m3.2.2.2.2.1.1.2.2.cmml">𝒫</mi><mn id="S3.9.p5.3.m3.2.2.2.2.1.1.2.3" xref="S3.9.p5.3.m3.2.2.2.2.1.1.2.3.cmml">1</mn><mo id="S3.9.p5.3.m3.2.2.2.2.1.1.3" xref="S3.9.p5.3.m3.2.2.2.2.1.1.3.cmml">⋆</mo></msubsup><mo id="S3.9.p5.3.m3.2.2.2.2.1.3" stretchy="false" xref="S3.9.p5.3.m3.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S3.9.p5.3.m3.3.3.4" xref="S3.9.p5.3.m3.3.3.4.cmml">≤</mo><mrow id="S3.9.p5.3.m3.3.3.3.1" xref="S3.9.p5.3.m3.3.3.3.2.cmml"><mo id="S3.9.p5.3.m3.3.3.3.1.2" stretchy="false" xref="S3.9.p5.3.m3.3.3.3.2.1.cmml">|</mo><mrow id="S3.9.p5.3.m3.3.3.3.1.1" xref="S3.9.p5.3.m3.3.3.3.1.1.cmml"><mi id="S3.9.p5.3.m3.3.3.3.1.1.2" xref="S3.9.p5.3.m3.3.3.3.1.1.2.cmml">W</mi><mo id="S3.9.p5.3.m3.3.3.3.1.1.1" xref="S3.9.p5.3.m3.3.3.3.1.1.1.cmml">∖</mo><mi id="S3.9.p5.3.m3.3.3.3.1.1.3" xref="S3.9.p5.3.m3.3.3.3.1.1.3.cmml">B</mi></mrow><mo id="S3.9.p5.3.m3.3.3.3.1.3" stretchy="false" xref="S3.9.p5.3.m3.3.3.3.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.3.m3.3b"><apply id="S3.9.p5.3.m3.3.3.cmml" xref="S3.9.p5.3.m3.3.3"><leq id="S3.9.p5.3.m3.3.3.4.cmml" xref="S3.9.p5.3.m3.3.3.4"></leq><apply id="S3.9.p5.3.m3.2.2.2.cmml" xref="S3.9.p5.3.m3.2.2.2"><plus id="S3.9.p5.3.m3.2.2.2.3.cmml" xref="S3.9.p5.3.m3.2.2.2.3"></plus><apply id="S3.9.p5.3.m3.1.1.1.1.2.cmml" xref="S3.9.p5.3.m3.1.1.1.1.1"><abs id="S3.9.p5.3.m3.1.1.1.1.2.1.cmml" xref="S3.9.p5.3.m3.1.1.1.1.1.2"></abs><apply id="S3.9.p5.3.m3.1.1.1.1.1.1.cmml" xref="S3.9.p5.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.9.p5.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.9.p5.3.m3.1.1.1.1.1.1">superscript</csymbol><apply id="S3.9.p5.3.m3.1.1.1.1.1.1.2.cmml" xref="S3.9.p5.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.9.p5.3.m3.1.1.1.1.1.1.2.1.cmml" xref="S3.9.p5.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S3.9.p5.3.m3.1.1.1.1.1.1.2.2.cmml" xref="S3.9.p5.3.m3.1.1.1.1.1.1.2.2">𝒫</ci><cn id="S3.9.p5.3.m3.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S3.9.p5.3.m3.1.1.1.1.1.1.2.3">0</cn></apply><ci id="S3.9.p5.3.m3.1.1.1.1.1.1.3.cmml" xref="S3.9.p5.3.m3.1.1.1.1.1.1.3">⋆</ci></apply></apply><apply id="S3.9.p5.3.m3.2.2.2.2.2.cmml" xref="S3.9.p5.3.m3.2.2.2.2.1"><abs id="S3.9.p5.3.m3.2.2.2.2.2.1.cmml" xref="S3.9.p5.3.m3.2.2.2.2.1.2"></abs><apply id="S3.9.p5.3.m3.2.2.2.2.1.1.cmml" xref="S3.9.p5.3.m3.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.9.p5.3.m3.2.2.2.2.1.1.1.cmml" xref="S3.9.p5.3.m3.2.2.2.2.1.1">superscript</csymbol><apply id="S3.9.p5.3.m3.2.2.2.2.1.1.2.cmml" xref="S3.9.p5.3.m3.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.9.p5.3.m3.2.2.2.2.1.1.2.1.cmml" xref="S3.9.p5.3.m3.2.2.2.2.1.1">subscript</csymbol><ci id="S3.9.p5.3.m3.2.2.2.2.1.1.2.2.cmml" xref="S3.9.p5.3.m3.2.2.2.2.1.1.2.2">𝒫</ci><cn id="S3.9.p5.3.m3.2.2.2.2.1.1.2.3.cmml" type="integer" xref="S3.9.p5.3.m3.2.2.2.2.1.1.2.3">1</cn></apply><ci id="S3.9.p5.3.m3.2.2.2.2.1.1.3.cmml" xref="S3.9.p5.3.m3.2.2.2.2.1.1.3">⋆</ci></apply></apply></apply><apply id="S3.9.p5.3.m3.3.3.3.2.cmml" xref="S3.9.p5.3.m3.3.3.3.1"><abs id="S3.9.p5.3.m3.3.3.3.2.1.cmml" xref="S3.9.p5.3.m3.3.3.3.1.2"></abs><apply id="S3.9.p5.3.m3.3.3.3.1.1.cmml" xref="S3.9.p5.3.m3.3.3.3.1.1"><setdiff id="S3.9.p5.3.m3.3.3.3.1.1.1.cmml" xref="S3.9.p5.3.m3.3.3.3.1.1.1"></setdiff><ci id="S3.9.p5.3.m3.3.3.3.1.1.2.cmml" xref="S3.9.p5.3.m3.3.3.3.1.1.2">𝑊</ci><ci id="S3.9.p5.3.m3.3.3.3.1.1.3.cmml" xref="S3.9.p5.3.m3.3.3.3.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.3.m3.3c">|\mathcal{P}_{0}^{\star}|+|\mathcal{P}_{1}^{\star}|\leq|W\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.3.m3.3d">| caligraphic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | + | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ≤ | italic_W ∖ italic_B |</annotation></semantics></math>. Each path in <math alttext="\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{\star}" class="ltx_Math" display="inline" id="S3.9.p5.4.m4.1"><semantics id="S3.9.p5.4.m4.1a"><mrow id="S3.9.p5.4.m4.1.1" xref="S3.9.p5.4.m4.1.1.cmml"><msubsup id="S3.9.p5.4.m4.1.1.2" xref="S3.9.p5.4.m4.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.4.m4.1.1.2.2.2" xref="S3.9.p5.4.m4.1.1.2.2.2.cmml">𝒫</mi><mn id="S3.9.p5.4.m4.1.1.2.2.3" xref="S3.9.p5.4.m4.1.1.2.2.3.cmml">1</mn><mo id="S3.9.p5.4.m4.1.1.2.3" xref="S3.9.p5.4.m4.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S3.9.p5.4.m4.1.1.1" xref="S3.9.p5.4.m4.1.1.1.cmml">∪</mo><msubsup id="S3.9.p5.4.m4.1.1.3" xref="S3.9.p5.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.4.m4.1.1.3.2.2" xref="S3.9.p5.4.m4.1.1.3.2.2.cmml">𝒫</mi><mn id="S3.9.p5.4.m4.1.1.3.2.3" xref="S3.9.p5.4.m4.1.1.3.2.3.cmml">2</mn><mo id="S3.9.p5.4.m4.1.1.3.3" xref="S3.9.p5.4.m4.1.1.3.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.4.m4.1b"><apply id="S3.9.p5.4.m4.1.1.cmml" xref="S3.9.p5.4.m4.1.1"><union id="S3.9.p5.4.m4.1.1.1.cmml" xref="S3.9.p5.4.m4.1.1.1"></union><apply id="S3.9.p5.4.m4.1.1.2.cmml" xref="S3.9.p5.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.4.m4.1.1.2.1.cmml" xref="S3.9.p5.4.m4.1.1.2">superscript</csymbol><apply id="S3.9.p5.4.m4.1.1.2.2.cmml" xref="S3.9.p5.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.4.m4.1.1.2.2.1.cmml" xref="S3.9.p5.4.m4.1.1.2">subscript</csymbol><ci id="S3.9.p5.4.m4.1.1.2.2.2.cmml" xref="S3.9.p5.4.m4.1.1.2.2.2">𝒫</ci><cn id="S3.9.p5.4.m4.1.1.2.2.3.cmml" type="integer" xref="S3.9.p5.4.m4.1.1.2.2.3">1</cn></apply><ci id="S3.9.p5.4.m4.1.1.2.3.cmml" xref="S3.9.p5.4.m4.1.1.2.3">⋆</ci></apply><apply id="S3.9.p5.4.m4.1.1.3.cmml" xref="S3.9.p5.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.4.m4.1.1.3.1.cmml" xref="S3.9.p5.4.m4.1.1.3">superscript</csymbol><apply id="S3.9.p5.4.m4.1.1.3.2.cmml" xref="S3.9.p5.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.4.m4.1.1.3.2.1.cmml" xref="S3.9.p5.4.m4.1.1.3">subscript</csymbol><ci id="S3.9.p5.4.m4.1.1.3.2.2.cmml" xref="S3.9.p5.4.m4.1.1.3.2.2">𝒫</ci><cn id="S3.9.p5.4.m4.1.1.3.2.3.cmml" type="integer" xref="S3.9.p5.4.m4.1.1.3.2.3">2</cn></apply><ci id="S3.9.p5.4.m4.1.1.3.3.cmml" xref="S3.9.p5.4.m4.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.4.m4.1c">\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.4.m4.1d">caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∪ caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> contains a vertex in <math alttext="Z\setminus B" class="ltx_Math" display="inline" id="S3.9.p5.5.m5.1"><semantics id="S3.9.p5.5.m5.1a"><mrow id="S3.9.p5.5.m5.1.1" xref="S3.9.p5.5.m5.1.1.cmml"><mi id="S3.9.p5.5.m5.1.1.2" xref="S3.9.p5.5.m5.1.1.2.cmml">Z</mi><mo id="S3.9.p5.5.m5.1.1.1" xref="S3.9.p5.5.m5.1.1.1.cmml">∖</mo><mi id="S3.9.p5.5.m5.1.1.3" xref="S3.9.p5.5.m5.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.5.m5.1b"><apply id="S3.9.p5.5.m5.1.1.cmml" xref="S3.9.p5.5.m5.1.1"><setdiff id="S3.9.p5.5.m5.1.1.1.cmml" xref="S3.9.p5.5.m5.1.1.1"></setdiff><ci id="S3.9.p5.5.m5.1.1.2.cmml" xref="S3.9.p5.5.m5.1.1.2">𝑍</ci><ci id="S3.9.p5.5.m5.1.1.3.cmml" xref="S3.9.p5.5.m5.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.5.m5.1c">Z\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.5.m5.1d">italic_Z ∖ italic_B</annotation></semantics></math> and a vertex in <math alttext="B" class="ltx_Math" display="inline" id="S3.9.p5.6.m6.1"><semantics id="S3.9.p5.6.m6.1a"><mi id="S3.9.p5.6.m6.1.1" xref="S3.9.p5.6.m6.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.9.p5.6.m6.1b"><ci id="S3.9.p5.6.m6.1.1.cmml" xref="S3.9.p5.6.m6.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.6.m6.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.6.m6.1d">italic_B</annotation></semantics></math>. Since <math alttext="(A,B)" class="ltx_Math" display="inline" id="S3.9.p5.7.m7.2"><semantics id="S3.9.p5.7.m7.2a"><mrow id="S3.9.p5.7.m7.2.3.2" xref="S3.9.p5.7.m7.2.3.1.cmml"><mo id="S3.9.p5.7.m7.2.3.2.1" stretchy="false" xref="S3.9.p5.7.m7.2.3.1.cmml">(</mo><mi id="S3.9.p5.7.m7.1.1" xref="S3.9.p5.7.m7.1.1.cmml">A</mi><mo id="S3.9.p5.7.m7.2.3.2.2" xref="S3.9.p5.7.m7.2.3.1.cmml">,</mo><mi id="S3.9.p5.7.m7.2.2" xref="S3.9.p5.7.m7.2.2.cmml">B</mi><mo id="S3.9.p5.7.m7.2.3.2.3" stretchy="false" xref="S3.9.p5.7.m7.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.7.m7.2b"><interval closure="open" id="S3.9.p5.7.m7.2.3.1.cmml" xref="S3.9.p5.7.m7.2.3.2"><ci id="S3.9.p5.7.m7.1.1.cmml" xref="S3.9.p5.7.m7.1.1">𝐴</ci><ci id="S3.9.p5.7.m7.2.2.cmml" xref="S3.9.p5.7.m7.2.2">𝐵</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.7.m7.2c">(A,B)</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.7.m7.2d">( italic_A , italic_B )</annotation></semantics></math> is a separation of <math alttext="G[W_{\ell+1}]" class="ltx_Math" display="inline" id="S3.9.p5.8.m8.1"><semantics id="S3.9.p5.8.m8.1a"><mrow id="S3.9.p5.8.m8.1.1" xref="S3.9.p5.8.m8.1.1.cmml"><mi id="S3.9.p5.8.m8.1.1.3" xref="S3.9.p5.8.m8.1.1.3.cmml">G</mi><mo id="S3.9.p5.8.m8.1.1.2" xref="S3.9.p5.8.m8.1.1.2.cmml">⁢</mo><mrow id="S3.9.p5.8.m8.1.1.1.1" xref="S3.9.p5.8.m8.1.1.1.2.cmml"><mo id="S3.9.p5.8.m8.1.1.1.1.2" stretchy="false" xref="S3.9.p5.8.m8.1.1.1.2.1.cmml">[</mo><msub id="S3.9.p5.8.m8.1.1.1.1.1" xref="S3.9.p5.8.m8.1.1.1.1.1.cmml"><mi id="S3.9.p5.8.m8.1.1.1.1.1.2" xref="S3.9.p5.8.m8.1.1.1.1.1.2.cmml">W</mi><mrow id="S3.9.p5.8.m8.1.1.1.1.1.3" xref="S3.9.p5.8.m8.1.1.1.1.1.3.cmml"><mi id="S3.9.p5.8.m8.1.1.1.1.1.3.2" mathvariant="normal" xref="S3.9.p5.8.m8.1.1.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.9.p5.8.m8.1.1.1.1.1.3.1" xref="S3.9.p5.8.m8.1.1.1.1.1.3.1.cmml">+</mo><mn id="S3.9.p5.8.m8.1.1.1.1.1.3.3" xref="S3.9.p5.8.m8.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.9.p5.8.m8.1.1.1.1.3" stretchy="false" xref="S3.9.p5.8.m8.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.8.m8.1b"><apply id="S3.9.p5.8.m8.1.1.cmml" xref="S3.9.p5.8.m8.1.1"><times id="S3.9.p5.8.m8.1.1.2.cmml" xref="S3.9.p5.8.m8.1.1.2"></times><ci id="S3.9.p5.8.m8.1.1.3.cmml" xref="S3.9.p5.8.m8.1.1.3">𝐺</ci><apply id="S3.9.p5.8.m8.1.1.1.2.cmml" xref="S3.9.p5.8.m8.1.1.1.1"><csymbol cd="latexml" id="S3.9.p5.8.m8.1.1.1.2.1.cmml" xref="S3.9.p5.8.m8.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.9.p5.8.m8.1.1.1.1.1.cmml" xref="S3.9.p5.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.9.p5.8.m8.1.1.1.1.1.1.cmml" xref="S3.9.p5.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S3.9.p5.8.m8.1.1.1.1.1.2.cmml" xref="S3.9.p5.8.m8.1.1.1.1.1.2">𝑊</ci><apply id="S3.9.p5.8.m8.1.1.1.1.1.3.cmml" xref="S3.9.p5.8.m8.1.1.1.1.1.3"><plus id="S3.9.p5.8.m8.1.1.1.1.1.3.1.cmml" xref="S3.9.p5.8.m8.1.1.1.1.1.3.1"></plus><ci id="S3.9.p5.8.m8.1.1.1.1.1.3.2.cmml" xref="S3.9.p5.8.m8.1.1.1.1.1.3.2">ℓ</ci><cn id="S3.9.p5.8.m8.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.9.p5.8.m8.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.8.m8.1c">G[W_{\ell+1}]</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.8.m8.1d">italic_G [ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math> this implies that each path in <math alttext="\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{\star}" class="ltx_Math" display="inline" id="S3.9.p5.9.m9.1"><semantics id="S3.9.p5.9.m9.1a"><mrow id="S3.9.p5.9.m9.1.1" xref="S3.9.p5.9.m9.1.1.cmml"><msubsup id="S3.9.p5.9.m9.1.1.2" xref="S3.9.p5.9.m9.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.9.m9.1.1.2.2.2" xref="S3.9.p5.9.m9.1.1.2.2.2.cmml">𝒫</mi><mn id="S3.9.p5.9.m9.1.1.2.2.3" xref="S3.9.p5.9.m9.1.1.2.2.3.cmml">1</mn><mo id="S3.9.p5.9.m9.1.1.2.3" xref="S3.9.p5.9.m9.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S3.9.p5.9.m9.1.1.1" xref="S3.9.p5.9.m9.1.1.1.cmml">∪</mo><msubsup id="S3.9.p5.9.m9.1.1.3" xref="S3.9.p5.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.9.m9.1.1.3.2.2" xref="S3.9.p5.9.m9.1.1.3.2.2.cmml">𝒫</mi><mn id="S3.9.p5.9.m9.1.1.3.2.3" xref="S3.9.p5.9.m9.1.1.3.2.3.cmml">2</mn><mo id="S3.9.p5.9.m9.1.1.3.3" xref="S3.9.p5.9.m9.1.1.3.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.9.m9.1b"><apply id="S3.9.p5.9.m9.1.1.cmml" xref="S3.9.p5.9.m9.1.1"><union id="S3.9.p5.9.m9.1.1.1.cmml" xref="S3.9.p5.9.m9.1.1.1"></union><apply id="S3.9.p5.9.m9.1.1.2.cmml" xref="S3.9.p5.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.9.m9.1.1.2.1.cmml" xref="S3.9.p5.9.m9.1.1.2">superscript</csymbol><apply id="S3.9.p5.9.m9.1.1.2.2.cmml" xref="S3.9.p5.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.9.m9.1.1.2.2.1.cmml" xref="S3.9.p5.9.m9.1.1.2">subscript</csymbol><ci id="S3.9.p5.9.m9.1.1.2.2.2.cmml" xref="S3.9.p5.9.m9.1.1.2.2.2">𝒫</ci><cn id="S3.9.p5.9.m9.1.1.2.2.3.cmml" type="integer" xref="S3.9.p5.9.m9.1.1.2.2.3">1</cn></apply><ci id="S3.9.p5.9.m9.1.1.2.3.cmml" xref="S3.9.p5.9.m9.1.1.2.3">⋆</ci></apply><apply id="S3.9.p5.9.m9.1.1.3.cmml" xref="S3.9.p5.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.9.m9.1.1.3.1.cmml" xref="S3.9.p5.9.m9.1.1.3">superscript</csymbol><apply id="S3.9.p5.9.m9.1.1.3.2.cmml" xref="S3.9.p5.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.9.m9.1.1.3.2.1.cmml" xref="S3.9.p5.9.m9.1.1.3">subscript</csymbol><ci id="S3.9.p5.9.m9.1.1.3.2.2.cmml" xref="S3.9.p5.9.m9.1.1.3.2.2">𝒫</ci><cn id="S3.9.p5.9.m9.1.1.3.2.3.cmml" type="integer" xref="S3.9.p5.9.m9.1.1.3.2.3">2</cn></apply><ci id="S3.9.p5.9.m9.1.1.3.3.cmml" xref="S3.9.p5.9.m9.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.9.m9.1c">\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.9.m9.1d">caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∪ caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> contains a vertex in <math alttext="A\cap B" class="ltx_Math" display="inline" id="S3.9.p5.10.m10.1"><semantics id="S3.9.p5.10.m10.1a"><mrow id="S3.9.p5.10.m10.1.1" xref="S3.9.p5.10.m10.1.1.cmml"><mi id="S3.9.p5.10.m10.1.1.2" xref="S3.9.p5.10.m10.1.1.2.cmml">A</mi><mo id="S3.9.p5.10.m10.1.1.1" xref="S3.9.p5.10.m10.1.1.1.cmml">∩</mo><mi id="S3.9.p5.10.m10.1.1.3" xref="S3.9.p5.10.m10.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.10.m10.1b"><apply id="S3.9.p5.10.m10.1.1.cmml" xref="S3.9.p5.10.m10.1.1"><intersect id="S3.9.p5.10.m10.1.1.1.cmml" xref="S3.9.p5.10.m10.1.1.1"></intersect><ci id="S3.9.p5.10.m10.1.1.2.cmml" xref="S3.9.p5.10.m10.1.1.2">𝐴</ci><ci id="S3.9.p5.10.m10.1.1.3.cmml" xref="S3.9.p5.10.m10.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.10.m10.1c">A\cap B</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.10.m10.1d">italic_A ∩ italic_B</annotation></semantics></math>. Since the paths in <math alttext="\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{\star}" class="ltx_Math" display="inline" id="S3.9.p5.11.m11.1"><semantics id="S3.9.p5.11.m11.1a"><mrow id="S3.9.p5.11.m11.1.1" xref="S3.9.p5.11.m11.1.1.cmml"><msubsup id="S3.9.p5.11.m11.1.1.2" xref="S3.9.p5.11.m11.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.11.m11.1.1.2.2.2" xref="S3.9.p5.11.m11.1.1.2.2.2.cmml">𝒫</mi><mn id="S3.9.p5.11.m11.1.1.2.2.3" xref="S3.9.p5.11.m11.1.1.2.2.3.cmml">1</mn><mo id="S3.9.p5.11.m11.1.1.2.3" xref="S3.9.p5.11.m11.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S3.9.p5.11.m11.1.1.1" xref="S3.9.p5.11.m11.1.1.1.cmml">∪</mo><msubsup id="S3.9.p5.11.m11.1.1.3" xref="S3.9.p5.11.m11.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.11.m11.1.1.3.2.2" xref="S3.9.p5.11.m11.1.1.3.2.2.cmml">𝒫</mi><mn id="S3.9.p5.11.m11.1.1.3.2.3" xref="S3.9.p5.11.m11.1.1.3.2.3.cmml">2</mn><mo id="S3.9.p5.11.m11.1.1.3.3" xref="S3.9.p5.11.m11.1.1.3.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.11.m11.1b"><apply id="S3.9.p5.11.m11.1.1.cmml" xref="S3.9.p5.11.m11.1.1"><union id="S3.9.p5.11.m11.1.1.1.cmml" xref="S3.9.p5.11.m11.1.1.1"></union><apply id="S3.9.p5.11.m11.1.1.2.cmml" xref="S3.9.p5.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.11.m11.1.1.2.1.cmml" xref="S3.9.p5.11.m11.1.1.2">superscript</csymbol><apply id="S3.9.p5.11.m11.1.1.2.2.cmml" xref="S3.9.p5.11.m11.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.11.m11.1.1.2.2.1.cmml" xref="S3.9.p5.11.m11.1.1.2">subscript</csymbol><ci id="S3.9.p5.11.m11.1.1.2.2.2.cmml" xref="S3.9.p5.11.m11.1.1.2.2.2">𝒫</ci><cn id="S3.9.p5.11.m11.1.1.2.2.3.cmml" type="integer" xref="S3.9.p5.11.m11.1.1.2.2.3">1</cn></apply><ci id="S3.9.p5.11.m11.1.1.2.3.cmml" xref="S3.9.p5.11.m11.1.1.2.3">⋆</ci></apply><apply id="S3.9.p5.11.m11.1.1.3.cmml" xref="S3.9.p5.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.11.m11.1.1.3.1.cmml" xref="S3.9.p5.11.m11.1.1.3">superscript</csymbol><apply id="S3.9.p5.11.m11.1.1.3.2.cmml" xref="S3.9.p5.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.11.m11.1.1.3.2.1.cmml" xref="S3.9.p5.11.m11.1.1.3">subscript</csymbol><ci id="S3.9.p5.11.m11.1.1.3.2.2.cmml" xref="S3.9.p5.11.m11.1.1.3.2.2">𝒫</ci><cn id="S3.9.p5.11.m11.1.1.3.2.3.cmml" type="integer" xref="S3.9.p5.11.m11.1.1.3.2.3">2</cn></apply><ci id="S3.9.p5.11.m11.1.1.3.3.cmml" xref="S3.9.p5.11.m11.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.11.m11.1c">\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.11.m11.1d">caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∪ caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> are pairwise vertex-disjoint, this implies that <math alttext="|\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{\star}|\leq|A\cap B|" class="ltx_Math" display="inline" id="S3.9.p5.12.m12.2"><semantics id="S3.9.p5.12.m12.2a"><mrow id="S3.9.p5.12.m12.2.2" xref="S3.9.p5.12.m12.2.2.cmml"><mrow id="S3.9.p5.12.m12.1.1.1.1" xref="S3.9.p5.12.m12.1.1.1.2.cmml"><mo id="S3.9.p5.12.m12.1.1.1.1.2" stretchy="false" xref="S3.9.p5.12.m12.1.1.1.2.1.cmml">|</mo><mrow id="S3.9.p5.12.m12.1.1.1.1.1" xref="S3.9.p5.12.m12.1.1.1.1.1.cmml"><msubsup id="S3.9.p5.12.m12.1.1.1.1.1.2" xref="S3.9.p5.12.m12.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.12.m12.1.1.1.1.1.2.2.2" xref="S3.9.p5.12.m12.1.1.1.1.1.2.2.2.cmml">𝒫</mi><mn id="S3.9.p5.12.m12.1.1.1.1.1.2.2.3" xref="S3.9.p5.12.m12.1.1.1.1.1.2.2.3.cmml">1</mn><mo id="S3.9.p5.12.m12.1.1.1.1.1.2.3" xref="S3.9.p5.12.m12.1.1.1.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S3.9.p5.12.m12.1.1.1.1.1.1" xref="S3.9.p5.12.m12.1.1.1.1.1.1.cmml">∪</mo><msubsup id="S3.9.p5.12.m12.1.1.1.1.1.3" xref="S3.9.p5.12.m12.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.12.m12.1.1.1.1.1.3.2.2" xref="S3.9.p5.12.m12.1.1.1.1.1.3.2.2.cmml">𝒫</mi><mn id="S3.9.p5.12.m12.1.1.1.1.1.3.2.3" xref="S3.9.p5.12.m12.1.1.1.1.1.3.2.3.cmml">2</mn><mo id="S3.9.p5.12.m12.1.1.1.1.1.3.3" xref="S3.9.p5.12.m12.1.1.1.1.1.3.3.cmml">⋆</mo></msubsup></mrow><mo id="S3.9.p5.12.m12.1.1.1.1.3" stretchy="false" xref="S3.9.p5.12.m12.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.9.p5.12.m12.2.2.3" xref="S3.9.p5.12.m12.2.2.3.cmml">≤</mo><mrow id="S3.9.p5.12.m12.2.2.2.1" xref="S3.9.p5.12.m12.2.2.2.2.cmml"><mo id="S3.9.p5.12.m12.2.2.2.1.2" stretchy="false" xref="S3.9.p5.12.m12.2.2.2.2.1.cmml">|</mo><mrow id="S3.9.p5.12.m12.2.2.2.1.1" xref="S3.9.p5.12.m12.2.2.2.1.1.cmml"><mi id="S3.9.p5.12.m12.2.2.2.1.1.2" xref="S3.9.p5.12.m12.2.2.2.1.1.2.cmml">A</mi><mo id="S3.9.p5.12.m12.2.2.2.1.1.1" xref="S3.9.p5.12.m12.2.2.2.1.1.1.cmml">∩</mo><mi id="S3.9.p5.12.m12.2.2.2.1.1.3" xref="S3.9.p5.12.m12.2.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.9.p5.12.m12.2.2.2.1.3" stretchy="false" xref="S3.9.p5.12.m12.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.12.m12.2b"><apply id="S3.9.p5.12.m12.2.2.cmml" xref="S3.9.p5.12.m12.2.2"><leq id="S3.9.p5.12.m12.2.2.3.cmml" xref="S3.9.p5.12.m12.2.2.3"></leq><apply id="S3.9.p5.12.m12.1.1.1.2.cmml" xref="S3.9.p5.12.m12.1.1.1.1"><abs id="S3.9.p5.12.m12.1.1.1.2.1.cmml" xref="S3.9.p5.12.m12.1.1.1.1.2"></abs><apply id="S3.9.p5.12.m12.1.1.1.1.1.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1"><union id="S3.9.p5.12.m12.1.1.1.1.1.1.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.1"></union><apply id="S3.9.p5.12.m12.1.1.1.1.1.2.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.12.m12.1.1.1.1.1.2.1.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.2">superscript</csymbol><apply id="S3.9.p5.12.m12.1.1.1.1.1.2.2.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.12.m12.1.1.1.1.1.2.2.1.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.2">subscript</csymbol><ci id="S3.9.p5.12.m12.1.1.1.1.1.2.2.2.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.2.2.2">𝒫</ci><cn id="S3.9.p5.12.m12.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S3.9.p5.12.m12.1.1.1.1.1.2.2.3">1</cn></apply><ci id="S3.9.p5.12.m12.1.1.1.1.1.2.3.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.2.3">⋆</ci></apply><apply id="S3.9.p5.12.m12.1.1.1.1.1.3.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.12.m12.1.1.1.1.1.3.1.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.3">superscript</csymbol><apply id="S3.9.p5.12.m12.1.1.1.1.1.3.2.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.12.m12.1.1.1.1.1.3.2.1.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.3">subscript</csymbol><ci id="S3.9.p5.12.m12.1.1.1.1.1.3.2.2.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.3.2.2">𝒫</ci><cn id="S3.9.p5.12.m12.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S3.9.p5.12.m12.1.1.1.1.1.3.2.3">2</cn></apply><ci id="S3.9.p5.12.m12.1.1.1.1.1.3.3.cmml" xref="S3.9.p5.12.m12.1.1.1.1.1.3.3">⋆</ci></apply></apply></apply><apply id="S3.9.p5.12.m12.2.2.2.2.cmml" xref="S3.9.p5.12.m12.2.2.2.1"><abs id="S3.9.p5.12.m12.2.2.2.2.1.cmml" xref="S3.9.p5.12.m12.2.2.2.1.2"></abs><apply id="S3.9.p5.12.m12.2.2.2.1.1.cmml" xref="S3.9.p5.12.m12.2.2.2.1.1"><intersect id="S3.9.p5.12.m12.2.2.2.1.1.1.cmml" xref="S3.9.p5.12.m12.2.2.2.1.1.1"></intersect><ci id="S3.9.p5.12.m12.2.2.2.1.1.2.cmml" xref="S3.9.p5.12.m12.2.2.2.1.1.2">𝐴</ci><ci id="S3.9.p5.12.m12.2.2.2.1.1.3.cmml" xref="S3.9.p5.12.m12.2.2.2.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.12.m12.2c">|\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{\star}|\leq|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.12.m12.2d">| caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∪ caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ≤ | italic_A ∩ italic_B |</annotation></semantics></math>. Therefore,</p> <table class="ltx_equation ltx_eqn_table" id="S3.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|Z\setminus B|=|\mathcal{P}^{\star}|=|\mathcal{P}_{0}^{\star}|+|\mathcal{P}_{1% }^{\star}|+|\mathcal{P}_{2}^{\star}|\leq|W\setminus B|+|\mathcal{P}_{2}^{\star% }|\leq\frac{|A\setminus B|-|\Delta_{\ell+1}\setminus B|}{\ell+1}+|A\cap B|+|% \mathcal{P}_{2}^{\star}|\enspace," class="ltx_Math" display="block" id="S3.E2.m1.3"><semantics id="S3.E2.m1.3a"><mrow id="S3.E2.m1.3.3.1" xref="S3.E2.m1.3.3.1.1.cmml"><mrow id="S3.E2.m1.3.3.1.1" xref="S3.E2.m1.3.3.1.1.cmml"><mrow id="S3.E2.m1.3.3.1.1.1.1" xref="S3.E2.m1.3.3.1.1.1.2.cmml"><mo id="S3.E2.m1.3.3.1.1.1.1.2" stretchy="false" xref="S3.E2.m1.3.3.1.1.1.2.1.cmml">|</mo><mrow id="S3.E2.m1.3.3.1.1.1.1.1" xref="S3.E2.m1.3.3.1.1.1.1.1.cmml"><mi id="S3.E2.m1.3.3.1.1.1.1.1.2" xref="S3.E2.m1.3.3.1.1.1.1.1.2.cmml">Z</mi><mo id="S3.E2.m1.3.3.1.1.1.1.1.1" xref="S3.E2.m1.3.3.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.E2.m1.3.3.1.1.1.1.1.3" xref="S3.E2.m1.3.3.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E2.m1.3.3.1.1.1.1.3" stretchy="false" xref="S3.E2.m1.3.3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.E2.m1.3.3.1.1.11" xref="S3.E2.m1.3.3.1.1.11.cmml">=</mo><mrow id="S3.E2.m1.3.3.1.1.2.1" xref="S3.E2.m1.3.3.1.1.2.2.cmml"><mo id="S3.E2.m1.3.3.1.1.2.1.2" stretchy="false" xref="S3.E2.m1.3.3.1.1.2.2.1.cmml">|</mo><msup id="S3.E2.m1.3.3.1.1.2.1.1" xref="S3.E2.m1.3.3.1.1.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E2.m1.3.3.1.1.2.1.1.2" xref="S3.E2.m1.3.3.1.1.2.1.1.2.cmml">𝒫</mi><mo id="S3.E2.m1.3.3.1.1.2.1.1.3" xref="S3.E2.m1.3.3.1.1.2.1.1.3.cmml">⋆</mo></msup><mo id="S3.E2.m1.3.3.1.1.2.1.3" stretchy="false" xref="S3.E2.m1.3.3.1.1.2.2.1.cmml">|</mo></mrow><mo id="S3.E2.m1.3.3.1.1.12" xref="S3.E2.m1.3.3.1.1.12.cmml">=</mo><mrow id="S3.E2.m1.3.3.1.1.5" xref="S3.E2.m1.3.3.1.1.5.cmml"><mrow id="S3.E2.m1.3.3.1.1.3.1.1" xref="S3.E2.m1.3.3.1.1.3.1.2.cmml"><mo id="S3.E2.m1.3.3.1.1.3.1.1.2" stretchy="false" xref="S3.E2.m1.3.3.1.1.3.1.2.1.cmml">|</mo><msubsup id="S3.E2.m1.3.3.1.1.3.1.1.1" xref="S3.E2.m1.3.3.1.1.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E2.m1.3.3.1.1.3.1.1.1.2.2" xref="S3.E2.m1.3.3.1.1.3.1.1.1.2.2.cmml">𝒫</mi><mn id="S3.E2.m1.3.3.1.1.3.1.1.1.2.3" xref="S3.E2.m1.3.3.1.1.3.1.1.1.2.3.cmml">0</mn><mo id="S3.E2.m1.3.3.1.1.3.1.1.1.3" xref="S3.E2.m1.3.3.1.1.3.1.1.1.3.cmml">⋆</mo></msubsup><mo id="S3.E2.m1.3.3.1.1.3.1.1.3" stretchy="false" xref="S3.E2.m1.3.3.1.1.3.1.2.1.cmml">|</mo></mrow><mo id="S3.E2.m1.3.3.1.1.5.4" xref="S3.E2.m1.3.3.1.1.5.4.cmml">+</mo><mrow id="S3.E2.m1.3.3.1.1.4.2.1" xref="S3.E2.m1.3.3.1.1.4.2.2.cmml"><mo id="S3.E2.m1.3.3.1.1.4.2.1.2" stretchy="false" xref="S3.E2.m1.3.3.1.1.4.2.2.1.cmml">|</mo><msubsup id="S3.E2.m1.3.3.1.1.4.2.1.1" xref="S3.E2.m1.3.3.1.1.4.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E2.m1.3.3.1.1.4.2.1.1.2.2" xref="S3.E2.m1.3.3.1.1.4.2.1.1.2.2.cmml">𝒫</mi><mn id="S3.E2.m1.3.3.1.1.4.2.1.1.2.3" xref="S3.E2.m1.3.3.1.1.4.2.1.1.2.3.cmml">1</mn><mo id="S3.E2.m1.3.3.1.1.4.2.1.1.3" xref="S3.E2.m1.3.3.1.1.4.2.1.1.3.cmml">⋆</mo></msubsup><mo id="S3.E2.m1.3.3.1.1.4.2.1.3" stretchy="false" xref="S3.E2.m1.3.3.1.1.4.2.2.1.cmml">|</mo></mrow><mo id="S3.E2.m1.3.3.1.1.5.4a" xref="S3.E2.m1.3.3.1.1.5.4.cmml">+</mo><mrow id="S3.E2.m1.3.3.1.1.5.3.1" xref="S3.E2.m1.3.3.1.1.5.3.2.cmml"><mo id="S3.E2.m1.3.3.1.1.5.3.1.2" stretchy="false" xref="S3.E2.m1.3.3.1.1.5.3.2.1.cmml">|</mo><msubsup id="S3.E2.m1.3.3.1.1.5.3.1.1" xref="S3.E2.m1.3.3.1.1.5.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E2.m1.3.3.1.1.5.3.1.1.2.2" xref="S3.E2.m1.3.3.1.1.5.3.1.1.2.2.cmml">𝒫</mi><mn id="S3.E2.m1.3.3.1.1.5.3.1.1.2.3" xref="S3.E2.m1.3.3.1.1.5.3.1.1.2.3.cmml">2</mn><mo id="S3.E2.m1.3.3.1.1.5.3.1.1.3" xref="S3.E2.m1.3.3.1.1.5.3.1.1.3.cmml">⋆</mo></msubsup><mo id="S3.E2.m1.3.3.1.1.5.3.1.3" stretchy="false" xref="S3.E2.m1.3.3.1.1.5.3.2.1.cmml">|</mo></mrow></mrow><mo id="S3.E2.m1.3.3.1.1.13" xref="S3.E2.m1.3.3.1.1.13.cmml">≤</mo><mrow id="S3.E2.m1.3.3.1.1.7" xref="S3.E2.m1.3.3.1.1.7.cmml"><mrow id="S3.E2.m1.3.3.1.1.6.1.1" xref="S3.E2.m1.3.3.1.1.6.1.2.cmml"><mo id="S3.E2.m1.3.3.1.1.6.1.1.2" stretchy="false" xref="S3.E2.m1.3.3.1.1.6.1.2.1.cmml">|</mo><mrow id="S3.E2.m1.3.3.1.1.6.1.1.1" xref="S3.E2.m1.3.3.1.1.6.1.1.1.cmml"><mi id="S3.E2.m1.3.3.1.1.6.1.1.1.2" xref="S3.E2.m1.3.3.1.1.6.1.1.1.2.cmml">W</mi><mo id="S3.E2.m1.3.3.1.1.6.1.1.1.1" xref="S3.E2.m1.3.3.1.1.6.1.1.1.1.cmml">∖</mo><mi id="S3.E2.m1.3.3.1.1.6.1.1.1.3" xref="S3.E2.m1.3.3.1.1.6.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E2.m1.3.3.1.1.6.1.1.3" stretchy="false" xref="S3.E2.m1.3.3.1.1.6.1.2.1.cmml">|</mo></mrow><mo id="S3.E2.m1.3.3.1.1.7.3" xref="S3.E2.m1.3.3.1.1.7.3.cmml">+</mo><mrow id="S3.E2.m1.3.3.1.1.7.2.1" xref="S3.E2.m1.3.3.1.1.7.2.2.cmml"><mo id="S3.E2.m1.3.3.1.1.7.2.1.2" stretchy="false" xref="S3.E2.m1.3.3.1.1.7.2.2.1.cmml">|</mo><msubsup id="S3.E2.m1.3.3.1.1.7.2.1.1" xref="S3.E2.m1.3.3.1.1.7.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E2.m1.3.3.1.1.7.2.1.1.2.2" xref="S3.E2.m1.3.3.1.1.7.2.1.1.2.2.cmml">𝒫</mi><mn id="S3.E2.m1.3.3.1.1.7.2.1.1.2.3" xref="S3.E2.m1.3.3.1.1.7.2.1.1.2.3.cmml">2</mn><mo id="S3.E2.m1.3.3.1.1.7.2.1.1.3" xref="S3.E2.m1.3.3.1.1.7.2.1.1.3.cmml">⋆</mo></msubsup><mo id="S3.E2.m1.3.3.1.1.7.2.1.3" stretchy="false" xref="S3.E2.m1.3.3.1.1.7.2.2.1.cmml">|</mo></mrow></mrow><mo id="S3.E2.m1.3.3.1.1.14" xref="S3.E2.m1.3.3.1.1.14.cmml">≤</mo><mrow id="S3.E2.m1.3.3.1.1.9" xref="S3.E2.m1.3.3.1.1.9.cmml"><mfrac id="S3.E2.m1.2.2" xref="S3.E2.m1.2.2.cmml"><mrow id="S3.E2.m1.2.2.2" xref="S3.E2.m1.2.2.2.cmml"><mrow id="S3.E2.m1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.2.cmml"><mo id="S3.E2.m1.1.1.1.1.1.2" stretchy="false" xref="S3.E2.m1.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.E2.m1.1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.1.cmml"><mi id="S3.E2.m1.1.1.1.1.1.1.2" xref="S3.E2.m1.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.E2.m1.1.1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.E2.m1.1.1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E2.m1.1.1.1.1.1.3" stretchy="false" xref="S3.E2.m1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.E2.m1.2.2.2.3" xref="S3.E2.m1.2.2.2.3.cmml">−</mo><mrow id="S3.E2.m1.2.2.2.2.1" xref="S3.E2.m1.2.2.2.2.2.cmml"><mo id="S3.E2.m1.2.2.2.2.1.2" stretchy="false" xref="S3.E2.m1.2.2.2.2.2.1.cmml">|</mo><mrow id="S3.E2.m1.2.2.2.2.1.1" xref="S3.E2.m1.2.2.2.2.1.1.cmml"><msub id="S3.E2.m1.2.2.2.2.1.1.2" xref="S3.E2.m1.2.2.2.2.1.1.2.cmml"><mi id="S3.E2.m1.2.2.2.2.1.1.2.2" mathvariant="normal" xref="S3.E2.m1.2.2.2.2.1.1.2.2.cmml">Δ</mi><mrow id="S3.E2.m1.2.2.2.2.1.1.2.3" xref="S3.E2.m1.2.2.2.2.1.1.2.3.cmml"><mi id="S3.E2.m1.2.2.2.2.1.1.2.3.2" mathvariant="normal" xref="S3.E2.m1.2.2.2.2.1.1.2.3.2.cmml">ℓ</mi><mo id="S3.E2.m1.2.2.2.2.1.1.2.3.1" 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B|=|\mathcal{P}^{\star}|=|\mathcal{P}_{0}^{\star}|+|\mathcal{P}_{1% }^{\star}|+|\mathcal{P}_{2}^{\star}|\leq|W\setminus B|+|\mathcal{P}_{2}^{\star% }|\leq\frac{|A\setminus B|-|\Delta_{\ell+1}\setminus B|}{\ell+1}+|A\cap B|+|% \mathcal{P}_{2}^{\star}|\enspace,</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.3d">| italic_Z ∖ italic_B | = | caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | = | caligraphic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | + | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | + | caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ≤ | italic_W ∖ italic_B | + | caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ≤ divide start_ARG | italic_A ∖ italic_B | - | roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + | italic_A ∩ italic_B | + | caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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> <p class="ltx_p" id="S3.9.p5.15">where the last inequality is an application of inequality (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E1" title="Equation 1 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">1</span></a>). At this point, adding (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E1" title="Equation 1 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">1</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E2" title="Equation 2 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">2</span></a>) and using the inequality <math alttext="|\mathcal{P}_{2}^{\star}|\leq|\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{% \star}|\leq|A\cap B|" class="ltx_Math" display="inline" id="S3.9.p5.13.m1.3"><semantics id="S3.9.p5.13.m1.3a"><mrow id="S3.9.p5.13.m1.3.3" xref="S3.9.p5.13.m1.3.3.cmml"><mrow id="S3.9.p5.13.m1.1.1.1.1" xref="S3.9.p5.13.m1.1.1.1.2.cmml"><mo id="S3.9.p5.13.m1.1.1.1.1.2" stretchy="false" xref="S3.9.p5.13.m1.1.1.1.2.1.cmml">|</mo><msubsup id="S3.9.p5.13.m1.1.1.1.1.1" 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xref="S3.9.p5.13.m1.2.2.2.1.1.3.2.2">𝒫</ci><cn id="S3.9.p5.13.m1.2.2.2.1.1.3.2.3.cmml" type="integer" xref="S3.9.p5.13.m1.2.2.2.1.1.3.2.3">2</cn></apply><ci id="S3.9.p5.13.m1.2.2.2.1.1.3.3.cmml" xref="S3.9.p5.13.m1.2.2.2.1.1.3.3">⋆</ci></apply></apply></apply></apply><apply id="S3.9.p5.13.m1.3.3c.cmml" xref="S3.9.p5.13.m1.3.3"><leq id="S3.9.p5.13.m1.3.3.6.cmml" xref="S3.9.p5.13.m1.3.3.6"></leq><share href="https://arxiv.org/html/2503.17112v1#S3.9.p5.13.m1.2.2.2.cmml" id="S3.9.p5.13.m1.3.3d.cmml" xref="S3.9.p5.13.m1.3.3"></share><apply id="S3.9.p5.13.m1.3.3.3.2.cmml" xref="S3.9.p5.13.m1.3.3.3.1"><abs id="S3.9.p5.13.m1.3.3.3.2.1.cmml" xref="S3.9.p5.13.m1.3.3.3.1.2"></abs><apply id="S3.9.p5.13.m1.3.3.3.1.1.cmml" xref="S3.9.p5.13.m1.3.3.3.1.1"><intersect id="S3.9.p5.13.m1.3.3.3.1.1.1.cmml" xref="S3.9.p5.13.m1.3.3.3.1.1.1"></intersect><ci id="S3.9.p5.13.m1.3.3.3.1.1.2.cmml" xref="S3.9.p5.13.m1.3.3.3.1.1.2">𝐴</ci><ci id="S3.9.p5.13.m1.3.3.3.1.1.3.cmml" xref="S3.9.p5.13.m1.3.3.3.1.1.3">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.13.m1.3c">|\mathcal{P}_{2}^{\star}|\leq|\mathcal{P}_{1}^{\star}\cup\mathcal{P}_{2}^{% \star}|\leq|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.13.m1.3d">| caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ≤ | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∪ caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ≤ | italic_A ∩ italic_B |</annotation></semantics></math> immediately gives the bound discussed in <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm8" title="Remark 8. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Remark</span> <span class="ltx_text ltx_ref_tag">8</span></a>. With a bit more work, we can do better. Since each path in <math alttext="\mathcal{P}_{0}^{\star}\cup\mathcal{P}^{\star}_{2}" class="ltx_Math" display="inline" id="S3.9.p5.14.m2.1"><semantics id="S3.9.p5.14.m2.1a"><mrow id="S3.9.p5.14.m2.1.1" xref="S3.9.p5.14.m2.1.1.cmml"><msubsup id="S3.9.p5.14.m2.1.1.2" xref="S3.9.p5.14.m2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.14.m2.1.1.2.2.2" xref="S3.9.p5.14.m2.1.1.2.2.2.cmml">𝒫</mi><mn id="S3.9.p5.14.m2.1.1.2.2.3" xref="S3.9.p5.14.m2.1.1.2.2.3.cmml">0</mn><mo id="S3.9.p5.14.m2.1.1.2.3" xref="S3.9.p5.14.m2.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S3.9.p5.14.m2.1.1.1" xref="S3.9.p5.14.m2.1.1.1.cmml">∪</mo><msubsup id="S3.9.p5.14.m2.1.1.3" xref="S3.9.p5.14.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.9.p5.14.m2.1.1.3.2.2" xref="S3.9.p5.14.m2.1.1.3.2.2.cmml">𝒫</mi><mn id="S3.9.p5.14.m2.1.1.3.3" xref="S3.9.p5.14.m2.1.1.3.3.cmml">2</mn><mo id="S3.9.p5.14.m2.1.1.3.2.3" xref="S3.9.p5.14.m2.1.1.3.2.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.14.m2.1b"><apply id="S3.9.p5.14.m2.1.1.cmml" xref="S3.9.p5.14.m2.1.1"><union id="S3.9.p5.14.m2.1.1.1.cmml" xref="S3.9.p5.14.m2.1.1.1"></union><apply id="S3.9.p5.14.m2.1.1.2.cmml" xref="S3.9.p5.14.m2.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.14.m2.1.1.2.1.cmml" xref="S3.9.p5.14.m2.1.1.2">superscript</csymbol><apply id="S3.9.p5.14.m2.1.1.2.2.cmml" xref="S3.9.p5.14.m2.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.14.m2.1.1.2.2.1.cmml" xref="S3.9.p5.14.m2.1.1.2">subscript</csymbol><ci id="S3.9.p5.14.m2.1.1.2.2.2.cmml" xref="S3.9.p5.14.m2.1.1.2.2.2">𝒫</ci><cn id="S3.9.p5.14.m2.1.1.2.2.3.cmml" type="integer" xref="S3.9.p5.14.m2.1.1.2.2.3">0</cn></apply><ci id="S3.9.p5.14.m2.1.1.2.3.cmml" xref="S3.9.p5.14.m2.1.1.2.3">⋆</ci></apply><apply id="S3.9.p5.14.m2.1.1.3.cmml" xref="S3.9.p5.14.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.14.m2.1.1.3.1.cmml" xref="S3.9.p5.14.m2.1.1.3">subscript</csymbol><apply id="S3.9.p5.14.m2.1.1.3.2.cmml" xref="S3.9.p5.14.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.14.m2.1.1.3.2.1.cmml" xref="S3.9.p5.14.m2.1.1.3">superscript</csymbol><ci id="S3.9.p5.14.m2.1.1.3.2.2.cmml" xref="S3.9.p5.14.m2.1.1.3.2.2">𝒫</ci><ci id="S3.9.p5.14.m2.1.1.3.2.3.cmml" xref="S3.9.p5.14.m2.1.1.3.2.3">⋆</ci></apply><cn id="S3.9.p5.14.m2.1.1.3.3.cmml" type="integer" xref="S3.9.p5.14.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.14.m2.1c">\mathcal{P}_{0}^{\star}\cup\mathcal{P}^{\star}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.14.m2.1d">caligraphic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∪ caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> starts at a distinct vertex in <math alttext="\Delta_{\ell+1}\setminus B" class="ltx_Math" display="inline" id="S3.9.p5.15.m3.1"><semantics id="S3.9.p5.15.m3.1a"><mrow id="S3.9.p5.15.m3.1.1" xref="S3.9.p5.15.m3.1.1.cmml"><msub id="S3.9.p5.15.m3.1.1.2" xref="S3.9.p5.15.m3.1.1.2.cmml"><mi id="S3.9.p5.15.m3.1.1.2.2" mathvariant="normal" xref="S3.9.p5.15.m3.1.1.2.2.cmml">Δ</mi><mrow id="S3.9.p5.15.m3.1.1.2.3" xref="S3.9.p5.15.m3.1.1.2.3.cmml"><mi id="S3.9.p5.15.m3.1.1.2.3.2" mathvariant="normal" xref="S3.9.p5.15.m3.1.1.2.3.2.cmml">ℓ</mi><mo id="S3.9.p5.15.m3.1.1.2.3.1" xref="S3.9.p5.15.m3.1.1.2.3.1.cmml">+</mo><mn id="S3.9.p5.15.m3.1.1.2.3.3" xref="S3.9.p5.15.m3.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.9.p5.15.m3.1.1.1" xref="S3.9.p5.15.m3.1.1.1.cmml">∖</mo><mi id="S3.9.p5.15.m3.1.1.3" xref="S3.9.p5.15.m3.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.15.m3.1b"><apply id="S3.9.p5.15.m3.1.1.cmml" xref="S3.9.p5.15.m3.1.1"><setdiff id="S3.9.p5.15.m3.1.1.1.cmml" xref="S3.9.p5.15.m3.1.1.1"></setdiff><apply id="S3.9.p5.15.m3.1.1.2.cmml" xref="S3.9.p5.15.m3.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.15.m3.1.1.2.1.cmml" xref="S3.9.p5.15.m3.1.1.2">subscript</csymbol><ci id="S3.9.p5.15.m3.1.1.2.2.cmml" xref="S3.9.p5.15.m3.1.1.2.2">Δ</ci><apply id="S3.9.p5.15.m3.1.1.2.3.cmml" xref="S3.9.p5.15.m3.1.1.2.3"><plus id="S3.9.p5.15.m3.1.1.2.3.1.cmml" xref="S3.9.p5.15.m3.1.1.2.3.1"></plus><ci id="S3.9.p5.15.m3.1.1.2.3.2.cmml" xref="S3.9.p5.15.m3.1.1.2.3.2">ℓ</ci><cn id="S3.9.p5.15.m3.1.1.2.3.3.cmml" type="integer" xref="S3.9.p5.15.m3.1.1.2.3.3">1</cn></apply></apply><ci id="S3.9.p5.15.m3.1.1.3.cmml" xref="S3.9.p5.15.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.15.m3.1c">\Delta_{\ell+1}\setminus B</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.15.m3.1d">roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∖ italic_B</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S3.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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xref="S3.E3.m1.1.1.1.1.2.1.1.1.2.3.cmml">0</mn><mo id="S3.E3.m1.1.1.1.1.2.1.1.1.3" xref="S3.E3.m1.1.1.1.1.2.1.1.1.3.cmml">⋆</mo></msubsup><mo id="S3.E3.m1.1.1.1.1.2.1.1.3" stretchy="false" xref="S3.E3.m1.1.1.1.1.2.1.2.1.cmml">|</mo></mrow><mo id="S3.E3.m1.1.1.1.1.3.3" xref="S3.E3.m1.1.1.1.1.3.3.cmml">+</mo><mrow id="S3.E3.m1.1.1.1.1.3.2.1" xref="S3.E3.m1.1.1.1.1.3.2.2.cmml"><mo id="S3.E3.m1.1.1.1.1.3.2.1.2" stretchy="false" xref="S3.E3.m1.1.1.1.1.3.2.2.1.cmml">|</mo><msubsup id="S3.E3.m1.1.1.1.1.3.2.1.1" xref="S3.E3.m1.1.1.1.1.3.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E3.m1.1.1.1.1.3.2.1.1.2.2" xref="S3.E3.m1.1.1.1.1.3.2.1.1.2.2.cmml">𝒫</mi><mn id="S3.E3.m1.1.1.1.1.3.2.1.1.3" xref="S3.E3.m1.1.1.1.1.3.2.1.1.3.cmml">2</mn><mo id="S3.E3.m1.1.1.1.1.3.2.1.1.2.3" xref="S3.E3.m1.1.1.1.1.3.2.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S3.E3.m1.1.1.1.1.3.2.1.3" stretchy="false" xref="S3.E3.m1.1.1.1.1.3.2.2.1.cmml">|</mo></mrow></mrow><mo id="S3.E3.m1.1.1.1.1.8" xref="S3.E3.m1.1.1.1.1.8.cmml">=</mo><mrow id="S3.E3.m1.1.1.1.1.5" xref="S3.E3.m1.1.1.1.1.5.cmml"><mrow id="S3.E3.m1.1.1.1.1.4.1.1" xref="S3.E3.m1.1.1.1.1.4.1.2.cmml"><mo id="S3.E3.m1.1.1.1.1.4.1.1.2" stretchy="false" xref="S3.E3.m1.1.1.1.1.4.1.2.1.cmml">|</mo><mrow id="S3.E3.m1.1.1.1.1.4.1.1.1" xref="S3.E3.m1.1.1.1.1.4.1.1.1.cmml"><mi id="S3.E3.m1.1.1.1.1.4.1.1.1.2" xref="S3.E3.m1.1.1.1.1.4.1.1.1.2.cmml">Z</mi><mo id="S3.E3.m1.1.1.1.1.4.1.1.1.1" xref="S3.E3.m1.1.1.1.1.4.1.1.1.1.cmml">∖</mo><mi id="S3.E3.m1.1.1.1.1.4.1.1.1.3" xref="S3.E3.m1.1.1.1.1.4.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E3.m1.1.1.1.1.4.1.1.3" stretchy="false" xref="S3.E3.m1.1.1.1.1.4.1.2.1.cmml">|</mo></mrow><mo id="S3.E3.m1.1.1.1.1.5.3" xref="S3.E3.m1.1.1.1.1.5.3.cmml">−</mo><mrow id="S3.E3.m1.1.1.1.1.5.2.1" xref="S3.E3.m1.1.1.1.1.5.2.2.cmml"><mo id="S3.E3.m1.1.1.1.1.5.2.1.2" stretchy="false" xref="S3.E3.m1.1.1.1.1.5.2.2.1.cmml">|</mo><msubsup id="S3.E3.m1.1.1.1.1.5.2.1.1" xref="S3.E3.m1.1.1.1.1.5.2.1.1.cmml"><mi 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xref="S3.E3.m1.1.1.1.1.5.2.1.1">subscript</csymbol><ci id="S3.E3.m1.1.1.1.1.5.2.1.1.2.2.cmml" xref="S3.E3.m1.1.1.1.1.5.2.1.1.2.2">𝒫</ci><cn id="S3.E3.m1.1.1.1.1.5.2.1.1.2.3.cmml" type="integer" xref="S3.E3.m1.1.1.1.1.5.2.1.1.2.3">1</cn></apply><ci id="S3.E3.m1.1.1.1.1.5.2.1.1.3.cmml" xref="S3.E3.m1.1.1.1.1.5.2.1.1.3">⋆</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m1.1c">|\Delta_{\ell+1}\setminus B|\geq|\mathcal{P}_{0}^{\star}|+|\mathcal{P}^{\star}% _{2}|=|Z\setminus B|-|\mathcal{P}_{1}^{\star}|\enspace.</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.1d">| roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∖ italic_B | ≥ | caligraphic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | + | caligraphic_P start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | = | italic_Z ∖ italic_B | - | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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> <p class="ltx_p" id="S3.9.p5.28">Using inequality (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E3" title="Equation 3 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">3</span></a>) in <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E2" title="In Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">2</span></a> we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|Z\setminus B|\leq\frac{|A\setminus B|-|Z\setminus B|+|\mathcal{P}_{1}^{\star}% |}{\ell+1}+|A\cap B|+|\mathcal{P}_{2}^{\star}|\enspace." class="ltx_Math" display="block" id="S3.Ex3.m1.4"><semantics id="S3.Ex3.m1.4a"><mrow id="S3.Ex3.m1.4.4.1" xref="S3.Ex3.m1.4.4.1.1.cmml"><mrow id="S3.Ex3.m1.4.4.1.1" xref="S3.Ex3.m1.4.4.1.1.cmml"><mrow id="S3.Ex3.m1.4.4.1.1.1.1" xref="S3.Ex3.m1.4.4.1.1.1.2.cmml"><mo id="S3.Ex3.m1.4.4.1.1.1.1.2" stretchy="false" xref="S3.Ex3.m1.4.4.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex3.m1.4.4.1.1.1.1.1" xref="S3.Ex3.m1.4.4.1.1.1.1.1.cmml"><mi id="S3.Ex3.m1.4.4.1.1.1.1.1.2" xref="S3.Ex3.m1.4.4.1.1.1.1.1.2.cmml">Z</mi><mo id="S3.Ex3.m1.4.4.1.1.1.1.1.1" xref="S3.Ex3.m1.4.4.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex3.m1.4.4.1.1.1.1.1.3" xref="S3.Ex3.m1.4.4.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex3.m1.4.4.1.1.1.1.3" stretchy="false" xref="S3.Ex3.m1.4.4.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex3.m1.4.4.1.1.4" xref="S3.Ex3.m1.4.4.1.1.4.cmml">≤</mo><mrow id="S3.Ex3.m1.4.4.1.1.3" xref="S3.Ex3.m1.4.4.1.1.3.cmml"><mfrac id="S3.Ex3.m1.3.3" xref="S3.Ex3.m1.3.3.cmml"><mrow id="S3.Ex3.m1.3.3.3" xref="S3.Ex3.m1.3.3.3.cmml"><mrow id="S3.Ex3.m1.2.2.2.2" xref="S3.Ex3.m1.2.2.2.2.cmml"><mrow id="S3.Ex3.m1.1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.2.cmml"><mo id="S3.Ex3.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex3.m1.1.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex3.m1.1.1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex3.m1.1.1.1.1.1.1.1.2" xref="S3.Ex3.m1.1.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex3.m1.1.1.1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex3.m1.1.1.1.1.1.1.1.3" xref="S3.Ex3.m1.1.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex3.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex3.m1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex3.m1.2.2.2.2.3" 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encoding="application/x-llamapun" id="S3.Ex3.m1.4d">| italic_Z ∖ italic_B | ≤ divide start_ARG | italic_A ∖ italic_B | - | italic_Z ∖ italic_B | + | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | end_ARG start_ARG roman_ℓ + 1 end_ARG + | italic_A ∩ italic_B | + | caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.9.p5.16">Rewriting this equation to isolate <math alttext="|Z\setminus B|" class="ltx_Math" display="inline" id="S3.9.p5.16.m1.1"><semantics id="S3.9.p5.16.m1.1a"><mrow id="S3.9.p5.16.m1.1.1.1" xref="S3.9.p5.16.m1.1.1.2.cmml"><mo id="S3.9.p5.16.m1.1.1.1.2" stretchy="false" xref="S3.9.p5.16.m1.1.1.2.1.cmml">|</mo><mrow id="S3.9.p5.16.m1.1.1.1.1" xref="S3.9.p5.16.m1.1.1.1.1.cmml"><mi id="S3.9.p5.16.m1.1.1.1.1.2" xref="S3.9.p5.16.m1.1.1.1.1.2.cmml">Z</mi><mo id="S3.9.p5.16.m1.1.1.1.1.1" xref="S3.9.p5.16.m1.1.1.1.1.1.cmml">∖</mo><mi id="S3.9.p5.16.m1.1.1.1.1.3" xref="S3.9.p5.16.m1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.9.p5.16.m1.1.1.1.3" stretchy="false" xref="S3.9.p5.16.m1.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.16.m1.1b"><apply id="S3.9.p5.16.m1.1.1.2.cmml" xref="S3.9.p5.16.m1.1.1.1"><abs id="S3.9.p5.16.m1.1.1.2.1.cmml" xref="S3.9.p5.16.m1.1.1.1.2"></abs><apply id="S3.9.p5.16.m1.1.1.1.1.cmml" xref="S3.9.p5.16.m1.1.1.1.1"><setdiff id="S3.9.p5.16.m1.1.1.1.1.1.cmml" xref="S3.9.p5.16.m1.1.1.1.1.1"></setdiff><ci id="S3.9.p5.16.m1.1.1.1.1.2.cmml" xref="S3.9.p5.16.m1.1.1.1.1.2">𝑍</ci><ci id="S3.9.p5.16.m1.1.1.1.1.3.cmml" xref="S3.9.p5.16.m1.1.1.1.1.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.16.m1.1c">|Z\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.16.m1.1d">| italic_Z ∖ italic_B |</annotation></semantics></math>, we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx2"> <tbody id="S3.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle|Z\setminus B|" class="ltx_Math" display="inline" id="S3.E4.m1.1"><semantics id="S3.E4.m1.1a"><mrow id="S3.E4.m1.1.1.1" xref="S3.E4.m1.1.1.2.cmml"><mo id="S3.E4.m1.1.1.1.2" stretchy="false" xref="S3.E4.m1.1.1.2.1.cmml">|</mo><mrow id="S3.E4.m1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.cmml"><mi id="S3.E4.m1.1.1.1.1.2" xref="S3.E4.m1.1.1.1.1.2.cmml">Z</mi><mo id="S3.E4.m1.1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.1.cmml">∖</mo><mi id="S3.E4.m1.1.1.1.1.3" xref="S3.E4.m1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E4.m1.1.1.1.3" stretchy="false" xref="S3.E4.m1.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E4.m1.1b"><apply id="S3.E4.m1.1.1.2.cmml" xref="S3.E4.m1.1.1.1"><abs id="S3.E4.m1.1.1.2.1.cmml" xref="S3.E4.m1.1.1.1.2"></abs><apply id="S3.E4.m1.1.1.1.1.cmml" xref="S3.E4.m1.1.1.1.1"><setdiff id="S3.E4.m1.1.1.1.1.1.cmml" xref="S3.E4.m1.1.1.1.1.1"></setdiff><ci id="S3.E4.m1.1.1.1.1.2.cmml" xref="S3.E4.m1.1.1.1.1.2">𝑍</ci><ci id="S3.E4.m1.1.1.1.1.3.cmml" xref="S3.E4.m1.1.1.1.1.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E4.m1.1c">\displaystyle|Z\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.1d">| italic_Z ∖ italic_B |</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{|A\setminus B|+|\mathcal{P}_{1}^{\star}|}{\ell+2}+\left% (\frac{\ell+1}{\ell+2}\right)\cdot\left(|A\cap B|+|\mathcal{P}_{2}^{\star}|\right)" class="ltx_Math" display="inline" id="S3.E4.m2.4"><semantics id="S3.E4.m2.4a"><mrow id="S3.E4.m2.4.4" xref="S3.E4.m2.4.4.cmml"><mi id="S3.E4.m2.4.4.3" 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type="integer" xref="S3.E4.m2.4.4.1.1.1.1.1.2.1.1.2.3">2</cn></apply><ci id="S3.E4.m2.4.4.1.1.1.1.1.2.1.1.3.cmml" xref="S3.E4.m2.4.4.1.1.1.1.1.2.1.1.3">⋆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E4.m2.4c">\displaystyle\leq\frac{|A\setminus B|+|\mathcal{P}_{1}^{\star}|}{\ell+2}+\left% (\frac{\ell+1}{\ell+2}\right)\cdot\left(|A\cap B|+|\mathcal{P}_{2}^{\star}|\right)</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m2.4d">≤ divide start_ARG | italic_A ∖ italic_B | + | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | end_ARG start_ARG roman_ℓ + 2 end_ARG + ( divide start_ARG roman_ℓ + 1 end_ARG start_ARG roman_ℓ + 2 end_ARG ) ⋅ ( | italic_A ∩ italic_B | + | caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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> <tbody id="S3.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle<\frac{|A\setminus B|}{\ell+2}+|A\cap B|+|\mathcal{P}_{2}^{\star}% |+\tfrac{1}{3}|\mathcal{P}_{1}^{\star}|\enspace." class="ltx_Math" display="inline" id="S3.E5.m1.2"><semantics id="S3.E5.m1.2a"><mrow id="S3.E5.m1.2.2.1" xref="S3.E5.m1.2.2.1.1.cmml"><mrow id="S3.E5.m1.2.2.1.1" xref="S3.E5.m1.2.2.1.1.cmml"><mi id="S3.E5.m1.2.2.1.1.5" xref="S3.E5.m1.2.2.1.1.5.cmml"></mi><mo id="S3.E5.m1.2.2.1.1.4" xref="S3.E5.m1.2.2.1.1.4.cmml"><</mo><mrow id="S3.E5.m1.2.2.1.1.3" xref="S3.E5.m1.2.2.1.1.3.cmml"><mstyle displaystyle="true" id="S3.E5.m1.1.1" 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id="S3.E5.m1.2.2.1.1.3.3.1.1.1.2.1.cmml" xref="S3.E5.m1.2.2.1.1.3.3.1.1.1">subscript</csymbol><ci id="S3.E5.m1.2.2.1.1.3.3.1.1.1.2.2.cmml" xref="S3.E5.m1.2.2.1.1.3.3.1.1.1.2.2">𝒫</ci><cn id="S3.E5.m1.2.2.1.1.3.3.1.1.1.2.3.cmml" type="integer" xref="S3.E5.m1.2.2.1.1.3.3.1.1.1.2.3">1</cn></apply><ci id="S3.E5.m1.2.2.1.1.3.3.1.1.1.3.cmml" xref="S3.E5.m1.2.2.1.1.3.3.1.1.1.3">⋆</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5.m1.2c">\displaystyle<\frac{|A\setminus B|}{\ell+2}+|A\cap B|+|\mathcal{P}_{2}^{\star}% |+\tfrac{1}{3}|\mathcal{P}_{1}^{\star}|\enspace.</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.2d">< divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG + | italic_A ∩ italic_B | + | caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | + divide start_ARG 1 end_ARG start_ARG 3 end_ARG | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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="S3.9.p5.17">(The second inequality uses the assumption that <math alttext="\ell\geq 1" class="ltx_Math" display="inline" id="S3.9.p5.17.m1.1"><semantics id="S3.9.p5.17.m1.1a"><mrow id="S3.9.p5.17.m1.1.1" xref="S3.9.p5.17.m1.1.1.cmml"><mi id="S3.9.p5.17.m1.1.1.2" mathvariant="normal" xref="S3.9.p5.17.m1.1.1.2.cmml">ℓ</mi><mo id="S3.9.p5.17.m1.1.1.1" xref="S3.9.p5.17.m1.1.1.1.cmml">≥</mo><mn id="S3.9.p5.17.m1.1.1.3" xref="S3.9.p5.17.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.17.m1.1b"><apply id="S3.9.p5.17.m1.1.1.cmml" xref="S3.9.p5.17.m1.1.1"><geq id="S3.9.p5.17.m1.1.1.1.cmml" xref="S3.9.p5.17.m1.1.1.1"></geq><ci id="S3.9.p5.17.m1.1.1.2.cmml" xref="S3.9.p5.17.m1.1.1.2">ℓ</ci><cn id="S3.9.p5.17.m1.1.1.3.cmml" type="integer" xref="S3.9.p5.17.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.17.m1.1c">\ell\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.17.m1.1d">roman_ℓ ≥ 1</annotation></semantics></math>.) Using inequality (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E3" title="Equation 3 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">3</span></a>) in <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E1" title="In Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">1</span></a> we obtain,</p> <table class="ltx_equation ltx_eqn_table" id="S3.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|W\setminus B|+\frac{|Z\setminus B|}{\ell+1}\leq\frac{|A\setminus B|+|\mathcal% {P}_{1}^{\star}|}{\ell+1}+|A\cap B|\leq\frac{|A\setminus B|}{\ell+1}+|A\cap B|% +\tfrac{1}{2}|\mathcal{P}_{1}^{\star}|\enspace." class="ltx_Math" display="block" id="S3.E6.m1.5"><semantics id="S3.E6.m1.5a"><mrow id="S3.E6.m1.5.5.1" xref="S3.E6.m1.5.5.1.1.cmml"><mrow id="S3.E6.m1.5.5.1.1" xref="S3.E6.m1.5.5.1.1.cmml"><mrow id="S3.E6.m1.5.5.1.1.1" xref="S3.E6.m1.5.5.1.1.1.cmml"><mrow id="S3.E6.m1.5.5.1.1.1.1.1" xref="S3.E6.m1.5.5.1.1.1.1.2.cmml"><mo id="S3.E6.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S3.E6.m1.5.5.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.E6.m1.5.5.1.1.1.1.1.1" xref="S3.E6.m1.5.5.1.1.1.1.1.1.cmml"><mi id="S3.E6.m1.5.5.1.1.1.1.1.1.2" xref="S3.E6.m1.5.5.1.1.1.1.1.1.2.cmml">W</mi><mo id="S3.E6.m1.5.5.1.1.1.1.1.1.1" xref="S3.E6.m1.5.5.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.E6.m1.5.5.1.1.1.1.1.1.3" xref="S3.E6.m1.5.5.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E6.m1.5.5.1.1.1.1.1.3" stretchy="false" xref="S3.E6.m1.5.5.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.E6.m1.5.5.1.1.1.2" xref="S3.E6.m1.5.5.1.1.1.2.cmml">+</mo><mfrac id="S3.E6.m1.1.1" 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id="S3.E6.m1.5.5.1.1.4.2.cmml" xref="S3.E6.m1.5.5.1.1.4.2"><times id="S3.E6.m1.5.5.1.1.4.2.2.cmml" xref="S3.E6.m1.5.5.1.1.4.2.2"></times><apply id="S3.E6.m1.5.5.1.1.4.2.3.cmml" xref="S3.E6.m1.5.5.1.1.4.2.3"><divide id="S3.E6.m1.5.5.1.1.4.2.3.1.cmml" xref="S3.E6.m1.5.5.1.1.4.2.3"></divide><cn id="S3.E6.m1.5.5.1.1.4.2.3.2.cmml" type="integer" xref="S3.E6.m1.5.5.1.1.4.2.3.2">1</cn><cn id="S3.E6.m1.5.5.1.1.4.2.3.3.cmml" type="integer" xref="S3.E6.m1.5.5.1.1.4.2.3.3">2</cn></apply><apply id="S3.E6.m1.5.5.1.1.4.2.1.2.cmml" xref="S3.E6.m1.5.5.1.1.4.2.1.1"><abs id="S3.E6.m1.5.5.1.1.4.2.1.2.1.cmml" xref="S3.E6.m1.5.5.1.1.4.2.1.1.2"></abs><apply id="S3.E6.m1.5.5.1.1.4.2.1.1.1.cmml" xref="S3.E6.m1.5.5.1.1.4.2.1.1.1"><csymbol cd="ambiguous" id="S3.E6.m1.5.5.1.1.4.2.1.1.1.1.cmml" xref="S3.E6.m1.5.5.1.1.4.2.1.1.1">superscript</csymbol><apply id="S3.E6.m1.5.5.1.1.4.2.1.1.1.2.cmml" xref="S3.E6.m1.5.5.1.1.4.2.1.1.1"><csymbol cd="ambiguous" id="S3.E6.m1.5.5.1.1.4.2.1.1.1.2.1.cmml" xref="S3.E6.m1.5.5.1.1.4.2.1.1.1">subscript</csymbol><ci id="S3.E6.m1.5.5.1.1.4.2.1.1.1.2.2.cmml" xref="S3.E6.m1.5.5.1.1.4.2.1.1.1.2.2">𝒫</ci><cn id="S3.E6.m1.5.5.1.1.4.2.1.1.1.2.3.cmml" type="integer" xref="S3.E6.m1.5.5.1.1.4.2.1.1.1.2.3">1</cn></apply><ci id="S3.E6.m1.5.5.1.1.4.2.1.1.1.3.cmml" xref="S3.E6.m1.5.5.1.1.4.2.1.1.1.3">⋆</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E6.m1.5c">|W\setminus B|+\frac{|Z\setminus B|}{\ell+1}\leq\frac{|A\setminus B|+|\mathcal% {P}_{1}^{\star}|}{\ell+1}+|A\cap B|\leq\frac{|A\setminus B|}{\ell+1}+|A\cap B|% +\tfrac{1}{2}|\mathcal{P}_{1}^{\star}|\enspace.</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.5d">| italic_W ∖ italic_B | + divide start_ARG | italic_Z ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG ≤ divide start_ARG | italic_A ∖ italic_B | + | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | end_ARG start_ARG roman_ℓ + 1 end_ARG + | italic_A ∩ italic_B | ≤ divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + | italic_A ∩ italic_B | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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> <p class="ltx_p" id="S3.9.p5.19">(The second inequality again uses the assumption that <math alttext="\ell\geq 1" class="ltx_Math" display="inline" id="S3.9.p5.18.m1.1"><semantics id="S3.9.p5.18.m1.1a"><mrow id="S3.9.p5.18.m1.1.1" xref="S3.9.p5.18.m1.1.1.cmml"><mi id="S3.9.p5.18.m1.1.1.2" mathvariant="normal" xref="S3.9.p5.18.m1.1.1.2.cmml">ℓ</mi><mo id="S3.9.p5.18.m1.1.1.1" xref="S3.9.p5.18.m1.1.1.1.cmml">≥</mo><mn id="S3.9.p5.18.m1.1.1.3" xref="S3.9.p5.18.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.18.m1.1b"><apply id="S3.9.p5.18.m1.1.1.cmml" xref="S3.9.p5.18.m1.1.1"><geq id="S3.9.p5.18.m1.1.1.1.cmml" xref="S3.9.p5.18.m1.1.1.1"></geq><ci id="S3.9.p5.18.m1.1.1.2.cmml" xref="S3.9.p5.18.m1.1.1.2">ℓ</ci><cn id="S3.9.p5.18.m1.1.1.3.cmml" type="integer" xref="S3.9.p5.18.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.18.m1.1c">\ell\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.18.m1.1d">roman_ℓ ≥ 1</annotation></semantics></math>.) Adding (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E6" title="Equation 6 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">6</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E5" title="Equation 5 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">5</span></a>) and using the fact that <math alttext="\tfrac{5}{6}|\mathcal{P}_{1}^{\star}|+|\mathcal{P}_{2}^{\star}|\leq|\mathcal{P% }_{1}^{\star}|+|\mathcal{P}_{2}^{\star}|\leq|A\cap B|" class="ltx_Math" display="inline" id="S3.9.p5.19.m2.5"><semantics id="S3.9.p5.19.m2.5a"><mrow id="S3.9.p5.19.m2.5.5" xref="S3.9.p5.19.m2.5.5.cmml"><mrow id="S3.9.p5.19.m2.2.2.2" xref="S3.9.p5.19.m2.2.2.2.cmml"><mrow id="S3.9.p5.19.m2.1.1.1.1" xref="S3.9.p5.19.m2.1.1.1.1.cmml"><mfrac id="S3.9.p5.19.m2.1.1.1.1.3" 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id="S3.9.p5.19.m2.4.4.4.2.1.1.2.1.cmml" xref="S3.9.p5.19.m2.4.4.4.2.1.1">subscript</csymbol><ci id="S3.9.p5.19.m2.4.4.4.2.1.1.2.2.cmml" xref="S3.9.p5.19.m2.4.4.4.2.1.1.2.2">𝒫</ci><cn id="S3.9.p5.19.m2.4.4.4.2.1.1.2.3.cmml" type="integer" xref="S3.9.p5.19.m2.4.4.4.2.1.1.2.3">2</cn></apply><ci id="S3.9.p5.19.m2.4.4.4.2.1.1.3.cmml" xref="S3.9.p5.19.m2.4.4.4.2.1.1.3">⋆</ci></apply></apply></apply></apply><apply id="S3.9.p5.19.m2.5.5c.cmml" xref="S3.9.p5.19.m2.5.5"><leq id="S3.9.p5.19.m2.5.5.8.cmml" xref="S3.9.p5.19.m2.5.5.8"></leq><share href="https://arxiv.org/html/2503.17112v1#S3.9.p5.19.m2.4.4.4.cmml" id="S3.9.p5.19.m2.5.5d.cmml" xref="S3.9.p5.19.m2.5.5"></share><apply id="S3.9.p5.19.m2.5.5.5.2.cmml" xref="S3.9.p5.19.m2.5.5.5.1"><abs id="S3.9.p5.19.m2.5.5.5.2.1.cmml" xref="S3.9.p5.19.m2.5.5.5.1.2"></abs><apply id="S3.9.p5.19.m2.5.5.5.1.1.cmml" xref="S3.9.p5.19.m2.5.5.5.1.1"><intersect id="S3.9.p5.19.m2.5.5.5.1.1.1.cmml" xref="S3.9.p5.19.m2.5.5.5.1.1.1"></intersect><ci id="S3.9.p5.19.m2.5.5.5.1.1.2.cmml" xref="S3.9.p5.19.m2.5.5.5.1.1.2">𝐴</ci><ci id="S3.9.p5.19.m2.5.5.5.1.1.3.cmml" xref="S3.9.p5.19.m2.5.5.5.1.1.3">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.19.m2.5c">\tfrac{5}{6}|\mathcal{P}_{1}^{\star}|+|\mathcal{P}_{2}^{\star}|\leq|\mathcal{P% }_{1}^{\star}|+|\mathcal{P}_{2}^{\star}|\leq|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.19.m2.5d">divide start_ARG 5 end_ARG start_ARG 6 end_ARG | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | + | caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ≤ | caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | + | caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | ≤ | italic_A ∩ italic_B |</annotation></semantics></math>, we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx3"> <tbody id="S3.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle|W\setminus B|+\left(\frac{\ell+2}{\ell+1}\right)\cdot|Z\setminus B|" class="ltx_Math" display="inline" id="S3.E7.m1.3"><semantics id="S3.E7.m1.3a"><mrow id="S3.E7.m1.3.3" xref="S3.E7.m1.3.3.cmml"><mrow id="S3.E7.m1.2.2.1.1" xref="S3.E7.m1.2.2.1.2.cmml"><mo id="S3.E7.m1.2.2.1.1.2" stretchy="false" xref="S3.E7.m1.2.2.1.2.1.cmml">|</mo><mrow id="S3.E7.m1.2.2.1.1.1" xref="S3.E7.m1.2.2.1.1.1.cmml"><mi id="S3.E7.m1.2.2.1.1.1.2" xref="S3.E7.m1.2.2.1.1.1.2.cmml">W</mi><mo id="S3.E7.m1.2.2.1.1.1.1" xref="S3.E7.m1.2.2.1.1.1.1.cmml">∖</mo><mi id="S3.E7.m1.2.2.1.1.1.3" xref="S3.E7.m1.2.2.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E7.m1.2.2.1.1.3" stretchy="false" xref="S3.E7.m1.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.E7.m1.3.3.3" xref="S3.E7.m1.3.3.3.cmml">+</mo><mrow id="S3.E7.m1.3.3.2" xref="S3.E7.m1.3.3.2.cmml"><mrow id="S3.E7.m1.3.3.2.3.2" xref="S3.E7.m1.1.1.cmml"><mo id="S3.E7.m1.3.3.2.3.2.1" xref="S3.E7.m1.1.1.cmml">(</mo><mstyle displaystyle="true" id="S3.E7.m1.1.1" xref="S3.E7.m1.1.1.cmml"><mfrac id="S3.E7.m1.1.1a" xref="S3.E7.m1.1.1.cmml"><mrow id="S3.E7.m1.1.1.2" xref="S3.E7.m1.1.1.2.cmml"><mi id="S3.E7.m1.1.1.2.2" mathvariant="normal" xref="S3.E7.m1.1.1.2.2.cmml">ℓ</mi><mo id="S3.E7.m1.1.1.2.1" xref="S3.E7.m1.1.1.2.1.cmml">+</mo><mn id="S3.E7.m1.1.1.2.3" xref="S3.E7.m1.1.1.2.3.cmml">2</mn></mrow><mrow id="S3.E7.m1.1.1.3" xref="S3.E7.m1.1.1.3.cmml"><mi id="S3.E7.m1.1.1.3.2" mathvariant="normal" xref="S3.E7.m1.1.1.3.2.cmml">ℓ</mi><mo id="S3.E7.m1.1.1.3.1" xref="S3.E7.m1.1.1.3.1.cmml">+</mo><mn id="S3.E7.m1.1.1.3.3" xref="S3.E7.m1.1.1.3.3.cmml">1</mn></mrow></mfrac></mstyle><mo id="S3.E7.m1.3.3.2.3.2.2" rspace="0.055em" xref="S3.E7.m1.1.1.cmml">)</mo></mrow><mo id="S3.E7.m1.3.3.2.2" rspace="0.222em" xref="S3.E7.m1.3.3.2.2.cmml">⋅</mo><mrow id="S3.E7.m1.3.3.2.1.1" xref="S3.E7.m1.3.3.2.1.2.cmml"><mo id="S3.E7.m1.3.3.2.1.1.2" stretchy="false" xref="S3.E7.m1.3.3.2.1.2.1.cmml">|</mo><mrow id="S3.E7.m1.3.3.2.1.1.1" xref="S3.E7.m1.3.3.2.1.1.1.cmml"><mi id="S3.E7.m1.3.3.2.1.1.1.2" xref="S3.E7.m1.3.3.2.1.1.1.2.cmml">Z</mi><mo id="S3.E7.m1.3.3.2.1.1.1.1" xref="S3.E7.m1.3.3.2.1.1.1.1.cmml">∖</mo><mi id="S3.E7.m1.3.3.2.1.1.1.3" xref="S3.E7.m1.3.3.2.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E7.m1.3.3.2.1.1.3" stretchy="false" xref="S3.E7.m1.3.3.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E7.m1.3b"><apply id="S3.E7.m1.3.3.cmml" xref="S3.E7.m1.3.3"><plus id="S3.E7.m1.3.3.3.cmml" xref="S3.E7.m1.3.3.3"></plus><apply id="S3.E7.m1.2.2.1.2.cmml" xref="S3.E7.m1.2.2.1.1"><abs id="S3.E7.m1.2.2.1.2.1.cmml" xref="S3.E7.m1.2.2.1.1.2"></abs><apply id="S3.E7.m1.2.2.1.1.1.cmml" xref="S3.E7.m1.2.2.1.1.1"><setdiff id="S3.E7.m1.2.2.1.1.1.1.cmml" xref="S3.E7.m1.2.2.1.1.1.1"></setdiff><ci id="S3.E7.m1.2.2.1.1.1.2.cmml" xref="S3.E7.m1.2.2.1.1.1.2">𝑊</ci><ci id="S3.E7.m1.2.2.1.1.1.3.cmml" xref="S3.E7.m1.2.2.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.E7.m1.3.3.2.cmml" xref="S3.E7.m1.3.3.2"><ci id="S3.E7.m1.3.3.2.2.cmml" xref="S3.E7.m1.3.3.2.2">⋅</ci><apply id="S3.E7.m1.1.1.cmml" xref="S3.E7.m1.3.3.2.3.2"><divide id="S3.E7.m1.1.1.1.cmml" xref="S3.E7.m1.3.3.2.3.2"></divide><apply id="S3.E7.m1.1.1.2.cmml" xref="S3.E7.m1.1.1.2"><plus id="S3.E7.m1.1.1.2.1.cmml" xref="S3.E7.m1.1.1.2.1"></plus><ci id="S3.E7.m1.1.1.2.2.cmml" xref="S3.E7.m1.1.1.2.2">ℓ</ci><cn id="S3.E7.m1.1.1.2.3.cmml" type="integer" xref="S3.E7.m1.1.1.2.3">2</cn></apply><apply id="S3.E7.m1.1.1.3.cmml" xref="S3.E7.m1.1.1.3"><plus id="S3.E7.m1.1.1.3.1.cmml" xref="S3.E7.m1.1.1.3.1"></plus><ci id="S3.E7.m1.1.1.3.2.cmml" xref="S3.E7.m1.1.1.3.2">ℓ</ci><cn id="S3.E7.m1.1.1.3.3.cmml" type="integer" xref="S3.E7.m1.1.1.3.3">1</cn></apply></apply><apply id="S3.E7.m1.3.3.2.1.2.cmml" xref="S3.E7.m1.3.3.2.1.1"><abs id="S3.E7.m1.3.3.2.1.2.1.cmml" xref="S3.E7.m1.3.3.2.1.1.2"></abs><apply id="S3.E7.m1.3.3.2.1.1.1.cmml" xref="S3.E7.m1.3.3.2.1.1.1"><setdiff id="S3.E7.m1.3.3.2.1.1.1.1.cmml" xref="S3.E7.m1.3.3.2.1.1.1.1"></setdiff><ci id="S3.E7.m1.3.3.2.1.1.1.2.cmml" xref="S3.E7.m1.3.3.2.1.1.1.2">𝑍</ci><ci id="S3.E7.m1.3.3.2.1.1.1.3.cmml" xref="S3.E7.m1.3.3.2.1.1.1.3">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.3c">\displaystyle|W\setminus B|+\left(\frac{\ell+2}{\ell+1}\right)\cdot|Z\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.3d">| italic_W ∖ italic_B | + ( divide start_ARG roman_ℓ + 2 end_ARG start_ARG roman_ℓ + 1 end_ARG ) ⋅ | italic_Z ∖ italic_B |</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{|A\setminus B|}{\ell+1}+\frac{|A\setminus B|}{\ell+2}+3% \,|A\cap B|" class="ltx_Math" display="inline" id="S3.E7.m2.3"><semantics id="S3.E7.m2.3a"><mrow id="S3.E7.m2.3.3" xref="S3.E7.m2.3.3.cmml"><mi id="S3.E7.m2.3.3.3" xref="S3.E7.m2.3.3.3.cmml"></mi><mo id="S3.E7.m2.3.3.2" xref="S3.E7.m2.3.3.2.cmml">≤</mo><mrow id="S3.E7.m2.3.3.1" xref="S3.E7.m2.3.3.1.cmml"><mstyle displaystyle="true" id="S3.E7.m2.1.1" xref="S3.E7.m2.1.1.cmml"><mfrac id="S3.E7.m2.1.1a" xref="S3.E7.m2.1.1.cmml"><mrow id="S3.E7.m2.1.1.1.1" xref="S3.E7.m2.1.1.1.2.cmml"><mo id="S3.E7.m2.1.1.1.1.2" stretchy="false" xref="S3.E7.m2.1.1.1.2.1.cmml">|</mo><mrow id="S3.E7.m2.1.1.1.1.1" xref="S3.E7.m2.1.1.1.1.1.cmml"><mi id="S3.E7.m2.1.1.1.1.1.2" xref="S3.E7.m2.1.1.1.1.1.2.cmml">A</mi><mo id="S3.E7.m2.1.1.1.1.1.1" xref="S3.E7.m2.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.E7.m2.1.1.1.1.1.3" xref="S3.E7.m2.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E7.m2.1.1.1.1.3" stretchy="false" xref="S3.E7.m2.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S3.E7.m2.1.1.3" xref="S3.E7.m2.1.1.3.cmml"><mi id="S3.E7.m2.1.1.3.2" mathvariant="normal" xref="S3.E7.m2.1.1.3.2.cmml">ℓ</mi><mo id="S3.E7.m2.1.1.3.1" xref="S3.E7.m2.1.1.3.1.cmml">+</mo><mn id="S3.E7.m2.1.1.3.3" xref="S3.E7.m2.1.1.3.3.cmml">1</mn></mrow></mfrac></mstyle><mo id="S3.E7.m2.3.3.1.2" xref="S3.E7.m2.3.3.1.2.cmml">+</mo><mstyle displaystyle="true" id="S3.E7.m2.2.2" xref="S3.E7.m2.2.2.cmml"><mfrac id="S3.E7.m2.2.2a" xref="S3.E7.m2.2.2.cmml"><mrow id="S3.E7.m2.2.2.1.1" xref="S3.E7.m2.2.2.1.2.cmml"><mo id="S3.E7.m2.2.2.1.1.2" stretchy="false" xref="S3.E7.m2.2.2.1.2.1.cmml">|</mo><mrow id="S3.E7.m2.2.2.1.1.1" xref="S3.E7.m2.2.2.1.1.1.cmml"><mi id="S3.E7.m2.2.2.1.1.1.2" xref="S3.E7.m2.2.2.1.1.1.2.cmml">A</mi><mo id="S3.E7.m2.2.2.1.1.1.1" xref="S3.E7.m2.2.2.1.1.1.1.cmml">∖</mo><mi id="S3.E7.m2.2.2.1.1.1.3" xref="S3.E7.m2.2.2.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E7.m2.2.2.1.1.3" stretchy="false" xref="S3.E7.m2.2.2.1.2.1.cmml">|</mo></mrow><mrow id="S3.E7.m2.2.2.3" xref="S3.E7.m2.2.2.3.cmml"><mi id="S3.E7.m2.2.2.3.2" mathvariant="normal" xref="S3.E7.m2.2.2.3.2.cmml">ℓ</mi><mo id="S3.E7.m2.2.2.3.1" xref="S3.E7.m2.2.2.3.1.cmml">+</mo><mn id="S3.E7.m2.2.2.3.3" xref="S3.E7.m2.2.2.3.3.cmml">2</mn></mrow></mfrac></mstyle><mo id="S3.E7.m2.3.3.1.2a" xref="S3.E7.m2.3.3.1.2.cmml">+</mo><mrow id="S3.E7.m2.3.3.1.1" xref="S3.E7.m2.3.3.1.1.cmml"><mn id="S3.E7.m2.3.3.1.1.3" xref="S3.E7.m2.3.3.1.1.3.cmml">3</mn><mo id="S3.E7.m2.3.3.1.1.2" lspace="0.170em" xref="S3.E7.m2.3.3.1.1.2.cmml">⁢</mo><mrow id="S3.E7.m2.3.3.1.1.1.1" xref="S3.E7.m2.3.3.1.1.1.2.cmml"><mo id="S3.E7.m2.3.3.1.1.1.1.2" stretchy="false" xref="S3.E7.m2.3.3.1.1.1.2.1.cmml">|</mo><mrow id="S3.E7.m2.3.3.1.1.1.1.1" xref="S3.E7.m2.3.3.1.1.1.1.1.cmml"><mi id="S3.E7.m2.3.3.1.1.1.1.1.2" xref="S3.E7.m2.3.3.1.1.1.1.1.2.cmml">A</mi><mo id="S3.E7.m2.3.3.1.1.1.1.1.1" xref="S3.E7.m2.3.3.1.1.1.1.1.1.cmml">∩</mo><mi id="S3.E7.m2.3.3.1.1.1.1.1.3" xref="S3.E7.m2.3.3.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.E7.m2.3.3.1.1.1.1.3" stretchy="false" xref="S3.E7.m2.3.3.1.1.1.2.1.cmml">|</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E7.m2.3b"><apply id="S3.E7.m2.3.3.cmml" xref="S3.E7.m2.3.3"><leq id="S3.E7.m2.3.3.2.cmml" xref="S3.E7.m2.3.3.2"></leq><csymbol cd="latexml" id="S3.E7.m2.3.3.3.cmml" xref="S3.E7.m2.3.3.3">absent</csymbol><apply id="S3.E7.m2.3.3.1.cmml" xref="S3.E7.m2.3.3.1"><plus id="S3.E7.m2.3.3.1.2.cmml" xref="S3.E7.m2.3.3.1.2"></plus><apply id="S3.E7.m2.1.1.cmml" xref="S3.E7.m2.1.1"><divide id="S3.E7.m2.1.1.2.cmml" xref="S3.E7.m2.1.1"></divide><apply id="S3.E7.m2.1.1.1.2.cmml" xref="S3.E7.m2.1.1.1.1"><abs id="S3.E7.m2.1.1.1.2.1.cmml" xref="S3.E7.m2.1.1.1.1.2"></abs><apply id="S3.E7.m2.1.1.1.1.1.cmml" xref="S3.E7.m2.1.1.1.1.1"><setdiff id="S3.E7.m2.1.1.1.1.1.1.cmml" xref="S3.E7.m2.1.1.1.1.1.1"></setdiff><ci id="S3.E7.m2.1.1.1.1.1.2.cmml" xref="S3.E7.m2.1.1.1.1.1.2">𝐴</ci><ci id="S3.E7.m2.1.1.1.1.1.3.cmml" xref="S3.E7.m2.1.1.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.E7.m2.1.1.3.cmml" xref="S3.E7.m2.1.1.3"><plus id="S3.E7.m2.1.1.3.1.cmml" xref="S3.E7.m2.1.1.3.1"></plus><ci id="S3.E7.m2.1.1.3.2.cmml" xref="S3.E7.m2.1.1.3.2">ℓ</ci><cn id="S3.E7.m2.1.1.3.3.cmml" type="integer" xref="S3.E7.m2.1.1.3.3">1</cn></apply></apply><apply id="S3.E7.m2.2.2.cmml" xref="S3.E7.m2.2.2"><divide id="S3.E7.m2.2.2.2.cmml" xref="S3.E7.m2.2.2"></divide><apply id="S3.E7.m2.2.2.1.2.cmml" xref="S3.E7.m2.2.2.1.1"><abs id="S3.E7.m2.2.2.1.2.1.cmml" xref="S3.E7.m2.2.2.1.1.2"></abs><apply id="S3.E7.m2.2.2.1.1.1.cmml" xref="S3.E7.m2.2.2.1.1.1"><setdiff id="S3.E7.m2.2.2.1.1.1.1.cmml" xref="S3.E7.m2.2.2.1.1.1.1"></setdiff><ci id="S3.E7.m2.2.2.1.1.1.2.cmml" xref="S3.E7.m2.2.2.1.1.1.2">𝐴</ci><ci id="S3.E7.m2.2.2.1.1.1.3.cmml" xref="S3.E7.m2.2.2.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.E7.m2.2.2.3.cmml" xref="S3.E7.m2.2.2.3"><plus id="S3.E7.m2.2.2.3.1.cmml" xref="S3.E7.m2.2.2.3.1"></plus><ci id="S3.E7.m2.2.2.3.2.cmml" xref="S3.E7.m2.2.2.3.2">ℓ</ci><cn id="S3.E7.m2.2.2.3.3.cmml" type="integer" xref="S3.E7.m2.2.2.3.3">2</cn></apply></apply><apply id="S3.E7.m2.3.3.1.1.cmml" xref="S3.E7.m2.3.3.1.1"><times id="S3.E7.m2.3.3.1.1.2.cmml" xref="S3.E7.m2.3.3.1.1.2"></times><cn id="S3.E7.m2.3.3.1.1.3.cmml" type="integer" xref="S3.E7.m2.3.3.1.1.3">3</cn><apply id="S3.E7.m2.3.3.1.1.1.2.cmml" xref="S3.E7.m2.3.3.1.1.1.1"><abs id="S3.E7.m2.3.3.1.1.1.2.1.cmml" xref="S3.E7.m2.3.3.1.1.1.1.2"></abs><apply id="S3.E7.m2.3.3.1.1.1.1.1.cmml" xref="S3.E7.m2.3.3.1.1.1.1.1"><intersect id="S3.E7.m2.3.3.1.1.1.1.1.1.cmml" xref="S3.E7.m2.3.3.1.1.1.1.1.1"></intersect><ci id="S3.E7.m2.3.3.1.1.1.1.1.2.cmml" xref="S3.E7.m2.3.3.1.1.1.1.1.2">𝐴</ci><ci id="S3.E7.m2.3.3.1.1.1.1.1.3.cmml" xref="S3.E7.m2.3.3.1.1.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m2.3c">\displaystyle\leq\frac{|A\setminus B|}{\ell+1}+\frac{|A\setminus B|}{\ell+2}+3% \,|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m2.3d">≤ divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG + 3 | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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> <p class="ltx_p" id="S3.9.p5.21">We can now upper bound <math alttext="|W\setminus B|+|Z\setminus B|" class="ltx_Math" display="inline" id="S3.9.p5.20.m1.2"><semantics id="S3.9.p5.20.m1.2a"><mrow id="S3.9.p5.20.m1.2.2" xref="S3.9.p5.20.m1.2.2.cmml"><mrow id="S3.9.p5.20.m1.1.1.1.1" xref="S3.9.p5.20.m1.1.1.1.2.cmml"><mo id="S3.9.p5.20.m1.1.1.1.1.2" stretchy="false" xref="S3.9.p5.20.m1.1.1.1.2.1.cmml">|</mo><mrow id="S3.9.p5.20.m1.1.1.1.1.1" xref="S3.9.p5.20.m1.1.1.1.1.1.cmml"><mi id="S3.9.p5.20.m1.1.1.1.1.1.2" xref="S3.9.p5.20.m1.1.1.1.1.1.2.cmml">W</mi><mo id="S3.9.p5.20.m1.1.1.1.1.1.1" xref="S3.9.p5.20.m1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.9.p5.20.m1.1.1.1.1.1.3" xref="S3.9.p5.20.m1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.9.p5.20.m1.1.1.1.1.3" stretchy="false" xref="S3.9.p5.20.m1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.9.p5.20.m1.2.2.3" xref="S3.9.p5.20.m1.2.2.3.cmml">+</mo><mrow id="S3.9.p5.20.m1.2.2.2.1" xref="S3.9.p5.20.m1.2.2.2.2.cmml"><mo id="S3.9.p5.20.m1.2.2.2.1.2" stretchy="false" xref="S3.9.p5.20.m1.2.2.2.2.1.cmml">|</mo><mrow id="S3.9.p5.20.m1.2.2.2.1.1" xref="S3.9.p5.20.m1.2.2.2.1.1.cmml"><mi id="S3.9.p5.20.m1.2.2.2.1.1.2" xref="S3.9.p5.20.m1.2.2.2.1.1.2.cmml">Z</mi><mo id="S3.9.p5.20.m1.2.2.2.1.1.1" xref="S3.9.p5.20.m1.2.2.2.1.1.1.cmml">∖</mo><mi id="S3.9.p5.20.m1.2.2.2.1.1.3" xref="S3.9.p5.20.m1.2.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.9.p5.20.m1.2.2.2.1.3" stretchy="false" xref="S3.9.p5.20.m1.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.20.m1.2b"><apply id="S3.9.p5.20.m1.2.2.cmml" xref="S3.9.p5.20.m1.2.2"><plus id="S3.9.p5.20.m1.2.2.3.cmml" xref="S3.9.p5.20.m1.2.2.3"></plus><apply id="S3.9.p5.20.m1.1.1.1.2.cmml" xref="S3.9.p5.20.m1.1.1.1.1"><abs id="S3.9.p5.20.m1.1.1.1.2.1.cmml" xref="S3.9.p5.20.m1.1.1.1.1.2"></abs><apply id="S3.9.p5.20.m1.1.1.1.1.1.cmml" xref="S3.9.p5.20.m1.1.1.1.1.1"><setdiff id="S3.9.p5.20.m1.1.1.1.1.1.1.cmml" xref="S3.9.p5.20.m1.1.1.1.1.1.1"></setdiff><ci id="S3.9.p5.20.m1.1.1.1.1.1.2.cmml" xref="S3.9.p5.20.m1.1.1.1.1.1.2">𝑊</ci><ci id="S3.9.p5.20.m1.1.1.1.1.1.3.cmml" xref="S3.9.p5.20.m1.1.1.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.9.p5.20.m1.2.2.2.2.cmml" xref="S3.9.p5.20.m1.2.2.2.1"><abs id="S3.9.p5.20.m1.2.2.2.2.1.cmml" xref="S3.9.p5.20.m1.2.2.2.1.2"></abs><apply id="S3.9.p5.20.m1.2.2.2.1.1.cmml" xref="S3.9.p5.20.m1.2.2.2.1.1"><setdiff id="S3.9.p5.20.m1.2.2.2.1.1.1.cmml" xref="S3.9.p5.20.m1.2.2.2.1.1.1"></setdiff><ci id="S3.9.p5.20.m1.2.2.2.1.1.2.cmml" xref="S3.9.p5.20.m1.2.2.2.1.1.2">𝑍</ci><ci id="S3.9.p5.20.m1.2.2.2.1.1.3.cmml" xref="S3.9.p5.20.m1.2.2.2.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.20.m1.2c">|W\setminus B|+|Z\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.20.m1.2d">| italic_W ∖ italic_B | + | italic_Z ∖ italic_B |</annotation></semantics></math> by maximizing <math alttext="x_{0}+x_{1}" class="ltx_Math" display="inline" id="S3.9.p5.21.m2.1"><semantics id="S3.9.p5.21.m2.1a"><mrow id="S3.9.p5.21.m2.1.1" xref="S3.9.p5.21.m2.1.1.cmml"><msub id="S3.9.p5.21.m2.1.1.2" xref="S3.9.p5.21.m2.1.1.2.cmml"><mi id="S3.9.p5.21.m2.1.1.2.2" xref="S3.9.p5.21.m2.1.1.2.2.cmml">x</mi><mn id="S3.9.p5.21.m2.1.1.2.3" xref="S3.9.p5.21.m2.1.1.2.3.cmml">0</mn></msub><mo id="S3.9.p5.21.m2.1.1.1" xref="S3.9.p5.21.m2.1.1.1.cmml">+</mo><msub id="S3.9.p5.21.m2.1.1.3" xref="S3.9.p5.21.m2.1.1.3.cmml"><mi id="S3.9.p5.21.m2.1.1.3.2" xref="S3.9.p5.21.m2.1.1.3.2.cmml">x</mi><mn id="S3.9.p5.21.m2.1.1.3.3" xref="S3.9.p5.21.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.21.m2.1b"><apply id="S3.9.p5.21.m2.1.1.cmml" xref="S3.9.p5.21.m2.1.1"><plus id="S3.9.p5.21.m2.1.1.1.cmml" xref="S3.9.p5.21.m2.1.1.1"></plus><apply id="S3.9.p5.21.m2.1.1.2.cmml" xref="S3.9.p5.21.m2.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.21.m2.1.1.2.1.cmml" xref="S3.9.p5.21.m2.1.1.2">subscript</csymbol><ci id="S3.9.p5.21.m2.1.1.2.2.cmml" xref="S3.9.p5.21.m2.1.1.2.2">𝑥</ci><cn id="S3.9.p5.21.m2.1.1.2.3.cmml" type="integer" xref="S3.9.p5.21.m2.1.1.2.3">0</cn></apply><apply id="S3.9.p5.21.m2.1.1.3.cmml" xref="S3.9.p5.21.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.21.m2.1.1.3.1.cmml" xref="S3.9.p5.21.m2.1.1.3">subscript</csymbol><ci id="S3.9.p5.21.m2.1.1.3.2.cmml" xref="S3.9.p5.21.m2.1.1.3.2">𝑥</ci><cn id="S3.9.p5.21.m2.1.1.3.3.cmml" type="integer" xref="S3.9.p5.21.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.21.m2.1c">x_{0}+x_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.21.m2.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> subject to</p> <table class="ltx_equation ltx_eqn_table" id="S3.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="x_{0}+\left(\frac{\ell+2}{\ell+1}\right)\cdot x_{1}\leq R\enspace," class="ltx_Math" display="block" id="S3.E8.m1.2"><semantics id="S3.E8.m1.2a"><mrow id="S3.E8.m1.2.2.1" xref="S3.E8.m1.2.2.1.1.cmml"><mrow id="S3.E8.m1.2.2.1.1" xref="S3.E8.m1.2.2.1.1.cmml"><mrow id="S3.E8.m1.2.2.1.1.2" xref="S3.E8.m1.2.2.1.1.2.cmml"><msub id="S3.E8.m1.2.2.1.1.2.2" xref="S3.E8.m1.2.2.1.1.2.2.cmml"><mi id="S3.E8.m1.2.2.1.1.2.2.2" xref="S3.E8.m1.2.2.1.1.2.2.2.cmml">x</mi><mn id="S3.E8.m1.2.2.1.1.2.2.3" xref="S3.E8.m1.2.2.1.1.2.2.3.cmml">0</mn></msub><mo id="S3.E8.m1.2.2.1.1.2.1" xref="S3.E8.m1.2.2.1.1.2.1.cmml">+</mo><mrow id="S3.E8.m1.2.2.1.1.2.3" xref="S3.E8.m1.2.2.1.1.2.3.cmml"><mrow id="S3.E8.m1.2.2.1.1.2.3.2.2" xref="S3.E8.m1.1.1.cmml"><mo id="S3.E8.m1.2.2.1.1.2.3.2.2.1" xref="S3.E8.m1.1.1.cmml">(</mo><mfrac id="S3.E8.m1.1.1" xref="S3.E8.m1.1.1.cmml"><mrow id="S3.E8.m1.1.1.2" xref="S3.E8.m1.1.1.2.cmml"><mi id="S3.E8.m1.1.1.2.2" mathvariant="normal" xref="S3.E8.m1.1.1.2.2.cmml">ℓ</mi><mo id="S3.E8.m1.1.1.2.1" xref="S3.E8.m1.1.1.2.1.cmml">+</mo><mn id="S3.E8.m1.1.1.2.3" xref="S3.E8.m1.1.1.2.3.cmml">2</mn></mrow><mrow id="S3.E8.m1.1.1.3" xref="S3.E8.m1.1.1.3.cmml"><mi id="S3.E8.m1.1.1.3.2" mathvariant="normal" xref="S3.E8.m1.1.1.3.2.cmml">ℓ</mi><mo id="S3.E8.m1.1.1.3.1" xref="S3.E8.m1.1.1.3.1.cmml">+</mo><mn id="S3.E8.m1.1.1.3.3" xref="S3.E8.m1.1.1.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.E8.m1.2.2.1.1.2.3.2.2.2" rspace="0.055em" xref="S3.E8.m1.1.1.cmml">)</mo></mrow><mo id="S3.E8.m1.2.2.1.1.2.3.1" rspace="0.222em" xref="S3.E8.m1.2.2.1.1.2.3.1.cmml">⋅</mo><msub id="S3.E8.m1.2.2.1.1.2.3.3" xref="S3.E8.m1.2.2.1.1.2.3.3.cmml"><mi id="S3.E8.m1.2.2.1.1.2.3.3.2" xref="S3.E8.m1.2.2.1.1.2.3.3.2.cmml">x</mi><mn id="S3.E8.m1.2.2.1.1.2.3.3.3" xref="S3.E8.m1.2.2.1.1.2.3.3.3.cmml">1</mn></msub></mrow></mrow><mo id="S3.E8.m1.2.2.1.1.1" xref="S3.E8.m1.2.2.1.1.1.cmml">≤</mo><mi id="S3.E8.m1.2.2.1.1.3" xref="S3.E8.m1.2.2.1.1.3.cmml">R</mi></mrow><mo id="S3.E8.m1.2.2.1.2" lspace="0.500em" xref="S3.E8.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E8.m1.2b"><apply id="S3.E8.m1.2.2.1.1.cmml" xref="S3.E8.m1.2.2.1"><leq id="S3.E8.m1.2.2.1.1.1.cmml" xref="S3.E8.m1.2.2.1.1.1"></leq><apply id="S3.E8.m1.2.2.1.1.2.cmml" xref="S3.E8.m1.2.2.1.1.2"><plus id="S3.E8.m1.2.2.1.1.2.1.cmml" xref="S3.E8.m1.2.2.1.1.2.1"></plus><apply id="S3.E8.m1.2.2.1.1.2.2.cmml" xref="S3.E8.m1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.1.1.2.2.1.cmml" xref="S3.E8.m1.2.2.1.1.2.2">subscript</csymbol><ci id="S3.E8.m1.2.2.1.1.2.2.2.cmml" xref="S3.E8.m1.2.2.1.1.2.2.2">𝑥</ci><cn id="S3.E8.m1.2.2.1.1.2.2.3.cmml" type="integer" xref="S3.E8.m1.2.2.1.1.2.2.3">0</cn></apply><apply id="S3.E8.m1.2.2.1.1.2.3.cmml" xref="S3.E8.m1.2.2.1.1.2.3"><ci id="S3.E8.m1.2.2.1.1.2.3.1.cmml" xref="S3.E8.m1.2.2.1.1.2.3.1">⋅</ci><apply id="S3.E8.m1.1.1.cmml" xref="S3.E8.m1.2.2.1.1.2.3.2.2"><divide id="S3.E8.m1.1.1.1.cmml" xref="S3.E8.m1.2.2.1.1.2.3.2.2"></divide><apply id="S3.E8.m1.1.1.2.cmml" xref="S3.E8.m1.1.1.2"><plus id="S3.E8.m1.1.1.2.1.cmml" xref="S3.E8.m1.1.1.2.1"></plus><ci id="S3.E8.m1.1.1.2.2.cmml" xref="S3.E8.m1.1.1.2.2">ℓ</ci><cn id="S3.E8.m1.1.1.2.3.cmml" type="integer" xref="S3.E8.m1.1.1.2.3">2</cn></apply><apply id="S3.E8.m1.1.1.3.cmml" xref="S3.E8.m1.1.1.3"><plus id="S3.E8.m1.1.1.3.1.cmml" xref="S3.E8.m1.1.1.3.1"></plus><ci id="S3.E8.m1.1.1.3.2.cmml" xref="S3.E8.m1.1.1.3.2">ℓ</ci><cn id="S3.E8.m1.1.1.3.3.cmml" type="integer" xref="S3.E8.m1.1.1.3.3">1</cn></apply></apply><apply id="S3.E8.m1.2.2.1.1.2.3.3.cmml" xref="S3.E8.m1.2.2.1.1.2.3.3"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.1.1.2.3.3.1.cmml" xref="S3.E8.m1.2.2.1.1.2.3.3">subscript</csymbol><ci id="S3.E8.m1.2.2.1.1.2.3.3.2.cmml" xref="S3.E8.m1.2.2.1.1.2.3.3.2">𝑥</ci><cn id="S3.E8.m1.2.2.1.1.2.3.3.3.cmml" type="integer" xref="S3.E8.m1.2.2.1.1.2.3.3.3">1</cn></apply></apply></apply><ci id="S3.E8.m1.2.2.1.1.3.cmml" xref="S3.E8.m1.2.2.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m1.2c">x_{0}+\left(\frac{\ell+2}{\ell+1}\right)\cdot x_{1}\leq R\enspace,</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.2d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + ( divide start_ARG roman_ℓ + 2 end_ARG start_ARG roman_ℓ + 1 end_ARG ) ⋅ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_R ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.9.p5.24">where <math alttext="R" class="ltx_Math" display="inline" id="S3.9.p5.22.m1.1"><semantics id="S3.9.p5.22.m1.1a"><mi id="S3.9.p5.22.m1.1.1" xref="S3.9.p5.22.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.9.p5.22.m1.1b"><ci id="S3.9.p5.22.m1.1.1.cmml" xref="S3.9.p5.22.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.22.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.22.m1.1d">italic_R</annotation></semantics></math> denotes the expression in (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E7" title="Equation 7 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">7</span></a>). For a fixed <math alttext="x_{0}=x^{\star}_{0}" class="ltx_Math" display="inline" id="S3.9.p5.23.m2.1"><semantics id="S3.9.p5.23.m2.1a"><mrow id="S3.9.p5.23.m2.1.1" xref="S3.9.p5.23.m2.1.1.cmml"><msub id="S3.9.p5.23.m2.1.1.2" xref="S3.9.p5.23.m2.1.1.2.cmml"><mi id="S3.9.p5.23.m2.1.1.2.2" xref="S3.9.p5.23.m2.1.1.2.2.cmml">x</mi><mn id="S3.9.p5.23.m2.1.1.2.3" xref="S3.9.p5.23.m2.1.1.2.3.cmml">0</mn></msub><mo id="S3.9.p5.23.m2.1.1.1" xref="S3.9.p5.23.m2.1.1.1.cmml">=</mo><msubsup id="S3.9.p5.23.m2.1.1.3" xref="S3.9.p5.23.m2.1.1.3.cmml"><mi id="S3.9.p5.23.m2.1.1.3.2.2" xref="S3.9.p5.23.m2.1.1.3.2.2.cmml">x</mi><mn id="S3.9.p5.23.m2.1.1.3.3" xref="S3.9.p5.23.m2.1.1.3.3.cmml">0</mn><mo id="S3.9.p5.23.m2.1.1.3.2.3" xref="S3.9.p5.23.m2.1.1.3.2.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.23.m2.1b"><apply id="S3.9.p5.23.m2.1.1.cmml" xref="S3.9.p5.23.m2.1.1"><eq id="S3.9.p5.23.m2.1.1.1.cmml" xref="S3.9.p5.23.m2.1.1.1"></eq><apply id="S3.9.p5.23.m2.1.1.2.cmml" xref="S3.9.p5.23.m2.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.23.m2.1.1.2.1.cmml" xref="S3.9.p5.23.m2.1.1.2">subscript</csymbol><ci id="S3.9.p5.23.m2.1.1.2.2.cmml" xref="S3.9.p5.23.m2.1.1.2.2">𝑥</ci><cn id="S3.9.p5.23.m2.1.1.2.3.cmml" type="integer" xref="S3.9.p5.23.m2.1.1.2.3">0</cn></apply><apply id="S3.9.p5.23.m2.1.1.3.cmml" xref="S3.9.p5.23.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.23.m2.1.1.3.1.cmml" xref="S3.9.p5.23.m2.1.1.3">subscript</csymbol><apply id="S3.9.p5.23.m2.1.1.3.2.cmml" xref="S3.9.p5.23.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.23.m2.1.1.3.2.1.cmml" xref="S3.9.p5.23.m2.1.1.3">superscript</csymbol><ci id="S3.9.p5.23.m2.1.1.3.2.2.cmml" xref="S3.9.p5.23.m2.1.1.3.2.2">𝑥</ci><ci id="S3.9.p5.23.m2.1.1.3.2.3.cmml" xref="S3.9.p5.23.m2.1.1.3.2.3">⋆</ci></apply><cn id="S3.9.p5.23.m2.1.1.3.3.cmml" type="integer" xref="S3.9.p5.23.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.23.m2.1c">x_{0}=x^{\star}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.23.m2.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> the maximum value of <math alttext="x_{1}" class="ltx_Math" display="inline" id="S3.9.p5.24.m3.1"><semantics id="S3.9.p5.24.m3.1a"><msub id="S3.9.p5.24.m3.1.1" xref="S3.9.p5.24.m3.1.1.cmml"><mi id="S3.9.p5.24.m3.1.1.2" xref="S3.9.p5.24.m3.1.1.2.cmml">x</mi><mn id="S3.9.p5.24.m3.1.1.3" xref="S3.9.p5.24.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.9.p5.24.m3.1b"><apply id="S3.9.p5.24.m3.1.1.cmml" xref="S3.9.p5.24.m3.1.1"><csymbol cd="ambiguous" id="S3.9.p5.24.m3.1.1.1.cmml" xref="S3.9.p5.24.m3.1.1">subscript</csymbol><ci id="S3.9.p5.24.m3.1.1.2.cmml" xref="S3.9.p5.24.m3.1.1.2">𝑥</ci><cn id="S3.9.p5.24.m3.1.1.3.cmml" type="integer" xref="S3.9.p5.24.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.24.m3.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.24.m3.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> that satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E8" title="Equation 8 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">8</span></a>) is</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="x_{1}=x_{1}^{\star}:=\left(\frac{\ell+1}{\ell+2}\right)\cdot(R-x^{\star}_{0})" class="ltx_Math" display="block" id="S3.Ex4.m1.2"><semantics id="S3.Ex4.m1.2a"><mrow id="S3.Ex4.m1.2.2" xref="S3.Ex4.m1.2.2.cmml"><msub id="S3.Ex4.m1.2.2.3" xref="S3.Ex4.m1.2.2.3.cmml"><mi id="S3.Ex4.m1.2.2.3.2" xref="S3.Ex4.m1.2.2.3.2.cmml">x</mi><mn id="S3.Ex4.m1.2.2.3.3" xref="S3.Ex4.m1.2.2.3.3.cmml">1</mn></msub><mo id="S3.Ex4.m1.2.2.4" xref="S3.Ex4.m1.2.2.4.cmml">=</mo><msubsup id="S3.Ex4.m1.2.2.5" xref="S3.Ex4.m1.2.2.5.cmml"><mi id="S3.Ex4.m1.2.2.5.2.2" xref="S3.Ex4.m1.2.2.5.2.2.cmml">x</mi><mn id="S3.Ex4.m1.2.2.5.2.3" xref="S3.Ex4.m1.2.2.5.2.3.cmml">1</mn><mo id="S3.Ex4.m1.2.2.5.3" xref="S3.Ex4.m1.2.2.5.3.cmml">⋆</mo></msubsup><mo id="S3.Ex4.m1.2.2.6" lspace="0.278em" rspace="0.278em" xref="S3.Ex4.m1.2.2.6.cmml">:=</mo><mrow id="S3.Ex4.m1.2.2.1" xref="S3.Ex4.m1.2.2.1.cmml"><mrow id="S3.Ex4.m1.2.2.1.3.2" xref="S3.Ex4.m1.1.1.cmml"><mo id="S3.Ex4.m1.2.2.1.3.2.1" xref="S3.Ex4.m1.1.1.cmml">(</mo><mfrac id="S3.Ex4.m1.1.1" xref="S3.Ex4.m1.1.1.cmml"><mrow id="S3.Ex4.m1.1.1.2" xref="S3.Ex4.m1.1.1.2.cmml"><mi id="S3.Ex4.m1.1.1.2.2" mathvariant="normal" xref="S3.Ex4.m1.1.1.2.2.cmml">ℓ</mi><mo id="S3.Ex4.m1.1.1.2.1" xref="S3.Ex4.m1.1.1.2.1.cmml">+</mo><mn id="S3.Ex4.m1.1.1.2.3" xref="S3.Ex4.m1.1.1.2.3.cmml">1</mn></mrow><mrow id="S3.Ex4.m1.1.1.3" xref="S3.Ex4.m1.1.1.3.cmml"><mi id="S3.Ex4.m1.1.1.3.2" mathvariant="normal" xref="S3.Ex4.m1.1.1.3.2.cmml">ℓ</mi><mo id="S3.Ex4.m1.1.1.3.1" xref="S3.Ex4.m1.1.1.3.1.cmml">+</mo><mn id="S3.Ex4.m1.1.1.3.3" xref="S3.Ex4.m1.1.1.3.3.cmml">2</mn></mrow></mfrac><mo id="S3.Ex4.m1.2.2.1.3.2.2" rspace="0.055em" xref="S3.Ex4.m1.1.1.cmml">)</mo></mrow><mo id="S3.Ex4.m1.2.2.1.2" rspace="0.222em" xref="S3.Ex4.m1.2.2.1.2.cmml">⋅</mo><mrow id="S3.Ex4.m1.2.2.1.1.1" xref="S3.Ex4.m1.2.2.1.1.1.1.cmml"><mo id="S3.Ex4.m1.2.2.1.1.1.2" stretchy="false" xref="S3.Ex4.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.Ex4.m1.2.2.1.1.1.1" xref="S3.Ex4.m1.2.2.1.1.1.1.cmml"><mi id="S3.Ex4.m1.2.2.1.1.1.1.2" xref="S3.Ex4.m1.2.2.1.1.1.1.2.cmml">R</mi><mo id="S3.Ex4.m1.2.2.1.1.1.1.1" xref="S3.Ex4.m1.2.2.1.1.1.1.1.cmml">−</mo><msubsup id="S3.Ex4.m1.2.2.1.1.1.1.3" xref="S3.Ex4.m1.2.2.1.1.1.1.3.cmml"><mi id="S3.Ex4.m1.2.2.1.1.1.1.3.2.2" xref="S3.Ex4.m1.2.2.1.1.1.1.3.2.2.cmml">x</mi><mn id="S3.Ex4.m1.2.2.1.1.1.1.3.3" xref="S3.Ex4.m1.2.2.1.1.1.1.3.3.cmml">0</mn><mo id="S3.Ex4.m1.2.2.1.1.1.1.3.2.3" xref="S3.Ex4.m1.2.2.1.1.1.1.3.2.3.cmml">⋆</mo></msubsup></mrow><mo id="S3.Ex4.m1.2.2.1.1.1.3" stretchy="false" xref="S3.Ex4.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex4.m1.2b"><apply id="S3.Ex4.m1.2.2.cmml" xref="S3.Ex4.m1.2.2"><and id="S3.Ex4.m1.2.2a.cmml" xref="S3.Ex4.m1.2.2"></and><apply id="S3.Ex4.m1.2.2b.cmml" xref="S3.Ex4.m1.2.2"><eq id="S3.Ex4.m1.2.2.4.cmml" xref="S3.Ex4.m1.2.2.4"></eq><apply id="S3.Ex4.m1.2.2.3.cmml" xref="S3.Ex4.m1.2.2.3"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.3.1.cmml" xref="S3.Ex4.m1.2.2.3">subscript</csymbol><ci id="S3.Ex4.m1.2.2.3.2.cmml" xref="S3.Ex4.m1.2.2.3.2">𝑥</ci><cn id="S3.Ex4.m1.2.2.3.3.cmml" type="integer" xref="S3.Ex4.m1.2.2.3.3">1</cn></apply><apply id="S3.Ex4.m1.2.2.5.cmml" 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id="S3.Ex4.m1.2.2.1.1.1.1.3.2.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.1.1.3.2.1.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.3">superscript</csymbol><ci id="S3.Ex4.m1.2.2.1.1.1.1.3.2.2.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.3.2.2">𝑥</ci><ci id="S3.Ex4.m1.2.2.1.1.1.1.3.2.3.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.3.2.3">⋆</ci></apply><cn id="S3.Ex4.m1.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S3.Ex4.m1.2.2.1.1.1.1.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex4.m1.2c">x_{1}=x_{1}^{\star}:=\left(\frac{\ell+1}{\ell+2}\right)\cdot(R-x^{\star}_{0})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex4.m1.2d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT := ( divide start_ARG roman_ℓ + 1 end_ARG start_ARG roman_ℓ + 2 end_ARG ) ⋅ ( italic_R - italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.9.p5.29">in which case</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="x_{0}+x_{1}=x^{\star}_{0}+x^{\star}_{1}=\left(\frac{\ell+1}{\ell+2}\right)% \cdot R+\frac{x^{\star}_{0}}{\ell+2}" class="ltx_Math" display="block" id="S3.Ex5.m1.1"><semantics id="S3.Ex5.m1.1a"><mrow id="S3.Ex5.m1.1.2" xref="S3.Ex5.m1.1.2.cmml"><mrow id="S3.Ex5.m1.1.2.2" xref="S3.Ex5.m1.1.2.2.cmml"><msub id="S3.Ex5.m1.1.2.2.2" xref="S3.Ex5.m1.1.2.2.2.cmml"><mi id="S3.Ex5.m1.1.2.2.2.2" xref="S3.Ex5.m1.1.2.2.2.2.cmml">x</mi><mn id="S3.Ex5.m1.1.2.2.2.3" xref="S3.Ex5.m1.1.2.2.2.3.cmml">0</mn></msub><mo id="S3.Ex5.m1.1.2.2.1" xref="S3.Ex5.m1.1.2.2.1.cmml">+</mo><msub id="S3.Ex5.m1.1.2.2.3" xref="S3.Ex5.m1.1.2.2.3.cmml"><mi id="S3.Ex5.m1.1.2.2.3.2" xref="S3.Ex5.m1.1.2.2.3.2.cmml">x</mi><mn id="S3.Ex5.m1.1.2.2.3.3" xref="S3.Ex5.m1.1.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S3.Ex5.m1.1.2.3" xref="S3.Ex5.m1.1.2.3.cmml">=</mo><mrow id="S3.Ex5.m1.1.2.4" xref="S3.Ex5.m1.1.2.4.cmml"><msubsup id="S3.Ex5.m1.1.2.4.2" xref="S3.Ex5.m1.1.2.4.2.cmml"><mi id="S3.Ex5.m1.1.2.4.2.2.2" xref="S3.Ex5.m1.1.2.4.2.2.2.cmml">x</mi><mn id="S3.Ex5.m1.1.2.4.2.3" xref="S3.Ex5.m1.1.2.4.2.3.cmml">0</mn><mo id="S3.Ex5.m1.1.2.4.2.2.3" xref="S3.Ex5.m1.1.2.4.2.2.3.cmml">⋆</mo></msubsup><mo id="S3.Ex5.m1.1.2.4.1" xref="S3.Ex5.m1.1.2.4.1.cmml">+</mo><msubsup id="S3.Ex5.m1.1.2.4.3" xref="S3.Ex5.m1.1.2.4.3.cmml"><mi id="S3.Ex5.m1.1.2.4.3.2.2" xref="S3.Ex5.m1.1.2.4.3.2.2.cmml">x</mi><mn id="S3.Ex5.m1.1.2.4.3.3" xref="S3.Ex5.m1.1.2.4.3.3.cmml">1</mn><mo id="S3.Ex5.m1.1.2.4.3.2.3" xref="S3.Ex5.m1.1.2.4.3.2.3.cmml">⋆</mo></msubsup></mrow><mo id="S3.Ex5.m1.1.2.5" xref="S3.Ex5.m1.1.2.5.cmml">=</mo><mrow id="S3.Ex5.m1.1.2.6" xref="S3.Ex5.m1.1.2.6.cmml"><mrow id="S3.Ex5.m1.1.2.6.2" xref="S3.Ex5.m1.1.2.6.2.cmml"><mrow id="S3.Ex5.m1.1.2.6.2.2.2" xref="S3.Ex5.m1.1.1.cmml"><mo id="S3.Ex5.m1.1.2.6.2.2.2.1" xref="S3.Ex5.m1.1.1.cmml">(</mo><mfrac id="S3.Ex5.m1.1.1" xref="S3.Ex5.m1.1.1.cmml"><mrow id="S3.Ex5.m1.1.1.2" xref="S3.Ex5.m1.1.1.2.cmml"><mi id="S3.Ex5.m1.1.1.2.2" mathvariant="normal" xref="S3.Ex5.m1.1.1.2.2.cmml">ℓ</mi><mo id="S3.Ex5.m1.1.1.2.1" xref="S3.Ex5.m1.1.1.2.1.cmml">+</mo><mn id="S3.Ex5.m1.1.1.2.3" xref="S3.Ex5.m1.1.1.2.3.cmml">1</mn></mrow><mrow id="S3.Ex5.m1.1.1.3" xref="S3.Ex5.m1.1.1.3.cmml"><mi id="S3.Ex5.m1.1.1.3.2" mathvariant="normal" xref="S3.Ex5.m1.1.1.3.2.cmml">ℓ</mi><mo id="S3.Ex5.m1.1.1.3.1" xref="S3.Ex5.m1.1.1.3.1.cmml">+</mo><mn id="S3.Ex5.m1.1.1.3.3" xref="S3.Ex5.m1.1.1.3.3.cmml">2</mn></mrow></mfrac><mo id="S3.Ex5.m1.1.2.6.2.2.2.2" rspace="0.055em" xref="S3.Ex5.m1.1.1.cmml">)</mo></mrow><mo id="S3.Ex5.m1.1.2.6.2.1" rspace="0.222em" xref="S3.Ex5.m1.1.2.6.2.1.cmml">⋅</mo><mi id="S3.Ex5.m1.1.2.6.2.3" xref="S3.Ex5.m1.1.2.6.2.3.cmml">R</mi></mrow><mo id="S3.Ex5.m1.1.2.6.1" xref="S3.Ex5.m1.1.2.6.1.cmml">+</mo><mfrac id="S3.Ex5.m1.1.2.6.3" xref="S3.Ex5.m1.1.2.6.3.cmml"><msubsup id="S3.Ex5.m1.1.2.6.3.2" xref="S3.Ex5.m1.1.2.6.3.2.cmml"><mi id="S3.Ex5.m1.1.2.6.3.2.2.2" xref="S3.Ex5.m1.1.2.6.3.2.2.2.cmml">x</mi><mn id="S3.Ex5.m1.1.2.6.3.2.3" xref="S3.Ex5.m1.1.2.6.3.2.3.cmml">0</mn><mo id="S3.Ex5.m1.1.2.6.3.2.2.3" xref="S3.Ex5.m1.1.2.6.3.2.2.3.cmml">⋆</mo></msubsup><mrow id="S3.Ex5.m1.1.2.6.3.3" xref="S3.Ex5.m1.1.2.6.3.3.cmml"><mi id="S3.Ex5.m1.1.2.6.3.3.2" mathvariant="normal" xref="S3.Ex5.m1.1.2.6.3.3.2.cmml">ℓ</mi><mo id="S3.Ex5.m1.1.2.6.3.3.1" xref="S3.Ex5.m1.1.2.6.3.3.1.cmml">+</mo><mn id="S3.Ex5.m1.1.2.6.3.3.3" xref="S3.Ex5.m1.1.2.6.3.3.3.cmml">2</mn></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex5.m1.1b"><apply id="S3.Ex5.m1.1.2.cmml" xref="S3.Ex5.m1.1.2"><and id="S3.Ex5.m1.1.2a.cmml" xref="S3.Ex5.m1.1.2"></and><apply id="S3.Ex5.m1.1.2b.cmml" xref="S3.Ex5.m1.1.2"><eq id="S3.Ex5.m1.1.2.3.cmml" xref="S3.Ex5.m1.1.2.3"></eq><apply id="S3.Ex5.m1.1.2.2.cmml" xref="S3.Ex5.m1.1.2.2"><plus id="S3.Ex5.m1.1.2.2.1.cmml" xref="S3.Ex5.m1.1.2.2.1"></plus><apply id="S3.Ex5.m1.1.2.2.2.cmml" xref="S3.Ex5.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.2.2.2.1.cmml" xref="S3.Ex5.m1.1.2.2.2">subscript</csymbol><ci id="S3.Ex5.m1.1.2.2.2.2.cmml" xref="S3.Ex5.m1.1.2.2.2.2">𝑥</ci><cn id="S3.Ex5.m1.1.2.2.2.3.cmml" type="integer" xref="S3.Ex5.m1.1.2.2.2.3">0</cn></apply><apply id="S3.Ex5.m1.1.2.2.3.cmml" xref="S3.Ex5.m1.1.2.2.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.2.2.3.1.cmml" xref="S3.Ex5.m1.1.2.2.3">subscript</csymbol><ci id="S3.Ex5.m1.1.2.2.3.2.cmml" xref="S3.Ex5.m1.1.2.2.3.2">𝑥</ci><cn id="S3.Ex5.m1.1.2.2.3.3.cmml" type="integer" xref="S3.Ex5.m1.1.2.2.3.3">1</cn></apply></apply><apply id="S3.Ex5.m1.1.2.4.cmml" xref="S3.Ex5.m1.1.2.4"><plus id="S3.Ex5.m1.1.2.4.1.cmml" xref="S3.Ex5.m1.1.2.4.1"></plus><apply id="S3.Ex5.m1.1.2.4.2.cmml" xref="S3.Ex5.m1.1.2.4.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.2.4.2.1.cmml" xref="S3.Ex5.m1.1.2.4.2">subscript</csymbol><apply id="S3.Ex5.m1.1.2.4.2.2.cmml" xref="S3.Ex5.m1.1.2.4.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.2.4.2.2.1.cmml" xref="S3.Ex5.m1.1.2.4.2">superscript</csymbol><ci id="S3.Ex5.m1.1.2.4.2.2.2.cmml" xref="S3.Ex5.m1.1.2.4.2.2.2">𝑥</ci><ci id="S3.Ex5.m1.1.2.4.2.2.3.cmml" xref="S3.Ex5.m1.1.2.4.2.2.3">⋆</ci></apply><cn id="S3.Ex5.m1.1.2.4.2.3.cmml" type="integer" xref="S3.Ex5.m1.1.2.4.2.3">0</cn></apply><apply id="S3.Ex5.m1.1.2.4.3.cmml" xref="S3.Ex5.m1.1.2.4.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.2.4.3.1.cmml" xref="S3.Ex5.m1.1.2.4.3">subscript</csymbol><apply id="S3.Ex5.m1.1.2.4.3.2.cmml" xref="S3.Ex5.m1.1.2.4.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.2.4.3.2.1.cmml" xref="S3.Ex5.m1.1.2.4.3">superscript</csymbol><ci id="S3.Ex5.m1.1.2.4.3.2.2.cmml" xref="S3.Ex5.m1.1.2.4.3.2.2">𝑥</ci><ci id="S3.Ex5.m1.1.2.4.3.2.3.cmml" xref="S3.Ex5.m1.1.2.4.3.2.3">⋆</ci></apply><cn id="S3.Ex5.m1.1.2.4.3.3.cmml" type="integer" xref="S3.Ex5.m1.1.2.4.3.3">1</cn></apply></apply></apply><apply id="S3.Ex5.m1.1.2c.cmml" xref="S3.Ex5.m1.1.2"><eq id="S3.Ex5.m1.1.2.5.cmml" xref="S3.Ex5.m1.1.2.5"></eq><share href="https://arxiv.org/html/2503.17112v1#S3.Ex5.m1.1.2.4.cmml" id="S3.Ex5.m1.1.2d.cmml" xref="S3.Ex5.m1.1.2"></share><apply id="S3.Ex5.m1.1.2.6.cmml" xref="S3.Ex5.m1.1.2.6"><plus id="S3.Ex5.m1.1.2.6.1.cmml" xref="S3.Ex5.m1.1.2.6.1"></plus><apply id="S3.Ex5.m1.1.2.6.2.cmml" xref="S3.Ex5.m1.1.2.6.2"><ci id="S3.Ex5.m1.1.2.6.2.1.cmml" xref="S3.Ex5.m1.1.2.6.2.1">⋅</ci><apply id="S3.Ex5.m1.1.1.cmml" xref="S3.Ex5.m1.1.2.6.2.2.2"><divide id="S3.Ex5.m1.1.1.1.cmml" xref="S3.Ex5.m1.1.2.6.2.2.2"></divide><apply id="S3.Ex5.m1.1.1.2.cmml" xref="S3.Ex5.m1.1.1.2"><plus id="S3.Ex5.m1.1.1.2.1.cmml" xref="S3.Ex5.m1.1.1.2.1"></plus><ci id="S3.Ex5.m1.1.1.2.2.cmml" xref="S3.Ex5.m1.1.1.2.2">ℓ</ci><cn id="S3.Ex5.m1.1.1.2.3.cmml" type="integer" xref="S3.Ex5.m1.1.1.2.3">1</cn></apply><apply id="S3.Ex5.m1.1.1.3.cmml" xref="S3.Ex5.m1.1.1.3"><plus id="S3.Ex5.m1.1.1.3.1.cmml" xref="S3.Ex5.m1.1.1.3.1"></plus><ci id="S3.Ex5.m1.1.1.3.2.cmml" xref="S3.Ex5.m1.1.1.3.2">ℓ</ci><cn id="S3.Ex5.m1.1.1.3.3.cmml" type="integer" xref="S3.Ex5.m1.1.1.3.3">2</cn></apply></apply><ci id="S3.Ex5.m1.1.2.6.2.3.cmml" xref="S3.Ex5.m1.1.2.6.2.3">𝑅</ci></apply><apply id="S3.Ex5.m1.1.2.6.3.cmml" xref="S3.Ex5.m1.1.2.6.3"><divide id="S3.Ex5.m1.1.2.6.3.1.cmml" xref="S3.Ex5.m1.1.2.6.3"></divide><apply id="S3.Ex5.m1.1.2.6.3.2.cmml" xref="S3.Ex5.m1.1.2.6.3.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.2.6.3.2.1.cmml" xref="S3.Ex5.m1.1.2.6.3.2">subscript</csymbol><apply id="S3.Ex5.m1.1.2.6.3.2.2.cmml" xref="S3.Ex5.m1.1.2.6.3.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.2.6.3.2.2.1.cmml" xref="S3.Ex5.m1.1.2.6.3.2">superscript</csymbol><ci id="S3.Ex5.m1.1.2.6.3.2.2.2.cmml" xref="S3.Ex5.m1.1.2.6.3.2.2.2">𝑥</ci><ci id="S3.Ex5.m1.1.2.6.3.2.2.3.cmml" xref="S3.Ex5.m1.1.2.6.3.2.2.3">⋆</ci></apply><cn id="S3.Ex5.m1.1.2.6.3.2.3.cmml" type="integer" xref="S3.Ex5.m1.1.2.6.3.2.3">0</cn></apply><apply id="S3.Ex5.m1.1.2.6.3.3.cmml" xref="S3.Ex5.m1.1.2.6.3.3"><plus id="S3.Ex5.m1.1.2.6.3.3.1.cmml" xref="S3.Ex5.m1.1.2.6.3.3.1"></plus><ci id="S3.Ex5.m1.1.2.6.3.3.2.cmml" xref="S3.Ex5.m1.1.2.6.3.3.2">ℓ</ci><cn id="S3.Ex5.m1.1.2.6.3.3.3.cmml" type="integer" xref="S3.Ex5.m1.1.2.6.3.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex5.m1.1c">x_{0}+x_{1}=x^{\star}_{0}+x^{\star}_{1}=\left(\frac{\ell+1}{\ell+2}\right)% \cdot R+\frac{x^{\star}_{0}}{\ell+2}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5.m1.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( divide start_ARG roman_ℓ + 1 end_ARG start_ARG roman_ℓ + 2 end_ARG ) ⋅ italic_R + divide start_ARG italic_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG roman_ℓ + 2 end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.9.p5.27">maximizes <math alttext="x_{0}+x_{1}" class="ltx_Math" display="inline" id="S3.9.p5.25.m1.1"><semantics id="S3.9.p5.25.m1.1a"><mrow id="S3.9.p5.25.m1.1.1" xref="S3.9.p5.25.m1.1.1.cmml"><msub id="S3.9.p5.25.m1.1.1.2" xref="S3.9.p5.25.m1.1.1.2.cmml"><mi id="S3.9.p5.25.m1.1.1.2.2" xref="S3.9.p5.25.m1.1.1.2.2.cmml">x</mi><mn id="S3.9.p5.25.m1.1.1.2.3" xref="S3.9.p5.25.m1.1.1.2.3.cmml">0</mn></msub><mo id="S3.9.p5.25.m1.1.1.1" xref="S3.9.p5.25.m1.1.1.1.cmml">+</mo><msub id="S3.9.p5.25.m1.1.1.3" xref="S3.9.p5.25.m1.1.1.3.cmml"><mi id="S3.9.p5.25.m1.1.1.3.2" xref="S3.9.p5.25.m1.1.1.3.2.cmml">x</mi><mn id="S3.9.p5.25.m1.1.1.3.3" xref="S3.9.p5.25.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.25.m1.1b"><apply id="S3.9.p5.25.m1.1.1.cmml" xref="S3.9.p5.25.m1.1.1"><plus id="S3.9.p5.25.m1.1.1.1.cmml" xref="S3.9.p5.25.m1.1.1.1"></plus><apply id="S3.9.p5.25.m1.1.1.2.cmml" xref="S3.9.p5.25.m1.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.25.m1.1.1.2.1.cmml" xref="S3.9.p5.25.m1.1.1.2">subscript</csymbol><ci id="S3.9.p5.25.m1.1.1.2.2.cmml" xref="S3.9.p5.25.m1.1.1.2.2">𝑥</ci><cn id="S3.9.p5.25.m1.1.1.2.3.cmml" type="integer" xref="S3.9.p5.25.m1.1.1.2.3">0</cn></apply><apply id="S3.9.p5.25.m1.1.1.3.cmml" xref="S3.9.p5.25.m1.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.25.m1.1.1.3.1.cmml" xref="S3.9.p5.25.m1.1.1.3">subscript</csymbol><ci id="S3.9.p5.25.m1.1.1.3.2.cmml" xref="S3.9.p5.25.m1.1.1.3.2">𝑥</ci><cn id="S3.9.p5.25.m1.1.1.3.3.cmml" type="integer" xref="S3.9.p5.25.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.25.m1.1c">x_{0}+x_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.25.m1.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> subject to fixed <math alttext="x_{0}=x_{0}^{\star}" class="ltx_Math" display="inline" id="S3.9.p5.26.m2.1"><semantics id="S3.9.p5.26.m2.1a"><mrow id="S3.9.p5.26.m2.1.1" xref="S3.9.p5.26.m2.1.1.cmml"><msub id="S3.9.p5.26.m2.1.1.2" xref="S3.9.p5.26.m2.1.1.2.cmml"><mi id="S3.9.p5.26.m2.1.1.2.2" xref="S3.9.p5.26.m2.1.1.2.2.cmml">x</mi><mn id="S3.9.p5.26.m2.1.1.2.3" xref="S3.9.p5.26.m2.1.1.2.3.cmml">0</mn></msub><mo id="S3.9.p5.26.m2.1.1.1" xref="S3.9.p5.26.m2.1.1.1.cmml">=</mo><msubsup id="S3.9.p5.26.m2.1.1.3" xref="S3.9.p5.26.m2.1.1.3.cmml"><mi id="S3.9.p5.26.m2.1.1.3.2.2" xref="S3.9.p5.26.m2.1.1.3.2.2.cmml">x</mi><mn id="S3.9.p5.26.m2.1.1.3.2.3" xref="S3.9.p5.26.m2.1.1.3.2.3.cmml">0</mn><mo id="S3.9.p5.26.m2.1.1.3.3" xref="S3.9.p5.26.m2.1.1.3.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p5.26.m2.1b"><apply id="S3.9.p5.26.m2.1.1.cmml" xref="S3.9.p5.26.m2.1.1"><eq id="S3.9.p5.26.m2.1.1.1.cmml" xref="S3.9.p5.26.m2.1.1.1"></eq><apply id="S3.9.p5.26.m2.1.1.2.cmml" xref="S3.9.p5.26.m2.1.1.2"><csymbol cd="ambiguous" id="S3.9.p5.26.m2.1.1.2.1.cmml" xref="S3.9.p5.26.m2.1.1.2">subscript</csymbol><ci id="S3.9.p5.26.m2.1.1.2.2.cmml" xref="S3.9.p5.26.m2.1.1.2.2">𝑥</ci><cn id="S3.9.p5.26.m2.1.1.2.3.cmml" type="integer" xref="S3.9.p5.26.m2.1.1.2.3">0</cn></apply><apply id="S3.9.p5.26.m2.1.1.3.cmml" xref="S3.9.p5.26.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.26.m2.1.1.3.1.cmml" xref="S3.9.p5.26.m2.1.1.3">superscript</csymbol><apply id="S3.9.p5.26.m2.1.1.3.2.cmml" xref="S3.9.p5.26.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p5.26.m2.1.1.3.2.1.cmml" xref="S3.9.p5.26.m2.1.1.3">subscript</csymbol><ci id="S3.9.p5.26.m2.1.1.3.2.2.cmml" xref="S3.9.p5.26.m2.1.1.3.2.2">𝑥</ci><cn id="S3.9.p5.26.m2.1.1.3.2.3.cmml" type="integer" xref="S3.9.p5.26.m2.1.1.3.2.3">0</cn></apply><ci id="S3.9.p5.26.m2.1.1.3.3.cmml" xref="S3.9.p5.26.m2.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.26.m2.1c">x_{0}=x_{0}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.26.m2.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. This is an increasing linear function of <math alttext="x_{0}^{\star}" class="ltx_Math" display="inline" id="S3.9.p5.27.m3.1"><semantics id="S3.9.p5.27.m3.1a"><msubsup id="S3.9.p5.27.m3.1.1" xref="S3.9.p5.27.m3.1.1.cmml"><mi id="S3.9.p5.27.m3.1.1.2.2" xref="S3.9.p5.27.m3.1.1.2.2.cmml">x</mi><mn id="S3.9.p5.27.m3.1.1.2.3" xref="S3.9.p5.27.m3.1.1.2.3.cmml">0</mn><mo id="S3.9.p5.27.m3.1.1.3" xref="S3.9.p5.27.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.9.p5.27.m3.1b"><apply id="S3.9.p5.27.m3.1.1.cmml" xref="S3.9.p5.27.m3.1.1"><csymbol cd="ambiguous" id="S3.9.p5.27.m3.1.1.1.cmml" xref="S3.9.p5.27.m3.1.1">superscript</csymbol><apply id="S3.9.p5.27.m3.1.1.2.cmml" xref="S3.9.p5.27.m3.1.1"><csymbol cd="ambiguous" id="S3.9.p5.27.m3.1.1.2.1.cmml" xref="S3.9.p5.27.m3.1.1">subscript</csymbol><ci id="S3.9.p5.27.m3.1.1.2.2.cmml" xref="S3.9.p5.27.m3.1.1.2.2">𝑥</ci><cn id="S3.9.p5.27.m3.1.1.2.3.cmml" type="integer" xref="S3.9.p5.27.m3.1.1.2.3">0</cn></apply><ci id="S3.9.p5.27.m3.1.1.3.cmml" xref="S3.9.p5.27.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p5.27.m3.1c">x_{0}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p5.27.m3.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. From (<a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#S3.E1" title="Equation 1 ‣ Proof. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">1</span></a>), we have the constraint</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="x_{0}^{\star}\leq\frac{|A\setminus B|-|\Delta_{\ell+1}\setminus B|}{\ell+1}+|A% \cap B|\leq\frac{|A\setminus B|}{\ell+1}+|A\cap B|" class="ltx_Math" display="block" id="S3.Ex6.m1.5"><semantics id="S3.Ex6.m1.5a"><mrow id="S3.Ex6.m1.5.5" xref="S3.Ex6.m1.5.5.cmml"><msubsup id="S3.Ex6.m1.5.5.4" xref="S3.Ex6.m1.5.5.4.cmml"><mi id="S3.Ex6.m1.5.5.4.2.2" xref="S3.Ex6.m1.5.5.4.2.2.cmml">x</mi><mn id="S3.Ex6.m1.5.5.4.2.3" xref="S3.Ex6.m1.5.5.4.2.3.cmml">0</mn><mo id="S3.Ex6.m1.5.5.4.3" xref="S3.Ex6.m1.5.5.4.3.cmml">⋆</mo></msubsup><mo id="S3.Ex6.m1.5.5.5" xref="S3.Ex6.m1.5.5.5.cmml">≤</mo><mrow id="S3.Ex6.m1.4.4.1" xref="S3.Ex6.m1.4.4.1.cmml"><mfrac id="S3.Ex6.m1.2.2" xref="S3.Ex6.m1.2.2.cmml"><mrow id="S3.Ex6.m1.2.2.2" xref="S3.Ex6.m1.2.2.2.cmml"><mrow id="S3.Ex6.m1.1.1.1.1.1" xref="S3.Ex6.m1.1.1.1.1.2.cmml"><mo id="S3.Ex6.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex6.m1.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex6.m1.1.1.1.1.1.1" xref="S3.Ex6.m1.1.1.1.1.1.1.cmml"><mi id="S3.Ex6.m1.1.1.1.1.1.1.2" xref="S3.Ex6.m1.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex6.m1.1.1.1.1.1.1.1" xref="S3.Ex6.m1.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex6.m1.1.1.1.1.1.1.3" xref="S3.Ex6.m1.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex6.m1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex6.m1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex6.m1.2.2.2.3" xref="S3.Ex6.m1.2.2.2.3.cmml">−</mo><mrow id="S3.Ex6.m1.2.2.2.2.1" xref="S3.Ex6.m1.2.2.2.2.2.cmml"><mo id="S3.Ex6.m1.2.2.2.2.1.2" stretchy="false" xref="S3.Ex6.m1.2.2.2.2.2.1.cmml">|</mo><mrow id="S3.Ex6.m1.2.2.2.2.1.1" xref="S3.Ex6.m1.2.2.2.2.1.1.cmml"><msub id="S3.Ex6.m1.2.2.2.2.1.1.2" xref="S3.Ex6.m1.2.2.2.2.1.1.2.cmml"><mi id="S3.Ex6.m1.2.2.2.2.1.1.2.2" mathvariant="normal" xref="S3.Ex6.m1.2.2.2.2.1.1.2.2.cmml">Δ</mi><mrow id="S3.Ex6.m1.2.2.2.2.1.1.2.3" xref="S3.Ex6.m1.2.2.2.2.1.1.2.3.cmml"><mi id="S3.Ex6.m1.2.2.2.2.1.1.2.3.2" mathvariant="normal" xref="S3.Ex6.m1.2.2.2.2.1.1.2.3.2.cmml">ℓ</mi><mo id="S3.Ex6.m1.2.2.2.2.1.1.2.3.1" xref="S3.Ex6.m1.2.2.2.2.1.1.2.3.1.cmml">+</mo><mn id="S3.Ex6.m1.2.2.2.2.1.1.2.3.3" xref="S3.Ex6.m1.2.2.2.2.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.Ex6.m1.2.2.2.2.1.1.1" xref="S3.Ex6.m1.2.2.2.2.1.1.1.cmml">∖</mo><mi id="S3.Ex6.m1.2.2.2.2.1.1.3" xref="S3.Ex6.m1.2.2.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex6.m1.2.2.2.2.1.3" stretchy="false" xref="S3.Ex6.m1.2.2.2.2.2.1.cmml">|</mo></mrow></mrow><mrow id="S3.Ex6.m1.2.2.4" xref="S3.Ex6.m1.2.2.4.cmml"><mi id="S3.Ex6.m1.2.2.4.2" mathvariant="normal" 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end_POSTSUBSCRIPT ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + | italic_A ∩ italic_B | ≤ divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.9.p5.30">from which we obtain the upper bound</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx4"> <tbody id="S3.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle|W\setminus B|+|Z\setminus B|" class="ltx_Math" display="inline" id="S3.Ex7.m1.2"><semantics id="S3.Ex7.m1.2a"><mrow id="S3.Ex7.m1.2.2" xref="S3.Ex7.m1.2.2.cmml"><mrow id="S3.Ex7.m1.1.1.1.1" xref="S3.Ex7.m1.1.1.1.2.cmml"><mo id="S3.Ex7.m1.1.1.1.1.2" stretchy="false" xref="S3.Ex7.m1.1.1.1.2.1.cmml">|</mo><mrow 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id="S3.Ex7.m1.2b"><apply id="S3.Ex7.m1.2.2.cmml" xref="S3.Ex7.m1.2.2"><plus id="S3.Ex7.m1.2.2.3.cmml" xref="S3.Ex7.m1.2.2.3"></plus><apply id="S3.Ex7.m1.1.1.1.2.cmml" xref="S3.Ex7.m1.1.1.1.1"><abs id="S3.Ex7.m1.1.1.1.2.1.cmml" xref="S3.Ex7.m1.1.1.1.1.2"></abs><apply id="S3.Ex7.m1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1"><setdiff id="S3.Ex7.m1.1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1"></setdiff><ci id="S3.Ex7.m1.1.1.1.1.1.2.cmml" xref="S3.Ex7.m1.1.1.1.1.1.2">𝑊</ci><ci id="S3.Ex7.m1.1.1.1.1.1.3.cmml" xref="S3.Ex7.m1.1.1.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.Ex7.m1.2.2.2.2.cmml" xref="S3.Ex7.m1.2.2.2.1"><abs id="S3.Ex7.m1.2.2.2.2.1.cmml" xref="S3.Ex7.m1.2.2.2.1.2"></abs><apply id="S3.Ex7.m1.2.2.2.1.1.cmml" xref="S3.Ex7.m1.2.2.2.1.1"><setdiff id="S3.Ex7.m1.2.2.2.1.1.1.cmml" xref="S3.Ex7.m1.2.2.2.1.1.1"></setdiff><ci id="S3.Ex7.m1.2.2.2.1.1.2.cmml" xref="S3.Ex7.m1.2.2.2.1.1.2">𝑍</ci><ci id="S3.Ex7.m1.2.2.2.1.1.3.cmml" xref="S3.Ex7.m1.2.2.2.1.1.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m1.2c">\displaystyle|W\setminus B|+|Z\setminus B|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m1.2d">| italic_W ∖ italic_B | + | italic_Z ∖ italic_B |</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\left(\frac{\ell+1}{\ell+2}\right)\cdot R+\frac{x_{0}^{\star}% }{\ell+2}" class="ltx_Math" display="inline" id="S3.Ex7.m2.1"><semantics id="S3.Ex7.m2.1a"><mrow id="S3.Ex7.m2.1.2" xref="S3.Ex7.m2.1.2.cmml"><mi id="S3.Ex7.m2.1.2.2" xref="S3.Ex7.m2.1.2.2.cmml"></mi><mo id="S3.Ex7.m2.1.2.1" xref="S3.Ex7.m2.1.2.1.cmml">≤</mo><mrow id="S3.Ex7.m2.1.2.3" xref="S3.Ex7.m2.1.2.3.cmml"><mrow id="S3.Ex7.m2.1.2.3.2" xref="S3.Ex7.m2.1.2.3.2.cmml"><mrow id="S3.Ex7.m2.1.2.3.2.2.2" xref="S3.Ex7.m2.1.1.cmml"><mo id="S3.Ex7.m2.1.2.3.2.2.2.1" xref="S3.Ex7.m2.1.1.cmml">(</mo><mstyle displaystyle="true" id="S3.Ex7.m2.1.1" xref="S3.Ex7.m2.1.1.cmml"><mfrac id="S3.Ex7.m2.1.1a" xref="S3.Ex7.m2.1.1.cmml"><mrow id="S3.Ex7.m2.1.1.2" xref="S3.Ex7.m2.1.1.2.cmml"><mi id="S3.Ex7.m2.1.1.2.2" mathvariant="normal" xref="S3.Ex7.m2.1.1.2.2.cmml">ℓ</mi><mo id="S3.Ex7.m2.1.1.2.1" xref="S3.Ex7.m2.1.1.2.1.cmml">+</mo><mn id="S3.Ex7.m2.1.1.2.3" xref="S3.Ex7.m2.1.1.2.3.cmml">1</mn></mrow><mrow id="S3.Ex7.m2.1.1.3" xref="S3.Ex7.m2.1.1.3.cmml"><mi id="S3.Ex7.m2.1.1.3.2" mathvariant="normal" xref="S3.Ex7.m2.1.1.3.2.cmml">ℓ</mi><mo id="S3.Ex7.m2.1.1.3.1" xref="S3.Ex7.m2.1.1.3.1.cmml">+</mo><mn id="S3.Ex7.m2.1.1.3.3" xref="S3.Ex7.m2.1.1.3.3.cmml">2</mn></mrow></mfrac></mstyle><mo id="S3.Ex7.m2.1.2.3.2.2.2.2" rspace="0.055em" xref="S3.Ex7.m2.1.1.cmml">)</mo></mrow><mo id="S3.Ex7.m2.1.2.3.2.1" rspace="0.222em" xref="S3.Ex7.m2.1.2.3.2.1.cmml">⋅</mo><mi id="S3.Ex7.m2.1.2.3.2.3" xref="S3.Ex7.m2.1.2.3.2.3.cmml">R</mi></mrow><mo id="S3.Ex7.m2.1.2.3.1" xref="S3.Ex7.m2.1.2.3.1.cmml">+</mo><mstyle displaystyle="true" id="S3.Ex7.m2.1.2.3.3" xref="S3.Ex7.m2.1.2.3.3.cmml"><mfrac id="S3.Ex7.m2.1.2.3.3a" xref="S3.Ex7.m2.1.2.3.3.cmml"><msubsup id="S3.Ex7.m2.1.2.3.3.2" xref="S3.Ex7.m2.1.2.3.3.2.cmml"><mi id="S3.Ex7.m2.1.2.3.3.2.2.2" xref="S3.Ex7.m2.1.2.3.3.2.2.2.cmml">x</mi><mn id="S3.Ex7.m2.1.2.3.3.2.2.3" xref="S3.Ex7.m2.1.2.3.3.2.2.3.cmml">0</mn><mo id="S3.Ex7.m2.1.2.3.3.2.3" xref="S3.Ex7.m2.1.2.3.3.2.3.cmml">⋆</mo></msubsup><mrow id="S3.Ex7.m2.1.2.3.3.3" xref="S3.Ex7.m2.1.2.3.3.3.cmml"><mi id="S3.Ex7.m2.1.2.3.3.3.2" mathvariant="normal" xref="S3.Ex7.m2.1.2.3.3.3.2.cmml">ℓ</mi><mo id="S3.Ex7.m2.1.2.3.3.3.1" xref="S3.Ex7.m2.1.2.3.3.3.1.cmml">+</mo><mn id="S3.Ex7.m2.1.2.3.3.3.3" xref="S3.Ex7.m2.1.2.3.3.3.3.cmml">2</mn></mrow></mfrac></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex7.m2.1b"><apply id="S3.Ex7.m2.1.2.cmml" xref="S3.Ex7.m2.1.2"><leq id="S3.Ex7.m2.1.2.1.cmml" xref="S3.Ex7.m2.1.2.1"></leq><csymbol cd="latexml" id="S3.Ex7.m2.1.2.2.cmml" xref="S3.Ex7.m2.1.2.2">absent</csymbol><apply id="S3.Ex7.m2.1.2.3.cmml" xref="S3.Ex7.m2.1.2.3"><plus id="S3.Ex7.m2.1.2.3.1.cmml" xref="S3.Ex7.m2.1.2.3.1"></plus><apply id="S3.Ex7.m2.1.2.3.2.cmml" xref="S3.Ex7.m2.1.2.3.2"><ci id="S3.Ex7.m2.1.2.3.2.1.cmml" xref="S3.Ex7.m2.1.2.3.2.1">⋅</ci><apply id="S3.Ex7.m2.1.1.cmml" xref="S3.Ex7.m2.1.2.3.2.2.2"><divide id="S3.Ex7.m2.1.1.1.cmml" xref="S3.Ex7.m2.1.2.3.2.2.2"></divide><apply id="S3.Ex7.m2.1.1.2.cmml" xref="S3.Ex7.m2.1.1.2"><plus id="S3.Ex7.m2.1.1.2.1.cmml" xref="S3.Ex7.m2.1.1.2.1"></plus><ci id="S3.Ex7.m2.1.1.2.2.cmml" xref="S3.Ex7.m2.1.1.2.2">ℓ</ci><cn id="S3.Ex7.m2.1.1.2.3.cmml" type="integer" xref="S3.Ex7.m2.1.1.2.3">1</cn></apply><apply id="S3.Ex7.m2.1.1.3.cmml" xref="S3.Ex7.m2.1.1.3"><plus id="S3.Ex7.m2.1.1.3.1.cmml" xref="S3.Ex7.m2.1.1.3.1"></plus><ci id="S3.Ex7.m2.1.1.3.2.cmml" xref="S3.Ex7.m2.1.1.3.2">ℓ</ci><cn id="S3.Ex7.m2.1.1.3.3.cmml" type="integer" xref="S3.Ex7.m2.1.1.3.3">2</cn></apply></apply><ci id="S3.Ex7.m2.1.2.3.2.3.cmml" xref="S3.Ex7.m2.1.2.3.2.3">𝑅</ci></apply><apply id="S3.Ex7.m2.1.2.3.3.cmml" xref="S3.Ex7.m2.1.2.3.3"><divide id="S3.Ex7.m2.1.2.3.3.1.cmml" xref="S3.Ex7.m2.1.2.3.3"></divide><apply id="S3.Ex7.m2.1.2.3.3.2.cmml" xref="S3.Ex7.m2.1.2.3.3.2"><csymbol cd="ambiguous" id="S3.Ex7.m2.1.2.3.3.2.1.cmml" xref="S3.Ex7.m2.1.2.3.3.2">superscript</csymbol><apply id="S3.Ex7.m2.1.2.3.3.2.2.cmml" xref="S3.Ex7.m2.1.2.3.3.2"><csymbol cd="ambiguous" id="S3.Ex7.m2.1.2.3.3.2.2.1.cmml" xref="S3.Ex7.m2.1.2.3.3.2">subscript</csymbol><ci id="S3.Ex7.m2.1.2.3.3.2.2.2.cmml" xref="S3.Ex7.m2.1.2.3.3.2.2.2">𝑥</ci><cn id="S3.Ex7.m2.1.2.3.3.2.2.3.cmml" type="integer" xref="S3.Ex7.m2.1.2.3.3.2.2.3">0</cn></apply><ci id="S3.Ex7.m2.1.2.3.3.2.3.cmml" xref="S3.Ex7.m2.1.2.3.3.2.3">⋆</ci></apply><apply id="S3.Ex7.m2.1.2.3.3.3.cmml" xref="S3.Ex7.m2.1.2.3.3.3"><plus id="S3.Ex7.m2.1.2.3.3.3.1.cmml" xref="S3.Ex7.m2.1.2.3.3.3.1"></plus><ci id="S3.Ex7.m2.1.2.3.3.3.2.cmml" xref="S3.Ex7.m2.1.2.3.3.3.2">ℓ</ci><cn id="S3.Ex7.m2.1.2.3.3.3.3.cmml" type="integer" xref="S3.Ex7.m2.1.2.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m2.1c">\displaystyle\leq\left(\frac{\ell+1}{\ell+2}\right)\cdot R+\frac{x_{0}^{\star}% }{\ell+2}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m2.1d">≤ ( divide start_ARG roman_ℓ + 1 end_ARG start_ARG roman_ℓ + 2 end_ARG ) ⋅ italic_R + divide start_ARG italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT end_ARG start_ARG roman_ℓ + 2 end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\left(\frac{\ell+1}{\ell+2}\right)\left(\frac{|A\setminus B|}% {\ell+1}+\frac{|A\setminus B|}{\ell+2}+3\,|A\cap B|\right)+\frac{|A\setminus B% |-|Z\setminus B|}{(\ell+1)(\ell+2)}+\frac{|A\cap B|}{\ell+2}" class="ltx_Math" display="inline" id="S3.Ex8.m1.9"><semantics id="S3.Ex8.m1.9a"><mrow id="S3.Ex8.m1.9.9" xref="S3.Ex8.m1.9.9.cmml"><mi id="S3.Ex8.m1.9.9.3" xref="S3.Ex8.m1.9.9.3.cmml"></mi><mo id="S3.Ex8.m1.9.9.2" xref="S3.Ex8.m1.9.9.2.cmml">≤</mo><mrow id="S3.Ex8.m1.9.9.1" xref="S3.Ex8.m1.9.9.1.cmml"><mrow id="S3.Ex8.m1.9.9.1.1" xref="S3.Ex8.m1.9.9.1.1.cmml"><mrow id="S3.Ex8.m1.9.9.1.1.3.2" xref="S3.Ex8.m1.8.8.cmml"><mo id="S3.Ex8.m1.9.9.1.1.3.2.1" xref="S3.Ex8.m1.8.8.cmml">(</mo><mstyle displaystyle="true" id="S3.Ex8.m1.8.8" xref="S3.Ex8.m1.8.8.cmml"><mfrac id="S3.Ex8.m1.8.8a" xref="S3.Ex8.m1.8.8.cmml"><mrow id="S3.Ex8.m1.8.8.2" xref="S3.Ex8.m1.8.8.2.cmml"><mi id="S3.Ex8.m1.8.8.2.2" mathvariant="normal" xref="S3.Ex8.m1.8.8.2.2.cmml">ℓ</mi><mo id="S3.Ex8.m1.8.8.2.1" xref="S3.Ex8.m1.8.8.2.1.cmml">+</mo><mn id="S3.Ex8.m1.8.8.2.3" xref="S3.Ex8.m1.8.8.2.3.cmml">1</mn></mrow><mrow id="S3.Ex8.m1.8.8.3" xref="S3.Ex8.m1.8.8.3.cmml"><mi id="S3.Ex8.m1.8.8.3.2" mathvariant="normal" xref="S3.Ex8.m1.8.8.3.2.cmml">ℓ</mi><mo id="S3.Ex8.m1.8.8.3.1" xref="S3.Ex8.m1.8.8.3.1.cmml">+</mo><mn id="S3.Ex8.m1.8.8.3.3" xref="S3.Ex8.m1.8.8.3.3.cmml">2</mn></mrow></mfrac></mstyle><mo id="S3.Ex8.m1.9.9.1.1.3.2.2" xref="S3.Ex8.m1.8.8.cmml">)</mo></mrow><mo id="S3.Ex8.m1.9.9.1.1.2" xref="S3.Ex8.m1.9.9.1.1.2.cmml">⁢</mo><mrow id="S3.Ex8.m1.9.9.1.1.1.1" xref="S3.Ex8.m1.9.9.1.1.1.1.1.cmml"><mo id="S3.Ex8.m1.9.9.1.1.1.1.2" xref="S3.Ex8.m1.9.9.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex8.m1.9.9.1.1.1.1.1" xref="S3.Ex8.m1.9.9.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S3.Ex8.m1.1.1" xref="S3.Ex8.m1.1.1.cmml"><mfrac id="S3.Ex8.m1.1.1a" xref="S3.Ex8.m1.1.1.cmml"><mrow id="S3.Ex8.m1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.2.cmml"><mo id="S3.Ex8.m1.1.1.1.1.2" stretchy="false" xref="S3.Ex8.m1.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex8.m1.1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.1.cmml"><mi id="S3.Ex8.m1.1.1.1.1.1.2" xref="S3.Ex8.m1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex8.m1.1.1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex8.m1.1.1.1.1.1.3" xref="S3.Ex8.m1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex8.m1.1.1.1.1.3" stretchy="false" xref="S3.Ex8.m1.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S3.Ex8.m1.1.1.3" xref="S3.Ex8.m1.1.1.3.cmml"><mi id="S3.Ex8.m1.1.1.3.2" mathvariant="normal" xref="S3.Ex8.m1.1.1.3.2.cmml">ℓ</mi><mo id="S3.Ex8.m1.1.1.3.1" xref="S3.Ex8.m1.1.1.3.1.cmml">+</mo><mn id="S3.Ex8.m1.1.1.3.3" xref="S3.Ex8.m1.1.1.3.3.cmml">1</mn></mrow></mfrac></mstyle><mo id="S3.Ex8.m1.9.9.1.1.1.1.1.2" xref="S3.Ex8.m1.9.9.1.1.1.1.1.2.cmml">+</mo><mstyle displaystyle="true" id="S3.Ex8.m1.2.2" xref="S3.Ex8.m1.2.2.cmml"><mfrac id="S3.Ex8.m1.2.2a" xref="S3.Ex8.m1.2.2.cmml"><mrow id="S3.Ex8.m1.2.2.1.1" xref="S3.Ex8.m1.2.2.1.2.cmml"><mo 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xref="S3.Ex8.m1.5.5.3.1.1.1.2">ℓ</ci><cn id="S3.Ex8.m1.5.5.3.1.1.1.3.cmml" type="integer" xref="S3.Ex8.m1.5.5.3.1.1.1.3">1</cn></apply><apply id="S3.Ex8.m1.6.6.4.2.1.1.cmml" xref="S3.Ex8.m1.6.6.4.2.1"><plus id="S3.Ex8.m1.6.6.4.2.1.1.1.cmml" xref="S3.Ex8.m1.6.6.4.2.1.1.1"></plus><ci id="S3.Ex8.m1.6.6.4.2.1.1.2.cmml" xref="S3.Ex8.m1.6.6.4.2.1.1.2">ℓ</ci><cn id="S3.Ex8.m1.6.6.4.2.1.1.3.cmml" type="integer" xref="S3.Ex8.m1.6.6.4.2.1.1.3">2</cn></apply></apply></apply><apply id="S3.Ex8.m1.7.7.cmml" xref="S3.Ex8.m1.7.7"><divide id="S3.Ex8.m1.7.7.2.cmml" xref="S3.Ex8.m1.7.7"></divide><apply id="S3.Ex8.m1.7.7.1.2.cmml" xref="S3.Ex8.m1.7.7.1.1"><abs id="S3.Ex8.m1.7.7.1.2.1.cmml" xref="S3.Ex8.m1.7.7.1.1.2"></abs><apply id="S3.Ex8.m1.7.7.1.1.1.cmml" xref="S3.Ex8.m1.7.7.1.1.1"><intersect id="S3.Ex8.m1.7.7.1.1.1.1.cmml" xref="S3.Ex8.m1.7.7.1.1.1.1"></intersect><ci id="S3.Ex8.m1.7.7.1.1.1.2.cmml" xref="S3.Ex8.m1.7.7.1.1.1.2">𝐴</ci><ci id="S3.Ex8.m1.7.7.1.1.1.3.cmml" xref="S3.Ex8.m1.7.7.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.Ex8.m1.7.7.3.cmml" xref="S3.Ex8.m1.7.7.3"><plus id="S3.Ex8.m1.7.7.3.1.cmml" xref="S3.Ex8.m1.7.7.3.1"></plus><ci id="S3.Ex8.m1.7.7.3.2.cmml" xref="S3.Ex8.m1.7.7.3.2">ℓ</ci><cn id="S3.Ex8.m1.7.7.3.3.cmml" type="integer" xref="S3.Ex8.m1.7.7.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex8.m1.9c">\displaystyle\leq\left(\frac{\ell+1}{\ell+2}\right)\left(\frac{|A\setminus B|}% {\ell+1}+\frac{|A\setminus B|}{\ell+2}+3\,|A\cap B|\right)+\frac{|A\setminus B% |-|Z\setminus B|}{(\ell+1)(\ell+2)}+\frac{|A\cap B|}{\ell+2}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m1.9d">≤ ( divide start_ARG roman_ℓ + 1 end_ARG start_ARG roman_ℓ + 2 end_ARG ) ( divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 1 end_ARG + divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG + 3 | italic_A ∩ italic_B | ) + divide start_ARG | italic_A ∖ italic_B | - | italic_Z ∖ italic_B | end_ARG start_ARG ( roman_ℓ + 1 ) ( roman_ℓ + 2 ) end_ARG + divide start_ARG | italic_A ∩ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{|A\setminus B|}{\ell+2}\cdot\left(\frac{\ell+1}{\ell+1}+% \frac{\ell+1}{\ell+2}+\frac{1}{\ell+1}\right)+\left(\frac{3\ell+4}{\ell+2}% \right)\cdot|A\cap B|" class="ltx_Math" display="inline" id="S3.Ex9.m1.4"><semantics id="S3.Ex9.m1.4a"><mrow id="S3.Ex9.m1.4.4" xref="S3.Ex9.m1.4.4.cmml"><mi id="S3.Ex9.m1.4.4.4" xref="S3.Ex9.m1.4.4.4.cmml"></mi><mo id="S3.Ex9.m1.4.4.3" xref="S3.Ex9.m1.4.4.3.cmml">=</mo><mrow id="S3.Ex9.m1.4.4.2" 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encoding="application/x-llamapun" id="S3.Ex9.m1.4d">= divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG ⋅ ( divide start_ARG roman_ℓ + 1 end_ARG start_ARG roman_ℓ + 1 end_ARG + divide start_ARG roman_ℓ + 1 end_ARG start_ARG roman_ℓ + 2 end_ARG + divide start_ARG 1 end_ARG start_ARG roman_ℓ + 1 end_ARG ) + ( divide start_ARG 3 roman_ℓ + 4 end_ARG start_ARG roman_ℓ + 2 end_ARG ) ⋅ | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{|A\setminus B|}{\ell+2}\cdot\left(2-\frac{1}{\ell+2}+\frac% {1}{\ell+1}\right)+\left(\frac{3\ell+4}{\ell+2}\right)\cdot|A\cap B|" class="ltx_Math" display="inline" id="S3.Ex10.m1.4"><semantics 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id="S3.Ex10.m1.4.4.2.2.1.1.1.2.cmml" xref="S3.Ex10.m1.4.4.2.2.1.1.1.2">𝐴</ci><ci id="S3.Ex10.m1.4.4.2.2.1.1.1.3.cmml" xref="S3.Ex10.m1.4.4.2.2.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.4c">\displaystyle=\frac{|A\setminus B|}{\ell+2}\cdot\left(2-\frac{1}{\ell+2}+\frac% {1}{\ell+1}\right)+\left(\frac{3\ell+4}{\ell+2}\right)\cdot|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.4d">= divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG ⋅ ( 2 - divide start_ARG 1 end_ARG start_ARG roman_ℓ + 2 end_ARG + divide start_ARG 1 end_ARG start_ARG roman_ℓ + 1 end_ARG ) + ( divide start_ARG 3 roman_ℓ + 4 end_ARG start_ARG roman_ℓ + 2 end_ARG ) ⋅ | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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xref="S3.Ex11.m1.6.6.2.2.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex11.m1.6c">\displaystyle=\frac{|A\setminus B|}{\ell+2}\cdot\left(2+\frac{1}{(\ell+1)(\ell% +2)}\right)+\left(\frac{3\ell+4}{\ell+2}\right)\cdot|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex11.m1.6d">= divide start_ARG | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG ⋅ ( 2 + divide start_ARG 1 end_ARG start_ARG ( roman_ℓ + 1 ) ( roman_ℓ + 2 ) end_ARG ) + ( divide start_ARG 3 roman_ℓ + 4 end_ARG start_ARG roman_ℓ + 2 end_ARG ) ⋅ | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle<\frac{(2+\tfrac{1}{6})\,|A\setminus B|}{\ell+2}+3\,|A\cap B|% \enspace.\qed" class="ltx_Math" display="inline" id="S3.Ex12.m1.3"><semantics id="S3.Ex12.m1.3a"><mrow id="S3.Ex12.m1.3.3.1" xref="S3.Ex12.m1.3.3.1.1.cmml"><mrow id="S3.Ex12.m1.3.3.1.1" xref="S3.Ex12.m1.3.3.1.1.cmml"><mi id="S3.Ex12.m1.3.3.1.1.3" xref="S3.Ex12.m1.3.3.1.1.3.cmml"></mi><mo id="S3.Ex12.m1.3.3.1.1.2" xref="S3.Ex12.m1.3.3.1.1.2.cmml"><</mo><mrow id="S3.Ex12.m1.3.3.1.1.1" xref="S3.Ex12.m1.3.3.1.1.1.cmml"><mstyle displaystyle="true" id="S3.Ex12.m1.2.2" xref="S3.Ex12.m1.2.2.cmml"><mfrac id="S3.Ex12.m1.2.2a" xref="S3.Ex12.m1.2.2.cmml"><mrow id="S3.Ex12.m1.2.2.2" xref="S3.Ex12.m1.2.2.2.cmml"><mrow id="S3.Ex12.m1.1.1.1.1.1" xref="S3.Ex12.m1.1.1.1.1.1.1.cmml"><mo id="S3.Ex12.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex12.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex12.m1.1.1.1.1.1.1" xref="S3.Ex12.m1.1.1.1.1.1.1.cmml"><mn id="S3.Ex12.m1.1.1.1.1.1.1.2" xref="S3.Ex12.m1.1.1.1.1.1.1.2.cmml">2</mn><mo id="S3.Ex12.m1.1.1.1.1.1.1.1" xref="S3.Ex12.m1.1.1.1.1.1.1.1.cmml">+</mo><mfrac id="S3.Ex12.m1.1.1.1.1.1.1.3" xref="S3.Ex12.m1.1.1.1.1.1.1.3.cmml"><mn id="S3.Ex12.m1.1.1.1.1.1.1.3.2" xref="S3.Ex12.m1.1.1.1.1.1.1.3.2.cmml">1</mn><mn id="S3.Ex12.m1.1.1.1.1.1.1.3.3" xref="S3.Ex12.m1.1.1.1.1.1.1.3.3.cmml">6</mn></mfrac></mrow><mo id="S3.Ex12.m1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex12.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex12.m1.2.2.2.3" lspace="0.170em" xref="S3.Ex12.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.Ex12.m1.2.2.2.2.1" xref="S3.Ex12.m1.2.2.2.2.2.cmml"><mo id="S3.Ex12.m1.2.2.2.2.1.2" stretchy="false" xref="S3.Ex12.m1.2.2.2.2.2.1.cmml">|</mo><mrow id="S3.Ex12.m1.2.2.2.2.1.1" xref="S3.Ex12.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex12.m1.2.2.2.2.1.1.2" xref="S3.Ex12.m1.2.2.2.2.1.1.2.cmml">A</mi><mo id="S3.Ex12.m1.2.2.2.2.1.1.1" xref="S3.Ex12.m1.2.2.2.2.1.1.1.cmml">∖</mo><mi id="S3.Ex12.m1.2.2.2.2.1.1.3" xref="S3.Ex12.m1.2.2.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex12.m1.2.2.2.2.1.3" 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xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.1" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.1.cmml">∩</mo><mi id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.3" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex12.m1.3.3.1.1.1.1.1.1.3" rspace="0.222em" stretchy="false" xref="S3.Ex12.m1.3.3.1.1.1.1.1.2.1.cmml">|</mo></mrow></mrow></mrow></mrow><mo id="S3.Ex12.m1.3.3.1.2" rspace="0.0835em" xref="S3.Ex12.m1.3.3.1.1.cmml">.</mo><mo class="ltx_mathvariant_italic" id="S3.Ex12.m1.3.3.1.3" lspace="0.0835em" mathvariant="italic" separator="true" xref="S3.Ex12.m1.3.3.1.1.cmml">∎</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex12.m1.3b"><apply id="S3.Ex12.m1.3.3.1.1.cmml" xref="S3.Ex12.m1.3.3.1"><lt id="S3.Ex12.m1.3.3.1.1.2.cmml" xref="S3.Ex12.m1.3.3.1.1.2"></lt><csymbol cd="latexml" id="S3.Ex12.m1.3.3.1.1.3.cmml" xref="S3.Ex12.m1.3.3.1.1.3">absent</csymbol><apply id="S3.Ex12.m1.3.3.1.1.1.cmml" xref="S3.Ex12.m1.3.3.1.1.1"><plus 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xref="S3.Ex12.m1.3.3.1.1.1.1.1.1"><abs id="S3.Ex12.m1.3.3.1.1.1.1.1.2.1.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.2"></abs><apply id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1"><intersect id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.1"></intersect><ci id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex12.m1.3c">\displaystyle<\frac{(2+\tfrac{1}{6})\,|A\setminus B|}{\ell+2}+3\,|A\cap B|% \enspace.\qed</annotation><annotation encoding="application/x-llamapun" id="S3.Ex12.m1.3d">< divide start_ARG ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG + 3 | italic_A ∩ italic_B | . italic_∎</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">We are now ready to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm1" title="Theorem 1. ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <div class="ltx_proof" id="S3.21"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm1" title="Theorem 1. ‣ 1 Introduction ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1</span></a>.</h6> <div class="ltx_para" id="S3.10.p1"> <p class="ltx_p" id="S3.10.p1.2">Let <math alttext="G" class="ltx_Math" display="inline" id="S3.10.p1.1.m1.1"><semantics id="S3.10.p1.1.m1.1a"><mi id="S3.10.p1.1.m1.1.1" xref="S3.10.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.10.p1.1.m1.1b"><ci id="S3.10.p1.1.m1.1.1.cmml" xref="S3.10.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.1.m1.1d">italic_G</annotation></semantics></math> be a graph and let <math alttext="a:=\operatorname{sn}(G)" class="ltx_Math" display="inline" id="S3.10.p1.2.m2.2"><semantics id="S3.10.p1.2.m2.2a"><mrow id="S3.10.p1.2.m2.2.3" xref="S3.10.p1.2.m2.2.3.cmml"><mi id="S3.10.p1.2.m2.2.3.2" xref="S3.10.p1.2.m2.2.3.2.cmml">a</mi><mo id="S3.10.p1.2.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.10.p1.2.m2.2.3.1.cmml">:=</mo><mrow id="S3.10.p1.2.m2.2.3.3.2" xref="S3.10.p1.2.m2.2.3.3.1.cmml"><mi id="S3.10.p1.2.m2.1.1" xref="S3.10.p1.2.m2.1.1.cmml">sn</mi><mo id="S3.10.p1.2.m2.2.3.3.2a" xref="S3.10.p1.2.m2.2.3.3.1.cmml">⁡</mo><mrow id="S3.10.p1.2.m2.2.3.3.2.1" xref="S3.10.p1.2.m2.2.3.3.1.cmml"><mo id="S3.10.p1.2.m2.2.3.3.2.1.1" stretchy="false" xref="S3.10.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S3.10.p1.2.m2.2.2" xref="S3.10.p1.2.m2.2.2.cmml">G</mi><mo id="S3.10.p1.2.m2.2.3.3.2.1.2" stretchy="false" xref="S3.10.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.2.m2.2b"><apply id="S3.10.p1.2.m2.2.3.cmml" xref="S3.10.p1.2.m2.2.3"><csymbol cd="latexml" id="S3.10.p1.2.m2.2.3.1.cmml" xref="S3.10.p1.2.m2.2.3.1">assign</csymbol><ci id="S3.10.p1.2.m2.2.3.2.cmml" xref="S3.10.p1.2.m2.2.3.2">𝑎</ci><apply id="S3.10.p1.2.m2.2.3.3.1.cmml" xref="S3.10.p1.2.m2.2.3.3.2"><ci id="S3.10.p1.2.m2.1.1.cmml" xref="S3.10.p1.2.m2.1.1">sn</ci><ci id="S3.10.p1.2.m2.2.2.cmml" xref="S3.10.p1.2.m2.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.2.m2.2c">a:=\operatorname{sn}(G)</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.2.m2.2d">italic_a := roman_sn ( italic_G )</annotation></semantics></math>. Let</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="h:=4\quad\text{and}\quad t:=\frac{4h}{1-(2+\tfrac{1}{6})\cdot(\tfrac{2}{3})^{h% }}=\frac{3888}{139}<27.972\enspace." class="ltx_Math" display="block" id="S3.Ex13.m1.5"><semantics id="S3.Ex13.m1.5a"><mrow id="S3.Ex13.m1.5.5.1"><mrow id="S3.Ex13.m1.5.5.1.1.2" xref="S3.Ex13.m1.5.5.1.1.3.cmml"><mrow id="S3.Ex13.m1.5.5.1.1.1.1" xref="S3.Ex13.m1.5.5.1.1.1.1.cmml"><mi id="S3.Ex13.m1.5.5.1.1.1.1.2" xref="S3.Ex13.m1.5.5.1.1.1.1.2.cmml">h</mi><mo id="S3.Ex13.m1.5.5.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Ex13.m1.5.5.1.1.1.1.1.cmml">:=</mo><mrow id="S3.Ex13.m1.5.5.1.1.1.1.3.2" xref="S3.Ex13.m1.5.5.1.1.1.1.3.1.cmml"><mn id="S3.Ex13.m1.3.3" xref="S3.Ex13.m1.3.3.cmml">4</mn><mspace id="S3.Ex13.m1.5.5.1.1.1.1.3.2.1" width="1em" xref="S3.Ex13.m1.5.5.1.1.1.1.3.1.cmml"></mspace><mtext id="S3.Ex13.m1.4.4" xref="S3.Ex13.m1.4.4a.cmml">and</mtext></mrow></mrow><mspace id="S3.Ex13.m1.5.5.1.1.2.3" width="1em" xref="S3.Ex13.m1.5.5.1.1.3a.cmml"></mspace><mrow id="S3.Ex13.m1.5.5.1.1.2.2" xref="S3.Ex13.m1.5.5.1.1.2.2.cmml"><mi id="S3.Ex13.m1.5.5.1.1.2.2.2" xref="S3.Ex13.m1.5.5.1.1.2.2.2.cmml">t</mi><mo id="S3.Ex13.m1.5.5.1.1.2.2.3" lspace="0.278em" rspace="0.278em" xref="S3.Ex13.m1.5.5.1.1.2.2.3.cmml">:=</mo><mfrac id="S3.Ex13.m1.2.2" xref="S3.Ex13.m1.2.2.cmml"><mrow id="S3.Ex13.m1.2.2.4" xref="S3.Ex13.m1.2.2.4.cmml"><mn id="S3.Ex13.m1.2.2.4.2" xref="S3.Ex13.m1.2.2.4.2.cmml">4</mn><mo id="S3.Ex13.m1.2.2.4.1" xref="S3.Ex13.m1.2.2.4.1.cmml">⁢</mo><mi id="S3.Ex13.m1.2.2.4.3" xref="S3.Ex13.m1.2.2.4.3.cmml">h</mi></mrow><mrow id="S3.Ex13.m1.2.2.2" xref="S3.Ex13.m1.2.2.2.cmml"><mn id="S3.Ex13.m1.2.2.2.4" xref="S3.Ex13.m1.2.2.2.4.cmml">1</mn><mo id="S3.Ex13.m1.2.2.2.3" xref="S3.Ex13.m1.2.2.2.3.cmml">−</mo><mrow 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id="S3.Ex13.m1.2.2.2.2.3.2.2" xref="S3.Ex13.m1.1.1.1.1.cmml"><mo id="S3.Ex13.m1.2.2.2.2.3.2.2.1" stretchy="false" xref="S3.Ex13.m1.1.1.1.1.cmml">(</mo><mfrac id="S3.Ex13.m1.1.1.1.1" xref="S3.Ex13.m1.1.1.1.1.cmml"><mn id="S3.Ex13.m1.1.1.1.1.2" xref="S3.Ex13.m1.1.1.1.1.2.cmml">2</mn><mn id="S3.Ex13.m1.1.1.1.1.3" xref="S3.Ex13.m1.1.1.1.1.3.cmml">3</mn></mfrac><mo id="S3.Ex13.m1.2.2.2.2.3.2.2.2" stretchy="false" xref="S3.Ex13.m1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.Ex13.m1.2.2.2.2.3.3" xref="S3.Ex13.m1.2.2.2.2.3.3.cmml">h</mi></msup></mrow></mrow></mfrac><mo id="S3.Ex13.m1.5.5.1.1.2.2.4" xref="S3.Ex13.m1.5.5.1.1.2.2.4.cmml">=</mo><mfrac id="S3.Ex13.m1.5.5.1.1.2.2.5" xref="S3.Ex13.m1.5.5.1.1.2.2.5.cmml"><mn id="S3.Ex13.m1.5.5.1.1.2.2.5.2" xref="S3.Ex13.m1.5.5.1.1.2.2.5.2.cmml">3888</mn><mn id="S3.Ex13.m1.5.5.1.1.2.2.5.3" xref="S3.Ex13.m1.5.5.1.1.2.2.5.3.cmml">139</mn></mfrac><mo id="S3.Ex13.m1.5.5.1.1.2.2.6" xref="S3.Ex13.m1.5.5.1.1.2.2.6.cmml"><</mo><mn id="S3.Ex13.m1.5.5.1.1.2.2.7" xref="S3.Ex13.m1.5.5.1.1.2.2.7.cmml">27.972</mn></mrow></mrow><mo id="S3.Ex13.m1.5.5.1.2" lspace="0.500em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex13.m1.5b"><apply id="S3.Ex13.m1.5.5.1.1.3.cmml" xref="S3.Ex13.m1.5.5.1.1.2"><csymbol cd="ambiguous" id="S3.Ex13.m1.5.5.1.1.3a.cmml" xref="S3.Ex13.m1.5.5.1.1.2.3">formulae-sequence</csymbol><apply id="S3.Ex13.m1.5.5.1.1.1.1.cmml" xref="S3.Ex13.m1.5.5.1.1.1.1"><csymbol cd="latexml" id="S3.Ex13.m1.5.5.1.1.1.1.1.cmml" xref="S3.Ex13.m1.5.5.1.1.1.1.1">assign</csymbol><ci id="S3.Ex13.m1.5.5.1.1.1.1.2.cmml" xref="S3.Ex13.m1.5.5.1.1.1.1.2">ℎ</ci><list id="S3.Ex13.m1.5.5.1.1.1.1.3.1.cmml" xref="S3.Ex13.m1.5.5.1.1.1.1.3.2"><cn id="S3.Ex13.m1.3.3.cmml" type="integer" xref="S3.Ex13.m1.3.3">4</cn><ci id="S3.Ex13.m1.4.4a.cmml" xref="S3.Ex13.m1.4.4"><mtext id="S3.Ex13.m1.4.4.cmml" xref="S3.Ex13.m1.4.4">and</mtext></ci></list></apply><apply id="S3.Ex13.m1.5.5.1.1.2.2.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2"><and 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xref="S3.Ex13.m1.2.2.2.2.3.2.2"></divide><cn id="S3.Ex13.m1.1.1.1.1.2.cmml" type="integer" xref="S3.Ex13.m1.1.1.1.1.2">2</cn><cn id="S3.Ex13.m1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex13.m1.1.1.1.1.3">3</cn></apply><ci id="S3.Ex13.m1.2.2.2.2.3.3.cmml" xref="S3.Ex13.m1.2.2.2.2.3.3">ℎ</ci></apply></apply></apply></apply></apply><apply id="S3.Ex13.m1.5.5.1.1.2.2c.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2"><eq id="S3.Ex13.m1.5.5.1.1.2.2.4.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2.4"></eq><share href="https://arxiv.org/html/2503.17112v1#S3.Ex13.m1.2.2.cmml" id="S3.Ex13.m1.5.5.1.1.2.2d.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2"></share><apply id="S3.Ex13.m1.5.5.1.1.2.2.5.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2.5"><divide id="S3.Ex13.m1.5.5.1.1.2.2.5.1.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2.5"></divide><cn id="S3.Ex13.m1.5.5.1.1.2.2.5.2.cmml" type="integer" xref="S3.Ex13.m1.5.5.1.1.2.2.5.2">3888</cn><cn id="S3.Ex13.m1.5.5.1.1.2.2.5.3.cmml" type="integer" xref="S3.Ex13.m1.5.5.1.1.2.2.5.3">139</cn></apply></apply><apply id="S3.Ex13.m1.5.5.1.1.2.2e.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2"><lt id="S3.Ex13.m1.5.5.1.1.2.2.6.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2.6"></lt><share href="https://arxiv.org/html/2503.17112v1#S3.Ex13.m1.5.5.1.1.2.2.5.cmml" id="S3.Ex13.m1.5.5.1.1.2.2f.cmml" xref="S3.Ex13.m1.5.5.1.1.2.2"></share><cn id="S3.Ex13.m1.5.5.1.1.2.2.7.cmml" type="float" xref="S3.Ex13.m1.5.5.1.1.2.2.7">27.972</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex13.m1.5c">h:=4\quad\text{and}\quad t:=\frac{4h}{1-(2+\tfrac{1}{6})\cdot(\tfrac{2}{3})^{h% }}=\frac{3888}{139}<27.972\enspace.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex13.m1.5d">italic_h := 4 and italic_t := divide start_ARG 4 italic_h end_ARG start_ARG 1 - ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT end_ARG = divide start_ARG 3888 end_ARG start_ARG 139 end_ARG < 27.972 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.10.p1.12">We will show that <math alttext="\operatorname{tw}(G)<(2t+1)a" class="ltx_Math" display="inline" id="S3.10.p1.3.m1.3"><semantics id="S3.10.p1.3.m1.3a"><mrow id="S3.10.p1.3.m1.3.3" xref="S3.10.p1.3.m1.3.3.cmml"><mrow id="S3.10.p1.3.m1.3.3.3.2" xref="S3.10.p1.3.m1.3.3.3.1.cmml"><mi id="S3.10.p1.3.m1.1.1" xref="S3.10.p1.3.m1.1.1.cmml">tw</mi><mo id="S3.10.p1.3.m1.3.3.3.2a" xref="S3.10.p1.3.m1.3.3.3.1.cmml">⁡</mo><mrow id="S3.10.p1.3.m1.3.3.3.2.1" xref="S3.10.p1.3.m1.3.3.3.1.cmml"><mo id="S3.10.p1.3.m1.3.3.3.2.1.1" stretchy="false" xref="S3.10.p1.3.m1.3.3.3.1.cmml">(</mo><mi id="S3.10.p1.3.m1.2.2" xref="S3.10.p1.3.m1.2.2.cmml">G</mi><mo id="S3.10.p1.3.m1.3.3.3.2.1.2" stretchy="false" xref="S3.10.p1.3.m1.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.10.p1.3.m1.3.3.2" xref="S3.10.p1.3.m1.3.3.2.cmml"><</mo><mrow id="S3.10.p1.3.m1.3.3.1" xref="S3.10.p1.3.m1.3.3.1.cmml"><mrow id="S3.10.p1.3.m1.3.3.1.1.1" xref="S3.10.p1.3.m1.3.3.1.1.1.1.cmml"><mo id="S3.10.p1.3.m1.3.3.1.1.1.2" stretchy="false" xref="S3.10.p1.3.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.10.p1.3.m1.3.3.1.1.1.1" xref="S3.10.p1.3.m1.3.3.1.1.1.1.cmml"><mrow id="S3.10.p1.3.m1.3.3.1.1.1.1.2" xref="S3.10.p1.3.m1.3.3.1.1.1.1.2.cmml"><mn id="S3.10.p1.3.m1.3.3.1.1.1.1.2.2" xref="S3.10.p1.3.m1.3.3.1.1.1.1.2.2.cmml">2</mn><mo id="S3.10.p1.3.m1.3.3.1.1.1.1.2.1" xref="S3.10.p1.3.m1.3.3.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.10.p1.3.m1.3.3.1.1.1.1.2.3" xref="S3.10.p1.3.m1.3.3.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.10.p1.3.m1.3.3.1.1.1.1.1" xref="S3.10.p1.3.m1.3.3.1.1.1.1.1.cmml">+</mo><mn id="S3.10.p1.3.m1.3.3.1.1.1.1.3" xref="S3.10.p1.3.m1.3.3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.10.p1.3.m1.3.3.1.1.1.3" stretchy="false" xref="S3.10.p1.3.m1.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S3.10.p1.3.m1.3.3.1.2" xref="S3.10.p1.3.m1.3.3.1.2.cmml">⁢</mo><mi id="S3.10.p1.3.m1.3.3.1.3" xref="S3.10.p1.3.m1.3.3.1.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.3.m1.3b"><apply id="S3.10.p1.3.m1.3.3.cmml" xref="S3.10.p1.3.m1.3.3"><lt id="S3.10.p1.3.m1.3.3.2.cmml" xref="S3.10.p1.3.m1.3.3.2"></lt><apply id="S3.10.p1.3.m1.3.3.3.1.cmml" xref="S3.10.p1.3.m1.3.3.3.2"><ci id="S3.10.p1.3.m1.1.1.cmml" xref="S3.10.p1.3.m1.1.1">tw</ci><ci id="S3.10.p1.3.m1.2.2.cmml" xref="S3.10.p1.3.m1.2.2">𝐺</ci></apply><apply id="S3.10.p1.3.m1.3.3.1.cmml" xref="S3.10.p1.3.m1.3.3.1"><times id="S3.10.p1.3.m1.3.3.1.2.cmml" xref="S3.10.p1.3.m1.3.3.1.2"></times><apply id="S3.10.p1.3.m1.3.3.1.1.1.1.cmml" xref="S3.10.p1.3.m1.3.3.1.1.1"><plus id="S3.10.p1.3.m1.3.3.1.1.1.1.1.cmml" xref="S3.10.p1.3.m1.3.3.1.1.1.1.1"></plus><apply id="S3.10.p1.3.m1.3.3.1.1.1.1.2.cmml" xref="S3.10.p1.3.m1.3.3.1.1.1.1.2"><times id="S3.10.p1.3.m1.3.3.1.1.1.1.2.1.cmml" xref="S3.10.p1.3.m1.3.3.1.1.1.1.2.1"></times><cn id="S3.10.p1.3.m1.3.3.1.1.1.1.2.2.cmml" type="integer" xref="S3.10.p1.3.m1.3.3.1.1.1.1.2.2">2</cn><ci id="S3.10.p1.3.m1.3.3.1.1.1.1.2.3.cmml" xref="S3.10.p1.3.m1.3.3.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.10.p1.3.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.10.p1.3.m1.3.3.1.1.1.1.3">1</cn></apply><ci id="S3.10.p1.3.m1.3.3.1.3.cmml" xref="S3.10.p1.3.m1.3.3.1.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.3.m1.3c">\operatorname{tw}(G)<(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.3.m1.3d">roman_tw ( italic_G ) < ( 2 italic_t + 1 ) italic_a</annotation></semantics></math>. The proof is by induction on the number of vertices of <math alttext="G" class="ltx_Math" display="inline" id="S3.10.p1.4.m2.1"><semantics id="S3.10.p1.4.m2.1a"><mi id="S3.10.p1.4.m2.1.1" xref="S3.10.p1.4.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.10.p1.4.m2.1b"><ci id="S3.10.p1.4.m2.1.1.cmml" xref="S3.10.p1.4.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.4.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.4.m2.1d">italic_G</annotation></semantics></math>. We will prove the following stronger statement: For any graph <math alttext="G" class="ltx_Math" display="inline" id="S3.10.p1.5.m3.1"><semantics id="S3.10.p1.5.m3.1a"><mi id="S3.10.p1.5.m3.1.1" xref="S3.10.p1.5.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.10.p1.5.m3.1b"><ci id="S3.10.p1.5.m3.1.1.cmml" xref="S3.10.p1.5.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.5.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.5.m3.1d">italic_G</annotation></semantics></math> and any non-empty subset <math alttext="W\subseteq V(G)" class="ltx_Math" display="inline" id="S3.10.p1.6.m4.1"><semantics id="S3.10.p1.6.m4.1a"><mrow id="S3.10.p1.6.m4.1.2" xref="S3.10.p1.6.m4.1.2.cmml"><mi id="S3.10.p1.6.m4.1.2.2" xref="S3.10.p1.6.m4.1.2.2.cmml">W</mi><mo id="S3.10.p1.6.m4.1.2.1" xref="S3.10.p1.6.m4.1.2.1.cmml">⊆</mo><mrow id="S3.10.p1.6.m4.1.2.3" xref="S3.10.p1.6.m4.1.2.3.cmml"><mi id="S3.10.p1.6.m4.1.2.3.2" xref="S3.10.p1.6.m4.1.2.3.2.cmml">V</mi><mo id="S3.10.p1.6.m4.1.2.3.1" xref="S3.10.p1.6.m4.1.2.3.1.cmml">⁢</mo><mrow id="S3.10.p1.6.m4.1.2.3.3.2" xref="S3.10.p1.6.m4.1.2.3.cmml"><mo id="S3.10.p1.6.m4.1.2.3.3.2.1" stretchy="false" xref="S3.10.p1.6.m4.1.2.3.cmml">(</mo><mi id="S3.10.p1.6.m4.1.1" xref="S3.10.p1.6.m4.1.1.cmml">G</mi><mo id="S3.10.p1.6.m4.1.2.3.3.2.2" stretchy="false" xref="S3.10.p1.6.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.6.m4.1b"><apply id="S3.10.p1.6.m4.1.2.cmml" xref="S3.10.p1.6.m4.1.2"><subset id="S3.10.p1.6.m4.1.2.1.cmml" xref="S3.10.p1.6.m4.1.2.1"></subset><ci id="S3.10.p1.6.m4.1.2.2.cmml" xref="S3.10.p1.6.m4.1.2.2">𝑊</ci><apply id="S3.10.p1.6.m4.1.2.3.cmml" xref="S3.10.p1.6.m4.1.2.3"><times id="S3.10.p1.6.m4.1.2.3.1.cmml" xref="S3.10.p1.6.m4.1.2.3.1"></times><ci id="S3.10.p1.6.m4.1.2.3.2.cmml" xref="S3.10.p1.6.m4.1.2.3.2">𝑉</ci><ci id="S3.10.p1.6.m4.1.1.cmml" xref="S3.10.p1.6.m4.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.6.m4.1c">W\subseteq V(G)</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.6.m4.1d">italic_W ⊆ italic_V ( italic_G )</annotation></semantics></math> of size at most <math alttext="ta" class="ltx_Math" display="inline" id="S3.10.p1.7.m5.1"><semantics id="S3.10.p1.7.m5.1a"><mrow id="S3.10.p1.7.m5.1.1" xref="S3.10.p1.7.m5.1.1.cmml"><mi id="S3.10.p1.7.m5.1.1.2" xref="S3.10.p1.7.m5.1.1.2.cmml">t</mi><mo id="S3.10.p1.7.m5.1.1.1" xref="S3.10.p1.7.m5.1.1.1.cmml">⁢</mo><mi id="S3.10.p1.7.m5.1.1.3" xref="S3.10.p1.7.m5.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.7.m5.1b"><apply id="S3.10.p1.7.m5.1.1.cmml" xref="S3.10.p1.7.m5.1.1"><times id="S3.10.p1.7.m5.1.1.1.cmml" xref="S3.10.p1.7.m5.1.1.1"></times><ci id="S3.10.p1.7.m5.1.1.2.cmml" xref="S3.10.p1.7.m5.1.1.2">𝑡</ci><ci id="S3.10.p1.7.m5.1.1.3.cmml" xref="S3.10.p1.7.m5.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.7.m5.1c">ta</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.7.m5.1d">italic_t italic_a</annotation></semantics></math>, <math alttext="G" class="ltx_Math" display="inline" id="S3.10.p1.8.m6.1"><semantics id="S3.10.p1.8.m6.1a"><mi id="S3.10.p1.8.m6.1.1" xref="S3.10.p1.8.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.10.p1.8.m6.1b"><ci id="S3.10.p1.8.m6.1.1.cmml" xref="S3.10.p1.8.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.8.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.8.m6.1d">italic_G</annotation></semantics></math> has a tree decomposition <math alttext="(B_{x}:x\in V(T))" class="ltx_Math" display="inline" id="S3.10.p1.9.m7.2"><semantics id="S3.10.p1.9.m7.2a"><mrow id="S3.10.p1.9.m7.2.2.1" xref="S3.10.p1.9.m7.2.2.1.1.cmml"><mo id="S3.10.p1.9.m7.2.2.1.2" stretchy="false" xref="S3.10.p1.9.m7.2.2.1.1.cmml">(</mo><mrow id="S3.10.p1.9.m7.2.2.1.1" xref="S3.10.p1.9.m7.2.2.1.1.cmml"><msub id="S3.10.p1.9.m7.2.2.1.1.2" xref="S3.10.p1.9.m7.2.2.1.1.2.cmml"><mi id="S3.10.p1.9.m7.2.2.1.1.2.2" xref="S3.10.p1.9.m7.2.2.1.1.2.2.cmml">B</mi><mi id="S3.10.p1.9.m7.2.2.1.1.2.3" xref="S3.10.p1.9.m7.2.2.1.1.2.3.cmml">x</mi></msub><mo id="S3.10.p1.9.m7.2.2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.10.p1.9.m7.2.2.1.1.1.cmml">:</mo><mrow id="S3.10.p1.9.m7.2.2.1.1.3" xref="S3.10.p1.9.m7.2.2.1.1.3.cmml"><mi id="S3.10.p1.9.m7.2.2.1.1.3.2" xref="S3.10.p1.9.m7.2.2.1.1.3.2.cmml">x</mi><mo id="S3.10.p1.9.m7.2.2.1.1.3.1" xref="S3.10.p1.9.m7.2.2.1.1.3.1.cmml">∈</mo><mrow id="S3.10.p1.9.m7.2.2.1.1.3.3" xref="S3.10.p1.9.m7.2.2.1.1.3.3.cmml"><mi id="S3.10.p1.9.m7.2.2.1.1.3.3.2" xref="S3.10.p1.9.m7.2.2.1.1.3.3.2.cmml">V</mi><mo id="S3.10.p1.9.m7.2.2.1.1.3.3.1" xref="S3.10.p1.9.m7.2.2.1.1.3.3.1.cmml">⁢</mo><mrow id="S3.10.p1.9.m7.2.2.1.1.3.3.3.2" xref="S3.10.p1.9.m7.2.2.1.1.3.3.cmml"><mo id="S3.10.p1.9.m7.2.2.1.1.3.3.3.2.1" stretchy="false" xref="S3.10.p1.9.m7.2.2.1.1.3.3.cmml">(</mo><mi id="S3.10.p1.9.m7.1.1" xref="S3.10.p1.9.m7.1.1.cmml">T</mi><mo id="S3.10.p1.9.m7.2.2.1.1.3.3.3.2.2" stretchy="false" xref="S3.10.p1.9.m7.2.2.1.1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.10.p1.9.m7.2.2.1.3" stretchy="false" xref="S3.10.p1.9.m7.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.9.m7.2b"><apply id="S3.10.p1.9.m7.2.2.1.1.cmml" xref="S3.10.p1.9.m7.2.2.1"><ci id="S3.10.p1.9.m7.2.2.1.1.1.cmml" xref="S3.10.p1.9.m7.2.2.1.1.1">:</ci><apply id="S3.10.p1.9.m7.2.2.1.1.2.cmml" xref="S3.10.p1.9.m7.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.10.p1.9.m7.2.2.1.1.2.1.cmml" xref="S3.10.p1.9.m7.2.2.1.1.2">subscript</csymbol><ci id="S3.10.p1.9.m7.2.2.1.1.2.2.cmml" xref="S3.10.p1.9.m7.2.2.1.1.2.2">𝐵</ci><ci id="S3.10.p1.9.m7.2.2.1.1.2.3.cmml" xref="S3.10.p1.9.m7.2.2.1.1.2.3">𝑥</ci></apply><apply id="S3.10.p1.9.m7.2.2.1.1.3.cmml" xref="S3.10.p1.9.m7.2.2.1.1.3"><in id="S3.10.p1.9.m7.2.2.1.1.3.1.cmml" xref="S3.10.p1.9.m7.2.2.1.1.3.1"></in><ci id="S3.10.p1.9.m7.2.2.1.1.3.2.cmml" xref="S3.10.p1.9.m7.2.2.1.1.3.2">𝑥</ci><apply id="S3.10.p1.9.m7.2.2.1.1.3.3.cmml" xref="S3.10.p1.9.m7.2.2.1.1.3.3"><times id="S3.10.p1.9.m7.2.2.1.1.3.3.1.cmml" xref="S3.10.p1.9.m7.2.2.1.1.3.3.1"></times><ci id="S3.10.p1.9.m7.2.2.1.1.3.3.2.cmml" xref="S3.10.p1.9.m7.2.2.1.1.3.3.2">𝑉</ci><ci id="S3.10.p1.9.m7.1.1.cmml" xref="S3.10.p1.9.m7.1.1">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.9.m7.2c">(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.9.m7.2d">( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math> of width less than <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="S3.10.p1.10.m8.1"><semantics id="S3.10.p1.10.m8.1a"><mrow id="S3.10.p1.10.m8.1.1" xref="S3.10.p1.10.m8.1.1.cmml"><mrow id="S3.10.p1.10.m8.1.1.1.1" xref="S3.10.p1.10.m8.1.1.1.1.1.cmml"><mo id="S3.10.p1.10.m8.1.1.1.1.2" stretchy="false" xref="S3.10.p1.10.m8.1.1.1.1.1.cmml">(</mo><mrow id="S3.10.p1.10.m8.1.1.1.1.1" xref="S3.10.p1.10.m8.1.1.1.1.1.cmml"><mrow id="S3.10.p1.10.m8.1.1.1.1.1.2" xref="S3.10.p1.10.m8.1.1.1.1.1.2.cmml"><mn id="S3.10.p1.10.m8.1.1.1.1.1.2.2" xref="S3.10.p1.10.m8.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.10.p1.10.m8.1.1.1.1.1.2.1" xref="S3.10.p1.10.m8.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.10.p1.10.m8.1.1.1.1.1.2.3" xref="S3.10.p1.10.m8.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.10.p1.10.m8.1.1.1.1.1.1" xref="S3.10.p1.10.m8.1.1.1.1.1.1.cmml">+</mo><mn id="S3.10.p1.10.m8.1.1.1.1.1.3" xref="S3.10.p1.10.m8.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.10.p1.10.m8.1.1.1.1.3" stretchy="false" xref="S3.10.p1.10.m8.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.10.p1.10.m8.1.1.2" xref="S3.10.p1.10.m8.1.1.2.cmml">⁢</mo><mi id="S3.10.p1.10.m8.1.1.3" xref="S3.10.p1.10.m8.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.10.m8.1b"><apply id="S3.10.p1.10.m8.1.1.cmml" xref="S3.10.p1.10.m8.1.1"><times id="S3.10.p1.10.m8.1.1.2.cmml" xref="S3.10.p1.10.m8.1.1.2"></times><apply id="S3.10.p1.10.m8.1.1.1.1.1.cmml" xref="S3.10.p1.10.m8.1.1.1.1"><plus id="S3.10.p1.10.m8.1.1.1.1.1.1.cmml" xref="S3.10.p1.10.m8.1.1.1.1.1.1"></plus><apply id="S3.10.p1.10.m8.1.1.1.1.1.2.cmml" xref="S3.10.p1.10.m8.1.1.1.1.1.2"><times id="S3.10.p1.10.m8.1.1.1.1.1.2.1.cmml" xref="S3.10.p1.10.m8.1.1.1.1.1.2.1"></times><cn id="S3.10.p1.10.m8.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.10.p1.10.m8.1.1.1.1.1.2.2">2</cn><ci id="S3.10.p1.10.m8.1.1.1.1.1.2.3.cmml" xref="S3.10.p1.10.m8.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.10.p1.10.m8.1.1.1.1.1.3.cmml" type="integer" xref="S3.10.p1.10.m8.1.1.1.1.1.3">1</cn></apply><ci id="S3.10.p1.10.m8.1.1.3.cmml" xref="S3.10.p1.10.m8.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.10.m8.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.10.m8.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math> in which <math alttext="W\subseteq B_{x}" class="ltx_Math" display="inline" id="S3.10.p1.11.m9.1"><semantics id="S3.10.p1.11.m9.1a"><mrow id="S3.10.p1.11.m9.1.1" xref="S3.10.p1.11.m9.1.1.cmml"><mi id="S3.10.p1.11.m9.1.1.2" xref="S3.10.p1.11.m9.1.1.2.cmml">W</mi><mo id="S3.10.p1.11.m9.1.1.1" xref="S3.10.p1.11.m9.1.1.1.cmml">⊆</mo><msub id="S3.10.p1.11.m9.1.1.3" xref="S3.10.p1.11.m9.1.1.3.cmml"><mi id="S3.10.p1.11.m9.1.1.3.2" xref="S3.10.p1.11.m9.1.1.3.2.cmml">B</mi><mi id="S3.10.p1.11.m9.1.1.3.3" xref="S3.10.p1.11.m9.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.11.m9.1b"><apply id="S3.10.p1.11.m9.1.1.cmml" xref="S3.10.p1.11.m9.1.1"><subset id="S3.10.p1.11.m9.1.1.1.cmml" xref="S3.10.p1.11.m9.1.1.1"></subset><ci id="S3.10.p1.11.m9.1.1.2.cmml" xref="S3.10.p1.11.m9.1.1.2">𝑊</ci><apply id="S3.10.p1.11.m9.1.1.3.cmml" xref="S3.10.p1.11.m9.1.1.3"><csymbol cd="ambiguous" id="S3.10.p1.11.m9.1.1.3.1.cmml" xref="S3.10.p1.11.m9.1.1.3">subscript</csymbol><ci id="S3.10.p1.11.m9.1.1.3.2.cmml" xref="S3.10.p1.11.m9.1.1.3.2">𝐵</ci><ci id="S3.10.p1.11.m9.1.1.3.3.cmml" xref="S3.10.p1.11.m9.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.11.m9.1c">W\subseteq B_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.11.m9.1d">italic_W ⊆ italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="x\in V(T)" class="ltx_Math" display="inline" id="S3.10.p1.12.m10.1"><semantics id="S3.10.p1.12.m10.1a"><mrow id="S3.10.p1.12.m10.1.2" xref="S3.10.p1.12.m10.1.2.cmml"><mi id="S3.10.p1.12.m10.1.2.2" xref="S3.10.p1.12.m10.1.2.2.cmml">x</mi><mo id="S3.10.p1.12.m10.1.2.1" xref="S3.10.p1.12.m10.1.2.1.cmml">∈</mo><mrow id="S3.10.p1.12.m10.1.2.3" xref="S3.10.p1.12.m10.1.2.3.cmml"><mi id="S3.10.p1.12.m10.1.2.3.2" xref="S3.10.p1.12.m10.1.2.3.2.cmml">V</mi><mo id="S3.10.p1.12.m10.1.2.3.1" xref="S3.10.p1.12.m10.1.2.3.1.cmml">⁢</mo><mrow id="S3.10.p1.12.m10.1.2.3.3.2" xref="S3.10.p1.12.m10.1.2.3.cmml"><mo id="S3.10.p1.12.m10.1.2.3.3.2.1" stretchy="false" xref="S3.10.p1.12.m10.1.2.3.cmml">(</mo><mi id="S3.10.p1.12.m10.1.1" xref="S3.10.p1.12.m10.1.1.cmml">T</mi><mo id="S3.10.p1.12.m10.1.2.3.3.2.2" stretchy="false" xref="S3.10.p1.12.m10.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.12.m10.1b"><apply id="S3.10.p1.12.m10.1.2.cmml" xref="S3.10.p1.12.m10.1.2"><in id="S3.10.p1.12.m10.1.2.1.cmml" xref="S3.10.p1.12.m10.1.2.1"></in><ci id="S3.10.p1.12.m10.1.2.2.cmml" xref="S3.10.p1.12.m10.1.2.2">𝑥</ci><apply id="S3.10.p1.12.m10.1.2.3.cmml" xref="S3.10.p1.12.m10.1.2.3"><times id="S3.10.p1.12.m10.1.2.3.1.cmml" xref="S3.10.p1.12.m10.1.2.3.1"></times><ci id="S3.10.p1.12.m10.1.2.3.2.cmml" xref="S3.10.p1.12.m10.1.2.3.2">𝑉</ci><ci id="S3.10.p1.12.m10.1.1.cmml" xref="S3.10.p1.12.m10.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.12.m10.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.12.m10.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.11.p2"> <p class="ltx_p" id="S3.11.p2.15">If <math alttext="G" class="ltx_Math" display="inline" id="S3.11.p2.1.m1.1"><semantics id="S3.11.p2.1.m1.1a"><mi id="S3.11.p2.1.m1.1.1" xref="S3.11.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.1.m1.1b"><ci id="S3.11.p2.1.m1.1.1.cmml" xref="S3.11.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.1.m1.1d">italic_G</annotation></semantics></math> has less than <math alttext="ta" class="ltx_Math" display="inline" id="S3.11.p2.2.m2.1"><semantics id="S3.11.p2.2.m2.1a"><mrow id="S3.11.p2.2.m2.1.1" xref="S3.11.p2.2.m2.1.1.cmml"><mi id="S3.11.p2.2.m2.1.1.2" xref="S3.11.p2.2.m2.1.1.2.cmml">t</mi><mo id="S3.11.p2.2.m2.1.1.1" xref="S3.11.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.11.p2.2.m2.1.1.3" xref="S3.11.p2.2.m2.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.2.m2.1b"><apply id="S3.11.p2.2.m2.1.1.cmml" xref="S3.11.p2.2.m2.1.1"><times id="S3.11.p2.2.m2.1.1.1.cmml" xref="S3.11.p2.2.m2.1.1.1"></times><ci id="S3.11.p2.2.m2.1.1.2.cmml" xref="S3.11.p2.2.m2.1.1.2">𝑡</ci><ci id="S3.11.p2.2.m2.1.1.3.cmml" xref="S3.11.p2.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.2.m2.1c">ta</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.2.m2.1d">italic_t italic_a</annotation></semantics></math> vertices, then the proof is trivial. We use a tree <math alttext="T" class="ltx_Math" display="inline" id="S3.11.p2.3.m3.1"><semantics id="S3.11.p2.3.m3.1a"><mi id="S3.11.p2.3.m3.1.1" xref="S3.11.p2.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.3.m3.1b"><ci id="S3.11.p2.3.m3.1.1.cmml" xref="S3.11.p2.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.3.m3.1d">italic_T</annotation></semantics></math> with a single vertex <math alttext="x" class="ltx_Math" display="inline" id="S3.11.p2.4.m4.1"><semantics id="S3.11.p2.4.m4.1a"><mi id="S3.11.p2.4.m4.1.1" xref="S3.11.p2.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.4.m4.1b"><ci id="S3.11.p2.4.m4.1.1.cmml" xref="S3.11.p2.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.4.m4.1d">italic_x</annotation></semantics></math> and set <math alttext="B_{x}:=V(G)" class="ltx_Math" display="inline" id="S3.11.p2.5.m5.1"><semantics id="S3.11.p2.5.m5.1a"><mrow id="S3.11.p2.5.m5.1.2" xref="S3.11.p2.5.m5.1.2.cmml"><msub id="S3.11.p2.5.m5.1.2.2" xref="S3.11.p2.5.m5.1.2.2.cmml"><mi id="S3.11.p2.5.m5.1.2.2.2" xref="S3.11.p2.5.m5.1.2.2.2.cmml">B</mi><mi id="S3.11.p2.5.m5.1.2.2.3" xref="S3.11.p2.5.m5.1.2.2.3.cmml">x</mi></msub><mo id="S3.11.p2.5.m5.1.2.1" lspace="0.278em" rspace="0.278em" xref="S3.11.p2.5.m5.1.2.1.cmml">:=</mo><mrow id="S3.11.p2.5.m5.1.2.3" xref="S3.11.p2.5.m5.1.2.3.cmml"><mi id="S3.11.p2.5.m5.1.2.3.2" xref="S3.11.p2.5.m5.1.2.3.2.cmml">V</mi><mo id="S3.11.p2.5.m5.1.2.3.1" xref="S3.11.p2.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S3.11.p2.5.m5.1.2.3.3.2" xref="S3.11.p2.5.m5.1.2.3.cmml"><mo id="S3.11.p2.5.m5.1.2.3.3.2.1" stretchy="false" xref="S3.11.p2.5.m5.1.2.3.cmml">(</mo><mi id="S3.11.p2.5.m5.1.1" xref="S3.11.p2.5.m5.1.1.cmml">G</mi><mo id="S3.11.p2.5.m5.1.2.3.3.2.2" stretchy="false" xref="S3.11.p2.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.5.m5.1b"><apply id="S3.11.p2.5.m5.1.2.cmml" xref="S3.11.p2.5.m5.1.2"><csymbol cd="latexml" id="S3.11.p2.5.m5.1.2.1.cmml" xref="S3.11.p2.5.m5.1.2.1">assign</csymbol><apply id="S3.11.p2.5.m5.1.2.2.cmml" xref="S3.11.p2.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.11.p2.5.m5.1.2.2.1.cmml" xref="S3.11.p2.5.m5.1.2.2">subscript</csymbol><ci id="S3.11.p2.5.m5.1.2.2.2.cmml" xref="S3.11.p2.5.m5.1.2.2.2">𝐵</ci><ci id="S3.11.p2.5.m5.1.2.2.3.cmml" xref="S3.11.p2.5.m5.1.2.2.3">𝑥</ci></apply><apply id="S3.11.p2.5.m5.1.2.3.cmml" xref="S3.11.p2.5.m5.1.2.3"><times id="S3.11.p2.5.m5.1.2.3.1.cmml" xref="S3.11.p2.5.m5.1.2.3.1"></times><ci id="S3.11.p2.5.m5.1.2.3.2.cmml" xref="S3.11.p2.5.m5.1.2.3.2">𝑉</ci><ci id="S3.11.p2.5.m5.1.1.cmml" xref="S3.11.p2.5.m5.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.5.m5.1c">B_{x}:=V(G)</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.5.m5.1d">italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT := italic_V ( italic_G )</annotation></semantics></math>. We now assume that <math alttext="|V(G)|\geq ta" class="ltx_Math" display="inline" id="S3.11.p2.6.m6.2"><semantics id="S3.11.p2.6.m6.2a"><mrow id="S3.11.p2.6.m6.2.2" xref="S3.11.p2.6.m6.2.2.cmml"><mrow id="S3.11.p2.6.m6.2.2.1.1" xref="S3.11.p2.6.m6.2.2.1.2.cmml"><mo id="S3.11.p2.6.m6.2.2.1.1.2" stretchy="false" xref="S3.11.p2.6.m6.2.2.1.2.1.cmml">|</mo><mrow id="S3.11.p2.6.m6.2.2.1.1.1" xref="S3.11.p2.6.m6.2.2.1.1.1.cmml"><mi id="S3.11.p2.6.m6.2.2.1.1.1.2" xref="S3.11.p2.6.m6.2.2.1.1.1.2.cmml">V</mi><mo id="S3.11.p2.6.m6.2.2.1.1.1.1" xref="S3.11.p2.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.11.p2.6.m6.2.2.1.1.1.3.2" xref="S3.11.p2.6.m6.2.2.1.1.1.cmml"><mo id="S3.11.p2.6.m6.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.11.p2.6.m6.2.2.1.1.1.cmml">(</mo><mi id="S3.11.p2.6.m6.1.1" xref="S3.11.p2.6.m6.1.1.cmml">G</mi><mo id="S3.11.p2.6.m6.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.11.p2.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.11.p2.6.m6.2.2.1.1.3" stretchy="false" xref="S3.11.p2.6.m6.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.11.p2.6.m6.2.2.2" xref="S3.11.p2.6.m6.2.2.2.cmml">≥</mo><mrow id="S3.11.p2.6.m6.2.2.3" xref="S3.11.p2.6.m6.2.2.3.cmml"><mi id="S3.11.p2.6.m6.2.2.3.2" xref="S3.11.p2.6.m6.2.2.3.2.cmml">t</mi><mo id="S3.11.p2.6.m6.2.2.3.1" xref="S3.11.p2.6.m6.2.2.3.1.cmml">⁢</mo><mi id="S3.11.p2.6.m6.2.2.3.3" xref="S3.11.p2.6.m6.2.2.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.6.m6.2b"><apply id="S3.11.p2.6.m6.2.2.cmml" xref="S3.11.p2.6.m6.2.2"><geq id="S3.11.p2.6.m6.2.2.2.cmml" xref="S3.11.p2.6.m6.2.2.2"></geq><apply id="S3.11.p2.6.m6.2.2.1.2.cmml" xref="S3.11.p2.6.m6.2.2.1.1"><abs id="S3.11.p2.6.m6.2.2.1.2.1.cmml" xref="S3.11.p2.6.m6.2.2.1.1.2"></abs><apply id="S3.11.p2.6.m6.2.2.1.1.1.cmml" xref="S3.11.p2.6.m6.2.2.1.1.1"><times id="S3.11.p2.6.m6.2.2.1.1.1.1.cmml" xref="S3.11.p2.6.m6.2.2.1.1.1.1"></times><ci id="S3.11.p2.6.m6.2.2.1.1.1.2.cmml" xref="S3.11.p2.6.m6.2.2.1.1.1.2">𝑉</ci><ci id="S3.11.p2.6.m6.1.1.cmml" xref="S3.11.p2.6.m6.1.1">𝐺</ci></apply></apply><apply id="S3.11.p2.6.m6.2.2.3.cmml" xref="S3.11.p2.6.m6.2.2.3"><times id="S3.11.p2.6.m6.2.2.3.1.cmml" xref="S3.11.p2.6.m6.2.2.3.1"></times><ci id="S3.11.p2.6.m6.2.2.3.2.cmml" xref="S3.11.p2.6.m6.2.2.3.2">𝑡</ci><ci id="S3.11.p2.6.m6.2.2.3.3.cmml" xref="S3.11.p2.6.m6.2.2.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.6.m6.2c">|V(G)|\geq ta</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.6.m6.2d">| italic_V ( italic_G ) | ≥ italic_t italic_a</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm6" title="Lemma 6. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a>, <math alttext="G" class="ltx_Math" display="inline" id="S3.11.p2.7.m7.1"><semantics id="S3.11.p2.7.m7.1a"><mi id="S3.11.p2.7.m7.1.1" xref="S3.11.p2.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.7.m7.1b"><ci id="S3.11.p2.7.m7.1.1.cmml" xref="S3.11.p2.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.7.m7.1d">italic_G</annotation></semantics></math> has a <math alttext="W" class="ltx_Math" display="inline" id="S3.11.p2.8.m8.1"><semantics id="S3.11.p2.8.m8.1a"><mi id="S3.11.p2.8.m8.1.1" xref="S3.11.p2.8.m8.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.8.m8.1b"><ci id="S3.11.p2.8.m8.1.1.cmml" xref="S3.11.p2.8.m8.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.8.m8.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.8.m8.1d">italic_W</annotation></semantics></math>-sequence <math alttext="W_{0},\ldots,W_{\ell+1}" class="ltx_Math" display="inline" id="S3.11.p2.9.m9.3"><semantics id="S3.11.p2.9.m9.3a"><mrow id="S3.11.p2.9.m9.3.3.2" xref="S3.11.p2.9.m9.3.3.3.cmml"><msub id="S3.11.p2.9.m9.2.2.1.1" xref="S3.11.p2.9.m9.2.2.1.1.cmml"><mi id="S3.11.p2.9.m9.2.2.1.1.2" xref="S3.11.p2.9.m9.2.2.1.1.2.cmml">W</mi><mn id="S3.11.p2.9.m9.2.2.1.1.3" xref="S3.11.p2.9.m9.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.11.p2.9.m9.3.3.2.3" xref="S3.11.p2.9.m9.3.3.3.cmml">,</mo><mi id="S3.11.p2.9.m9.1.1" mathvariant="normal" xref="S3.11.p2.9.m9.1.1.cmml">…</mi><mo id="S3.11.p2.9.m9.3.3.2.4" xref="S3.11.p2.9.m9.3.3.3.cmml">,</mo><msub id="S3.11.p2.9.m9.3.3.2.2" xref="S3.11.p2.9.m9.3.3.2.2.cmml"><mi id="S3.11.p2.9.m9.3.3.2.2.2" xref="S3.11.p2.9.m9.3.3.2.2.2.cmml">W</mi><mrow id="S3.11.p2.9.m9.3.3.2.2.3" xref="S3.11.p2.9.m9.3.3.2.2.3.cmml"><mi id="S3.11.p2.9.m9.3.3.2.2.3.2" mathvariant="normal" xref="S3.11.p2.9.m9.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.11.p2.9.m9.3.3.2.2.3.1" xref="S3.11.p2.9.m9.3.3.2.2.3.1.cmml">+</mo><mn id="S3.11.p2.9.m9.3.3.2.2.3.3" xref="S3.11.p2.9.m9.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.9.m9.3b"><list id="S3.11.p2.9.m9.3.3.3.cmml" xref="S3.11.p2.9.m9.3.3.2"><apply id="S3.11.p2.9.m9.2.2.1.1.cmml" xref="S3.11.p2.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S3.11.p2.9.m9.2.2.1.1.1.cmml" xref="S3.11.p2.9.m9.2.2.1.1">subscript</csymbol><ci id="S3.11.p2.9.m9.2.2.1.1.2.cmml" xref="S3.11.p2.9.m9.2.2.1.1.2">𝑊</ci><cn id="S3.11.p2.9.m9.2.2.1.1.3.cmml" type="integer" xref="S3.11.p2.9.m9.2.2.1.1.3">0</cn></apply><ci id="S3.11.p2.9.m9.1.1.cmml" xref="S3.11.p2.9.m9.1.1">…</ci><apply id="S3.11.p2.9.m9.3.3.2.2.cmml" xref="S3.11.p2.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="S3.11.p2.9.m9.3.3.2.2.1.cmml" xref="S3.11.p2.9.m9.3.3.2.2">subscript</csymbol><ci id="S3.11.p2.9.m9.3.3.2.2.2.cmml" xref="S3.11.p2.9.m9.3.3.2.2.2">𝑊</ci><apply id="S3.11.p2.9.m9.3.3.2.2.3.cmml" xref="S3.11.p2.9.m9.3.3.2.2.3"><plus id="S3.11.p2.9.m9.3.3.2.2.3.1.cmml" xref="S3.11.p2.9.m9.3.3.2.2.3.1"></plus><ci id="S3.11.p2.9.m9.3.3.2.2.3.2.cmml" xref="S3.11.p2.9.m9.3.3.2.2.3.2">ℓ</ci><cn id="S3.11.p2.9.m9.3.3.2.2.3.3.cmml" type="integer" xref="S3.11.p2.9.m9.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.9.m9.3c">W_{0},\ldots,W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.9.m9.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> of width <math alttext="|W|" class="ltx_Math" display="inline" id="S3.11.p2.10.m10.1"><semantics id="S3.11.p2.10.m10.1a"><mrow id="S3.11.p2.10.m10.1.2.2" xref="S3.11.p2.10.m10.1.2.1.cmml"><mo id="S3.11.p2.10.m10.1.2.2.1" stretchy="false" xref="S3.11.p2.10.m10.1.2.1.1.cmml">|</mo><mi id="S3.11.p2.10.m10.1.1" xref="S3.11.p2.10.m10.1.1.cmml">W</mi><mo id="S3.11.p2.10.m10.1.2.2.2" stretchy="false" xref="S3.11.p2.10.m10.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.10.m10.1b"><apply id="S3.11.p2.10.m10.1.2.1.cmml" xref="S3.11.p2.10.m10.1.2.2"><abs id="S3.11.p2.10.m10.1.2.1.1.cmml" xref="S3.11.p2.10.m10.1.2.2.1"></abs><ci id="S3.11.p2.10.m10.1.1.cmml" xref="S3.11.p2.10.m10.1.1">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.10.m10.1c">|W|</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.10.m10.1d">| italic_W |</annotation></semantics></math>. Let <math alttext="\Delta_{0},\ldots,\Delta_{\ell+1}" class="ltx_Math" display="inline" id="S3.11.p2.11.m11.3"><semantics id="S3.11.p2.11.m11.3a"><mrow id="S3.11.p2.11.m11.3.3.2" xref="S3.11.p2.11.m11.3.3.3.cmml"><msub id="S3.11.p2.11.m11.2.2.1.1" xref="S3.11.p2.11.m11.2.2.1.1.cmml"><mi id="S3.11.p2.11.m11.2.2.1.1.2" mathvariant="normal" xref="S3.11.p2.11.m11.2.2.1.1.2.cmml">Δ</mi><mn id="S3.11.p2.11.m11.2.2.1.1.3" xref="S3.11.p2.11.m11.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.11.p2.11.m11.3.3.2.3" xref="S3.11.p2.11.m11.3.3.3.cmml">,</mo><mi id="S3.11.p2.11.m11.1.1" mathvariant="normal" xref="S3.11.p2.11.m11.1.1.cmml">…</mi><mo id="S3.11.p2.11.m11.3.3.2.4" xref="S3.11.p2.11.m11.3.3.3.cmml">,</mo><msub id="S3.11.p2.11.m11.3.3.2.2" xref="S3.11.p2.11.m11.3.3.2.2.cmml"><mi id="S3.11.p2.11.m11.3.3.2.2.2" mathvariant="normal" xref="S3.11.p2.11.m11.3.3.2.2.2.cmml">Δ</mi><mrow id="S3.11.p2.11.m11.3.3.2.2.3" xref="S3.11.p2.11.m11.3.3.2.2.3.cmml"><mi id="S3.11.p2.11.m11.3.3.2.2.3.2" mathvariant="normal" xref="S3.11.p2.11.m11.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.11.p2.11.m11.3.3.2.2.3.1" xref="S3.11.p2.11.m11.3.3.2.2.3.1.cmml">+</mo><mn id="S3.11.p2.11.m11.3.3.2.2.3.3" xref="S3.11.p2.11.m11.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.11.m11.3b"><list id="S3.11.p2.11.m11.3.3.3.cmml" xref="S3.11.p2.11.m11.3.3.2"><apply id="S3.11.p2.11.m11.2.2.1.1.cmml" xref="S3.11.p2.11.m11.2.2.1.1"><csymbol cd="ambiguous" id="S3.11.p2.11.m11.2.2.1.1.1.cmml" xref="S3.11.p2.11.m11.2.2.1.1">subscript</csymbol><ci id="S3.11.p2.11.m11.2.2.1.1.2.cmml" xref="S3.11.p2.11.m11.2.2.1.1.2">Δ</ci><cn id="S3.11.p2.11.m11.2.2.1.1.3.cmml" type="integer" xref="S3.11.p2.11.m11.2.2.1.1.3">0</cn></apply><ci id="S3.11.p2.11.m11.1.1.cmml" xref="S3.11.p2.11.m11.1.1">…</ci><apply id="S3.11.p2.11.m11.3.3.2.2.cmml" xref="S3.11.p2.11.m11.3.3.2.2"><csymbol cd="ambiguous" id="S3.11.p2.11.m11.3.3.2.2.1.cmml" xref="S3.11.p2.11.m11.3.3.2.2">subscript</csymbol><ci id="S3.11.p2.11.m11.3.3.2.2.2.cmml" xref="S3.11.p2.11.m11.3.3.2.2.2">Δ</ci><apply id="S3.11.p2.11.m11.3.3.2.2.3.cmml" xref="S3.11.p2.11.m11.3.3.2.2.3"><plus id="S3.11.p2.11.m11.3.3.2.2.3.1.cmml" xref="S3.11.p2.11.m11.3.3.2.2.3.1"></plus><ci id="S3.11.p2.11.m11.3.3.2.2.3.2.cmml" xref="S3.11.p2.11.m11.3.3.2.2.3.2">ℓ</ci><cn id="S3.11.p2.11.m11.3.3.2.2.3.3.cmml" type="integer" xref="S3.11.p2.11.m11.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.11.m11.3c">\Delta_{0},\ldots,\Delta_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.11.m11.3d">roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Z" class="ltx_Math" display="inline" id="S3.11.p2.12.m12.1"><semantics id="S3.11.p2.12.m12.1a"><mi id="S3.11.p2.12.m12.1.1" xref="S3.11.p2.12.m12.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.12.m12.1b"><ci id="S3.11.p2.12.m12.1.1.cmml" xref="S3.11.p2.12.m12.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.12.m12.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.12.m12.1d">italic_Z</annotation></semantics></math> be as in the definition of <math alttext="W" class="ltx_Math" display="inline" id="S3.11.p2.13.m13.1"><semantics id="S3.11.p2.13.m13.1a"><mi id="S3.11.p2.13.m13.1.1" xref="S3.11.p2.13.m13.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.13.m13.1b"><ci id="S3.11.p2.13.m13.1.1.cmml" xref="S3.11.p2.13.m13.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.13.m13.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.13.m13.1d">italic_W</annotation></semantics></math>-sequence. Let <math alttext="(X,Y)" class="ltx_Math" display="inline" id="S3.11.p2.14.m14.2"><semantics id="S3.11.p2.14.m14.2a"><mrow id="S3.11.p2.14.m14.2.3.2" xref="S3.11.p2.14.m14.2.3.1.cmml"><mo id="S3.11.p2.14.m14.2.3.2.1" stretchy="false" xref="S3.11.p2.14.m14.2.3.1.cmml">(</mo><mi id="S3.11.p2.14.m14.1.1" xref="S3.11.p2.14.m14.1.1.cmml">X</mi><mo id="S3.11.p2.14.m14.2.3.2.2" xref="S3.11.p2.14.m14.2.3.1.cmml">,</mo><mi id="S3.11.p2.14.m14.2.2" xref="S3.11.p2.14.m14.2.2.cmml">Y</mi><mo id="S3.11.p2.14.m14.2.3.2.3" stretchy="false" xref="S3.11.p2.14.m14.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.14.m14.2b"><interval closure="open" id="S3.11.p2.14.m14.2.3.1.cmml" xref="S3.11.p2.14.m14.2.3.2"><ci id="S3.11.p2.14.m14.1.1.cmml" xref="S3.11.p2.14.m14.1.1">𝑋</ci><ci id="S3.11.p2.14.m14.2.2.cmml" xref="S3.11.p2.14.m14.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.14.m14.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.14.m14.2d">( italic_X , italic_Y )</annotation></semantics></math> be a <math alttext="(V(G)\setminus W_{\ell},Z,W)" class="ltx_Math" display="inline" id="S3.11.p2.15.m15.4"><semantics id="S3.11.p2.15.m15.4a"><mrow id="S3.11.p2.15.m15.4.4.1" xref="S3.11.p2.15.m15.4.4.2.cmml"><mo id="S3.11.p2.15.m15.4.4.1.2" stretchy="false" xref="S3.11.p2.15.m15.4.4.2.cmml">(</mo><mrow id="S3.11.p2.15.m15.4.4.1.1" xref="S3.11.p2.15.m15.4.4.1.1.cmml"><mrow id="S3.11.p2.15.m15.4.4.1.1.2" xref="S3.11.p2.15.m15.4.4.1.1.2.cmml"><mi id="S3.11.p2.15.m15.4.4.1.1.2.2" xref="S3.11.p2.15.m15.4.4.1.1.2.2.cmml">V</mi><mo id="S3.11.p2.15.m15.4.4.1.1.2.1" xref="S3.11.p2.15.m15.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S3.11.p2.15.m15.4.4.1.1.2.3.2" xref="S3.11.p2.15.m15.4.4.1.1.2.cmml"><mo id="S3.11.p2.15.m15.4.4.1.1.2.3.2.1" stretchy="false" xref="S3.11.p2.15.m15.4.4.1.1.2.cmml">(</mo><mi id="S3.11.p2.15.m15.1.1" xref="S3.11.p2.15.m15.1.1.cmml">G</mi><mo id="S3.11.p2.15.m15.4.4.1.1.2.3.2.2" stretchy="false" xref="S3.11.p2.15.m15.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.11.p2.15.m15.4.4.1.1.1" xref="S3.11.p2.15.m15.4.4.1.1.1.cmml">∖</mo><msub id="S3.11.p2.15.m15.4.4.1.1.3" xref="S3.11.p2.15.m15.4.4.1.1.3.cmml"><mi id="S3.11.p2.15.m15.4.4.1.1.3.2" xref="S3.11.p2.15.m15.4.4.1.1.3.2.cmml">W</mi><mi id="S3.11.p2.15.m15.4.4.1.1.3.3" mathvariant="normal" xref="S3.11.p2.15.m15.4.4.1.1.3.3.cmml">ℓ</mi></msub></mrow><mo id="S3.11.p2.15.m15.4.4.1.3" xref="S3.11.p2.15.m15.4.4.2.cmml">,</mo><mi id="S3.11.p2.15.m15.2.2" xref="S3.11.p2.15.m15.2.2.cmml">Z</mi><mo id="S3.11.p2.15.m15.4.4.1.4" xref="S3.11.p2.15.m15.4.4.2.cmml">,</mo><mi id="S3.11.p2.15.m15.3.3" xref="S3.11.p2.15.m15.3.3.cmml">W</mi><mo id="S3.11.p2.15.m15.4.4.1.5" stretchy="false" xref="S3.11.p2.15.m15.4.4.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.15.m15.4b"><vector id="S3.11.p2.15.m15.4.4.2.cmml" xref="S3.11.p2.15.m15.4.4.1"><apply id="S3.11.p2.15.m15.4.4.1.1.cmml" xref="S3.11.p2.15.m15.4.4.1.1"><setdiff id="S3.11.p2.15.m15.4.4.1.1.1.cmml" xref="S3.11.p2.15.m15.4.4.1.1.1"></setdiff><apply id="S3.11.p2.15.m15.4.4.1.1.2.cmml" xref="S3.11.p2.15.m15.4.4.1.1.2"><times id="S3.11.p2.15.m15.4.4.1.1.2.1.cmml" xref="S3.11.p2.15.m15.4.4.1.1.2.1"></times><ci id="S3.11.p2.15.m15.4.4.1.1.2.2.cmml" xref="S3.11.p2.15.m15.4.4.1.1.2.2">𝑉</ci><ci id="S3.11.p2.15.m15.1.1.cmml" xref="S3.11.p2.15.m15.1.1">𝐺</ci></apply><apply id="S3.11.p2.15.m15.4.4.1.1.3.cmml" xref="S3.11.p2.15.m15.4.4.1.1.3"><csymbol cd="ambiguous" id="S3.11.p2.15.m15.4.4.1.1.3.1.cmml" xref="S3.11.p2.15.m15.4.4.1.1.3">subscript</csymbol><ci id="S3.11.p2.15.m15.4.4.1.1.3.2.cmml" xref="S3.11.p2.15.m15.4.4.1.1.3.2">𝑊</ci><ci id="S3.11.p2.15.m15.4.4.1.1.3.3.cmml" xref="S3.11.p2.15.m15.4.4.1.1.3.3">ℓ</ci></apply></apply><ci id="S3.11.p2.15.m15.2.2.cmml" xref="S3.11.p2.15.m15.2.2">𝑍</ci><ci id="S3.11.p2.15.m15.3.3.cmml" xref="S3.11.p2.15.m15.3.3">𝑊</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.15.m15.4c">(V(G)\setminus W_{\ell},Z,W)</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.15.m15.4d">( italic_V ( italic_G ) ∖ italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_Z , italic_W )</annotation></semantics></math>-separation.</p> </div> <div class="ltx_para" id="S3.12.p3"> <p class="ltx_p" id="S3.12.p3.16">Since <math alttext="|X\cap Y|=|Z|<|W|\leq ta" class="ltx_Math" display="inline" id="S3.12.p3.1.m1.3"><semantics id="S3.12.p3.1.m1.3a"><mrow id="S3.12.p3.1.m1.3.3" xref="S3.12.p3.1.m1.3.3.cmml"><mrow id="S3.12.p3.1.m1.3.3.1.1" xref="S3.12.p3.1.m1.3.3.1.2.cmml"><mo id="S3.12.p3.1.m1.3.3.1.1.2" stretchy="false" xref="S3.12.p3.1.m1.3.3.1.2.1.cmml">|</mo><mrow id="S3.12.p3.1.m1.3.3.1.1.1" xref="S3.12.p3.1.m1.3.3.1.1.1.cmml"><mi id="S3.12.p3.1.m1.3.3.1.1.1.2" xref="S3.12.p3.1.m1.3.3.1.1.1.2.cmml">X</mi><mo id="S3.12.p3.1.m1.3.3.1.1.1.1" xref="S3.12.p3.1.m1.3.3.1.1.1.1.cmml">∩</mo><mi id="S3.12.p3.1.m1.3.3.1.1.1.3" xref="S3.12.p3.1.m1.3.3.1.1.1.3.cmml">Y</mi></mrow><mo id="S3.12.p3.1.m1.3.3.1.1.3" stretchy="false" xref="S3.12.p3.1.m1.3.3.1.2.1.cmml">|</mo></mrow><mo id="S3.12.p3.1.m1.3.3.3" xref="S3.12.p3.1.m1.3.3.3.cmml">=</mo><mrow id="S3.12.p3.1.m1.3.3.4.2" xref="S3.12.p3.1.m1.3.3.4.1.cmml"><mo id="S3.12.p3.1.m1.3.3.4.2.1" stretchy="false" xref="S3.12.p3.1.m1.3.3.4.1.1.cmml">|</mo><mi id="S3.12.p3.1.m1.1.1" xref="S3.12.p3.1.m1.1.1.cmml">Z</mi><mo id="S3.12.p3.1.m1.3.3.4.2.2" stretchy="false" xref="S3.12.p3.1.m1.3.3.4.1.1.cmml">|</mo></mrow><mo id="S3.12.p3.1.m1.3.3.5" xref="S3.12.p3.1.m1.3.3.5.cmml"><</mo><mrow id="S3.12.p3.1.m1.3.3.6.2" xref="S3.12.p3.1.m1.3.3.6.1.cmml"><mo id="S3.12.p3.1.m1.3.3.6.2.1" stretchy="false" xref="S3.12.p3.1.m1.3.3.6.1.1.cmml">|</mo><mi id="S3.12.p3.1.m1.2.2" xref="S3.12.p3.1.m1.2.2.cmml">W</mi><mo id="S3.12.p3.1.m1.3.3.6.2.2" stretchy="false" xref="S3.12.p3.1.m1.3.3.6.1.1.cmml">|</mo></mrow><mo id="S3.12.p3.1.m1.3.3.7" xref="S3.12.p3.1.m1.3.3.7.cmml">≤</mo><mrow id="S3.12.p3.1.m1.3.3.8" xref="S3.12.p3.1.m1.3.3.8.cmml"><mi id="S3.12.p3.1.m1.3.3.8.2" xref="S3.12.p3.1.m1.3.3.8.2.cmml">t</mi><mo id="S3.12.p3.1.m1.3.3.8.1" xref="S3.12.p3.1.m1.3.3.8.1.cmml">⁢</mo><mi id="S3.12.p3.1.m1.3.3.8.3" xref="S3.12.p3.1.m1.3.3.8.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.1.m1.3b"><apply id="S3.12.p3.1.m1.3.3.cmml" xref="S3.12.p3.1.m1.3.3"><and id="S3.12.p3.1.m1.3.3a.cmml" xref="S3.12.p3.1.m1.3.3"></and><apply id="S3.12.p3.1.m1.3.3b.cmml" xref="S3.12.p3.1.m1.3.3"><eq id="S3.12.p3.1.m1.3.3.3.cmml" xref="S3.12.p3.1.m1.3.3.3"></eq><apply id="S3.12.p3.1.m1.3.3.1.2.cmml" xref="S3.12.p3.1.m1.3.3.1.1"><abs id="S3.12.p3.1.m1.3.3.1.2.1.cmml" xref="S3.12.p3.1.m1.3.3.1.1.2"></abs><apply id="S3.12.p3.1.m1.3.3.1.1.1.cmml" xref="S3.12.p3.1.m1.3.3.1.1.1"><intersect id="S3.12.p3.1.m1.3.3.1.1.1.1.cmml" xref="S3.12.p3.1.m1.3.3.1.1.1.1"></intersect><ci id="S3.12.p3.1.m1.3.3.1.1.1.2.cmml" xref="S3.12.p3.1.m1.3.3.1.1.1.2">𝑋</ci><ci id="S3.12.p3.1.m1.3.3.1.1.1.3.cmml" xref="S3.12.p3.1.m1.3.3.1.1.1.3">𝑌</ci></apply></apply><apply id="S3.12.p3.1.m1.3.3.4.1.cmml" xref="S3.12.p3.1.m1.3.3.4.2"><abs id="S3.12.p3.1.m1.3.3.4.1.1.cmml" xref="S3.12.p3.1.m1.3.3.4.2.1"></abs><ci id="S3.12.p3.1.m1.1.1.cmml" xref="S3.12.p3.1.m1.1.1">𝑍</ci></apply></apply><apply id="S3.12.p3.1.m1.3.3c.cmml" xref="S3.12.p3.1.m1.3.3"><lt id="S3.12.p3.1.m1.3.3.5.cmml" xref="S3.12.p3.1.m1.3.3.5"></lt><share href="https://arxiv.org/html/2503.17112v1#S3.12.p3.1.m1.3.3.4.cmml" id="S3.12.p3.1.m1.3.3d.cmml" xref="S3.12.p3.1.m1.3.3"></share><apply id="S3.12.p3.1.m1.3.3.6.1.cmml" xref="S3.12.p3.1.m1.3.3.6.2"><abs id="S3.12.p3.1.m1.3.3.6.1.1.cmml" xref="S3.12.p3.1.m1.3.3.6.2.1"></abs><ci id="S3.12.p3.1.m1.2.2.cmml" xref="S3.12.p3.1.m1.2.2">𝑊</ci></apply></apply><apply id="S3.12.p3.1.m1.3.3e.cmml" xref="S3.12.p3.1.m1.3.3"><leq id="S3.12.p3.1.m1.3.3.7.cmml" xref="S3.12.p3.1.m1.3.3.7"></leq><share href="https://arxiv.org/html/2503.17112v1#S3.12.p3.1.m1.3.3.6.cmml" id="S3.12.p3.1.m1.3.3f.cmml" xref="S3.12.p3.1.m1.3.3"></share><apply id="S3.12.p3.1.m1.3.3.8.cmml" xref="S3.12.p3.1.m1.3.3.8"><times id="S3.12.p3.1.m1.3.3.8.1.cmml" xref="S3.12.p3.1.m1.3.3.8.1"></times><ci id="S3.12.p3.1.m1.3.3.8.2.cmml" xref="S3.12.p3.1.m1.3.3.8.2">𝑡</ci><ci id="S3.12.p3.1.m1.3.3.8.3.cmml" xref="S3.12.p3.1.m1.3.3.8.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.1.m1.3c">|X\cap Y|=|Z|<|W|\leq ta</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.1.m1.3d">| italic_X ∩ italic_Y | = | italic_Z | < | italic_W | ≤ italic_t italic_a</annotation></semantics></math>, the inductive hypothesis implies that <math alttext="G[X]" class="ltx_Math" display="inline" id="S3.12.p3.2.m2.1"><semantics id="S3.12.p3.2.m2.1a"><mrow id="S3.12.p3.2.m2.1.2" xref="S3.12.p3.2.m2.1.2.cmml"><mi id="S3.12.p3.2.m2.1.2.2" xref="S3.12.p3.2.m2.1.2.2.cmml">G</mi><mo id="S3.12.p3.2.m2.1.2.1" xref="S3.12.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.12.p3.2.m2.1.2.3.2" xref="S3.12.p3.2.m2.1.2.3.1.cmml"><mo id="S3.12.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S3.12.p3.2.m2.1.2.3.1.1.cmml">[</mo><mi id="S3.12.p3.2.m2.1.1" xref="S3.12.p3.2.m2.1.1.cmml">X</mi><mo id="S3.12.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S3.12.p3.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.2.m2.1b"><apply id="S3.12.p3.2.m2.1.2.cmml" xref="S3.12.p3.2.m2.1.2"><times id="S3.12.p3.2.m2.1.2.1.cmml" xref="S3.12.p3.2.m2.1.2.1"></times><ci id="S3.12.p3.2.m2.1.2.2.cmml" xref="S3.12.p3.2.m2.1.2.2">𝐺</ci><apply id="S3.12.p3.2.m2.1.2.3.1.cmml" xref="S3.12.p3.2.m2.1.2.3.2"><csymbol cd="latexml" id="S3.12.p3.2.m2.1.2.3.1.1.cmml" xref="S3.12.p3.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.12.p3.2.m2.1.1.cmml" xref="S3.12.p3.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.2.m2.1c">G[X]</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.2.m2.1d">italic_G [ italic_X ]</annotation></semantics></math> has a tree decomposition <math alttext="\mathcal{T}_{X}:=(B_{x}:x\in V(T_{X}))" class="ltx_math_unparsed" display="inline" id="S3.12.p3.3.m3.1"><semantics id="S3.12.p3.3.m3.1a"><mrow id="S3.12.p3.3.m3.1b"><msub id="S3.12.p3.3.m3.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.12.p3.3.m3.1.1.2">𝒯</mi><mi id="S3.12.p3.3.m3.1.1.3">X</mi></msub><mo id="S3.12.p3.3.m3.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S3.12.p3.3.m3.1.3"><mo id="S3.12.p3.3.m3.1.3.1" stretchy="false">(</mo><msub id="S3.12.p3.3.m3.1.3.2"><mi id="S3.12.p3.3.m3.1.3.2.2">B</mi><mi id="S3.12.p3.3.m3.1.3.2.3">x</mi></msub><mo id="S3.12.p3.3.m3.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.12.p3.3.m3.1.3.4">x</mi><mo id="S3.12.p3.3.m3.1.3.5">∈</mo><mi id="S3.12.p3.3.m3.1.3.6">V</mi><mrow id="S3.12.p3.3.m3.1.3.7"><mo id="S3.12.p3.3.m3.1.3.7.1" stretchy="false">(</mo><msub id="S3.12.p3.3.m3.1.3.7.2"><mi id="S3.12.p3.3.m3.1.3.7.2.2">T</mi><mi id="S3.12.p3.3.m3.1.3.7.2.3">X</mi></msub><mo id="S3.12.p3.3.m3.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S3.12.p3.3.m3.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.12.p3.3.m3.1c">\mathcal{T}_{X}:=(B_{x}:x\in V(T_{X}))</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.3.m3.1d">caligraphic_T start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) )</annotation></semantics></math> of width less than <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="S3.12.p3.4.m4.1"><semantics id="S3.12.p3.4.m4.1a"><mrow id="S3.12.p3.4.m4.1.1" xref="S3.12.p3.4.m4.1.1.cmml"><mrow id="S3.12.p3.4.m4.1.1.1.1" xref="S3.12.p3.4.m4.1.1.1.1.1.cmml"><mo id="S3.12.p3.4.m4.1.1.1.1.2" stretchy="false" xref="S3.12.p3.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.12.p3.4.m4.1.1.1.1.1" xref="S3.12.p3.4.m4.1.1.1.1.1.cmml"><mrow id="S3.12.p3.4.m4.1.1.1.1.1.2" xref="S3.12.p3.4.m4.1.1.1.1.1.2.cmml"><mn id="S3.12.p3.4.m4.1.1.1.1.1.2.2" xref="S3.12.p3.4.m4.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.12.p3.4.m4.1.1.1.1.1.2.1" xref="S3.12.p3.4.m4.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.12.p3.4.m4.1.1.1.1.1.2.3" xref="S3.12.p3.4.m4.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.12.p3.4.m4.1.1.1.1.1.1" xref="S3.12.p3.4.m4.1.1.1.1.1.1.cmml">+</mo><mn id="S3.12.p3.4.m4.1.1.1.1.1.3" xref="S3.12.p3.4.m4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.12.p3.4.m4.1.1.1.1.3" stretchy="false" xref="S3.12.p3.4.m4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.12.p3.4.m4.1.1.2" xref="S3.12.p3.4.m4.1.1.2.cmml">⁢</mo><mi id="S3.12.p3.4.m4.1.1.3" xref="S3.12.p3.4.m4.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.4.m4.1b"><apply id="S3.12.p3.4.m4.1.1.cmml" xref="S3.12.p3.4.m4.1.1"><times id="S3.12.p3.4.m4.1.1.2.cmml" xref="S3.12.p3.4.m4.1.1.2"></times><apply id="S3.12.p3.4.m4.1.1.1.1.1.cmml" xref="S3.12.p3.4.m4.1.1.1.1"><plus id="S3.12.p3.4.m4.1.1.1.1.1.1.cmml" xref="S3.12.p3.4.m4.1.1.1.1.1.1"></plus><apply id="S3.12.p3.4.m4.1.1.1.1.1.2.cmml" xref="S3.12.p3.4.m4.1.1.1.1.1.2"><times id="S3.12.p3.4.m4.1.1.1.1.1.2.1.cmml" xref="S3.12.p3.4.m4.1.1.1.1.1.2.1"></times><cn id="S3.12.p3.4.m4.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.12.p3.4.m4.1.1.1.1.1.2.2">2</cn><ci id="S3.12.p3.4.m4.1.1.1.1.1.2.3.cmml" xref="S3.12.p3.4.m4.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.12.p3.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S3.12.p3.4.m4.1.1.1.1.1.3">1</cn></apply><ci id="S3.12.p3.4.m4.1.1.3.cmml" xref="S3.12.p3.4.m4.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.4.m4.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.4.m4.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math> in which <math alttext="Z\subseteq B_{x}" class="ltx_Math" display="inline" id="S3.12.p3.5.m5.1"><semantics id="S3.12.p3.5.m5.1a"><mrow id="S3.12.p3.5.m5.1.1" xref="S3.12.p3.5.m5.1.1.cmml"><mi id="S3.12.p3.5.m5.1.1.2" xref="S3.12.p3.5.m5.1.1.2.cmml">Z</mi><mo id="S3.12.p3.5.m5.1.1.1" xref="S3.12.p3.5.m5.1.1.1.cmml">⊆</mo><msub id="S3.12.p3.5.m5.1.1.3" xref="S3.12.p3.5.m5.1.1.3.cmml"><mi id="S3.12.p3.5.m5.1.1.3.2" xref="S3.12.p3.5.m5.1.1.3.2.cmml">B</mi><mi id="S3.12.p3.5.m5.1.1.3.3" xref="S3.12.p3.5.m5.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.5.m5.1b"><apply id="S3.12.p3.5.m5.1.1.cmml" xref="S3.12.p3.5.m5.1.1"><subset id="S3.12.p3.5.m5.1.1.1.cmml" xref="S3.12.p3.5.m5.1.1.1"></subset><ci id="S3.12.p3.5.m5.1.1.2.cmml" xref="S3.12.p3.5.m5.1.1.2">𝑍</ci><apply id="S3.12.p3.5.m5.1.1.3.cmml" xref="S3.12.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.12.p3.5.m5.1.1.3.1.cmml" xref="S3.12.p3.5.m5.1.1.3">subscript</csymbol><ci id="S3.12.p3.5.m5.1.1.3.2.cmml" xref="S3.12.p3.5.m5.1.1.3.2">𝐵</ci><ci id="S3.12.p3.5.m5.1.1.3.3.cmml" xref="S3.12.p3.5.m5.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.5.m5.1c">Z\subseteq B_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.5.m5.1d">italic_Z ⊆ italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="x\in V(T_{X})" class="ltx_Math" display="inline" id="S3.12.p3.6.m6.1"><semantics id="S3.12.p3.6.m6.1a"><mrow id="S3.12.p3.6.m6.1.1" xref="S3.12.p3.6.m6.1.1.cmml"><mi id="S3.12.p3.6.m6.1.1.3" xref="S3.12.p3.6.m6.1.1.3.cmml">x</mi><mo id="S3.12.p3.6.m6.1.1.2" xref="S3.12.p3.6.m6.1.1.2.cmml">∈</mo><mrow id="S3.12.p3.6.m6.1.1.1" xref="S3.12.p3.6.m6.1.1.1.cmml"><mi id="S3.12.p3.6.m6.1.1.1.3" xref="S3.12.p3.6.m6.1.1.1.3.cmml">V</mi><mo id="S3.12.p3.6.m6.1.1.1.2" xref="S3.12.p3.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S3.12.p3.6.m6.1.1.1.1.1" xref="S3.12.p3.6.m6.1.1.1.1.1.1.cmml"><mo id="S3.12.p3.6.m6.1.1.1.1.1.2" stretchy="false" xref="S3.12.p3.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="S3.12.p3.6.m6.1.1.1.1.1.1" xref="S3.12.p3.6.m6.1.1.1.1.1.1.cmml"><mi id="S3.12.p3.6.m6.1.1.1.1.1.1.2" xref="S3.12.p3.6.m6.1.1.1.1.1.1.2.cmml">T</mi><mi id="S3.12.p3.6.m6.1.1.1.1.1.1.3" xref="S3.12.p3.6.m6.1.1.1.1.1.1.3.cmml">X</mi></msub><mo id="S3.12.p3.6.m6.1.1.1.1.1.3" stretchy="false" xref="S3.12.p3.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.6.m6.1b"><apply id="S3.12.p3.6.m6.1.1.cmml" xref="S3.12.p3.6.m6.1.1"><in id="S3.12.p3.6.m6.1.1.2.cmml" xref="S3.12.p3.6.m6.1.1.2"></in><ci id="S3.12.p3.6.m6.1.1.3.cmml" xref="S3.12.p3.6.m6.1.1.3">𝑥</ci><apply id="S3.12.p3.6.m6.1.1.1.cmml" xref="S3.12.p3.6.m6.1.1.1"><times id="S3.12.p3.6.m6.1.1.1.2.cmml" xref="S3.12.p3.6.m6.1.1.1.2"></times><ci id="S3.12.p3.6.m6.1.1.1.3.cmml" xref="S3.12.p3.6.m6.1.1.1.3">𝑉</ci><apply id="S3.12.p3.6.m6.1.1.1.1.1.1.cmml" xref="S3.12.p3.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.12.p3.6.m6.1.1.1.1.1.1.1.cmml" xref="S3.12.p3.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S3.12.p3.6.m6.1.1.1.1.1.1.2.cmml" xref="S3.12.p3.6.m6.1.1.1.1.1.1.2">𝑇</ci><ci id="S3.12.p3.6.m6.1.1.1.1.1.1.3.cmml" xref="S3.12.p3.6.m6.1.1.1.1.1.1.3">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.6.m6.1c">x\in V(T_{X})</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.6.m6.1d">italic_x ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT )</annotation></semantics></math>. To finish the proof, we construct a tree decomposition <math alttext="\mathcal{T}_{Y}:=(B_{y}:y\in V(T_{Y}))" class="ltx_math_unparsed" display="inline" id="S3.12.p3.7.m7.1"><semantics id="S3.12.p3.7.m7.1a"><mrow id="S3.12.p3.7.m7.1b"><msub id="S3.12.p3.7.m7.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.12.p3.7.m7.1.1.2">𝒯</mi><mi id="S3.12.p3.7.m7.1.1.3">Y</mi></msub><mo id="S3.12.p3.7.m7.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S3.12.p3.7.m7.1.3"><mo id="S3.12.p3.7.m7.1.3.1" stretchy="false">(</mo><msub id="S3.12.p3.7.m7.1.3.2"><mi id="S3.12.p3.7.m7.1.3.2.2">B</mi><mi id="S3.12.p3.7.m7.1.3.2.3">y</mi></msub><mo id="S3.12.p3.7.m7.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.12.p3.7.m7.1.3.4">y</mi><mo id="S3.12.p3.7.m7.1.3.5">∈</mo><mi id="S3.12.p3.7.m7.1.3.6">V</mi><mrow id="S3.12.p3.7.m7.1.3.7"><mo id="S3.12.p3.7.m7.1.3.7.1" stretchy="false">(</mo><msub id="S3.12.p3.7.m7.1.3.7.2"><mi id="S3.12.p3.7.m7.1.3.7.2.2">T</mi><mi id="S3.12.p3.7.m7.1.3.7.2.3">Y</mi></msub><mo id="S3.12.p3.7.m7.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S3.12.p3.7.m7.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.12.p3.7.m7.1c">\mathcal{T}_{Y}:=(B_{y}:y\in V(T_{Y}))</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.7.m7.1d">caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT := ( italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT : italic_y ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) )</annotation></semantics></math> of <math alttext="G[Y]" class="ltx_Math" display="inline" id="S3.12.p3.8.m8.1"><semantics id="S3.12.p3.8.m8.1a"><mrow id="S3.12.p3.8.m8.1.2" xref="S3.12.p3.8.m8.1.2.cmml"><mi id="S3.12.p3.8.m8.1.2.2" xref="S3.12.p3.8.m8.1.2.2.cmml">G</mi><mo id="S3.12.p3.8.m8.1.2.1" xref="S3.12.p3.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3.12.p3.8.m8.1.2.3.2" xref="S3.12.p3.8.m8.1.2.3.1.cmml"><mo id="S3.12.p3.8.m8.1.2.3.2.1" stretchy="false" xref="S3.12.p3.8.m8.1.2.3.1.1.cmml">[</mo><mi id="S3.12.p3.8.m8.1.1" xref="S3.12.p3.8.m8.1.1.cmml">Y</mi><mo id="S3.12.p3.8.m8.1.2.3.2.2" stretchy="false" xref="S3.12.p3.8.m8.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.8.m8.1b"><apply id="S3.12.p3.8.m8.1.2.cmml" xref="S3.12.p3.8.m8.1.2"><times id="S3.12.p3.8.m8.1.2.1.cmml" xref="S3.12.p3.8.m8.1.2.1"></times><ci id="S3.12.p3.8.m8.1.2.2.cmml" xref="S3.12.p3.8.m8.1.2.2">𝐺</ci><apply id="S3.12.p3.8.m8.1.2.3.1.cmml" xref="S3.12.p3.8.m8.1.2.3.2"><csymbol cd="latexml" id="S3.12.p3.8.m8.1.2.3.1.1.cmml" xref="S3.12.p3.8.m8.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.12.p3.8.m8.1.1.cmml" xref="S3.12.p3.8.m8.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.8.m8.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.8.m8.1d">italic_G [ italic_Y ]</annotation></semantics></math> of width less than <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="S3.12.p3.9.m9.1"><semantics id="S3.12.p3.9.m9.1a"><mrow id="S3.12.p3.9.m9.1.1" xref="S3.12.p3.9.m9.1.1.cmml"><mrow id="S3.12.p3.9.m9.1.1.1.1" xref="S3.12.p3.9.m9.1.1.1.1.1.cmml"><mo id="S3.12.p3.9.m9.1.1.1.1.2" stretchy="false" xref="S3.12.p3.9.m9.1.1.1.1.1.cmml">(</mo><mrow id="S3.12.p3.9.m9.1.1.1.1.1" xref="S3.12.p3.9.m9.1.1.1.1.1.cmml"><mrow id="S3.12.p3.9.m9.1.1.1.1.1.2" xref="S3.12.p3.9.m9.1.1.1.1.1.2.cmml"><mn id="S3.12.p3.9.m9.1.1.1.1.1.2.2" xref="S3.12.p3.9.m9.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.12.p3.9.m9.1.1.1.1.1.2.1" xref="S3.12.p3.9.m9.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.12.p3.9.m9.1.1.1.1.1.2.3" xref="S3.12.p3.9.m9.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.12.p3.9.m9.1.1.1.1.1.1" xref="S3.12.p3.9.m9.1.1.1.1.1.1.cmml">+</mo><mn id="S3.12.p3.9.m9.1.1.1.1.1.3" xref="S3.12.p3.9.m9.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.12.p3.9.m9.1.1.1.1.3" stretchy="false" xref="S3.12.p3.9.m9.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.12.p3.9.m9.1.1.2" xref="S3.12.p3.9.m9.1.1.2.cmml">⁢</mo><mi id="S3.12.p3.9.m9.1.1.3" xref="S3.12.p3.9.m9.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.9.m9.1b"><apply id="S3.12.p3.9.m9.1.1.cmml" xref="S3.12.p3.9.m9.1.1"><times id="S3.12.p3.9.m9.1.1.2.cmml" xref="S3.12.p3.9.m9.1.1.2"></times><apply id="S3.12.p3.9.m9.1.1.1.1.1.cmml" xref="S3.12.p3.9.m9.1.1.1.1"><plus id="S3.12.p3.9.m9.1.1.1.1.1.1.cmml" xref="S3.12.p3.9.m9.1.1.1.1.1.1"></plus><apply id="S3.12.p3.9.m9.1.1.1.1.1.2.cmml" xref="S3.12.p3.9.m9.1.1.1.1.1.2"><times id="S3.12.p3.9.m9.1.1.1.1.1.2.1.cmml" xref="S3.12.p3.9.m9.1.1.1.1.1.2.1"></times><cn id="S3.12.p3.9.m9.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.12.p3.9.m9.1.1.1.1.1.2.2">2</cn><ci id="S3.12.p3.9.m9.1.1.1.1.1.2.3.cmml" xref="S3.12.p3.9.m9.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.12.p3.9.m9.1.1.1.1.1.3.cmml" type="integer" xref="S3.12.p3.9.m9.1.1.1.1.1.3">1</cn></apply><ci id="S3.12.p3.9.m9.1.1.3.cmml" xref="S3.12.p3.9.m9.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.9.m9.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.9.m9.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math> in which some bag <math alttext="B_{y}" class="ltx_Math" display="inline" id="S3.12.p3.10.m10.1"><semantics id="S3.12.p3.10.m10.1a"><msub id="S3.12.p3.10.m10.1.1" xref="S3.12.p3.10.m10.1.1.cmml"><mi id="S3.12.p3.10.m10.1.1.2" xref="S3.12.p3.10.m10.1.1.2.cmml">B</mi><mi id="S3.12.p3.10.m10.1.1.3" xref="S3.12.p3.10.m10.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S3.12.p3.10.m10.1b"><apply id="S3.12.p3.10.m10.1.1.cmml" xref="S3.12.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S3.12.p3.10.m10.1.1.1.cmml" xref="S3.12.p3.10.m10.1.1">subscript</csymbol><ci id="S3.12.p3.10.m10.1.1.2.cmml" xref="S3.12.p3.10.m10.1.1.2">𝐵</ci><ci id="S3.12.p3.10.m10.1.1.3.cmml" xref="S3.12.p3.10.m10.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.10.m10.1c">B_{y}</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.10.m10.1d">italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> contains <math alttext="W\cup Z" class="ltx_Math" display="inline" id="S3.12.p3.11.m11.1"><semantics id="S3.12.p3.11.m11.1a"><mrow id="S3.12.p3.11.m11.1.1" xref="S3.12.p3.11.m11.1.1.cmml"><mi id="S3.12.p3.11.m11.1.1.2" xref="S3.12.p3.11.m11.1.1.2.cmml">W</mi><mo id="S3.12.p3.11.m11.1.1.1" xref="S3.12.p3.11.m11.1.1.1.cmml">∪</mo><mi id="S3.12.p3.11.m11.1.1.3" xref="S3.12.p3.11.m11.1.1.3.cmml">Z</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.11.m11.1b"><apply id="S3.12.p3.11.m11.1.1.cmml" xref="S3.12.p3.11.m11.1.1"><union id="S3.12.p3.11.m11.1.1.1.cmml" xref="S3.12.p3.11.m11.1.1.1"></union><ci id="S3.12.p3.11.m11.1.1.2.cmml" xref="S3.12.p3.11.m11.1.1.2">𝑊</ci><ci id="S3.12.p3.11.m11.1.1.3.cmml" xref="S3.12.p3.11.m11.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.11.m11.1c">W\cup Z</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.11.m11.1d">italic_W ∪ italic_Z</annotation></semantics></math>. Then the tree <math alttext="T" class="ltx_Math" display="inline" id="S3.12.p3.12.m12.1"><semantics id="S3.12.p3.12.m12.1a"><mi id="S3.12.p3.12.m12.1.1" xref="S3.12.p3.12.m12.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.12.p3.12.m12.1b"><ci id="S3.12.p3.12.m12.1.1.cmml" xref="S3.12.p3.12.m12.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.12.m12.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.12.m12.1d">italic_T</annotation></semantics></math> obtained by joining <math alttext="T_{Y}" class="ltx_Math" display="inline" id="S3.12.p3.13.m13.1"><semantics id="S3.12.p3.13.m13.1a"><msub id="S3.12.p3.13.m13.1.1" xref="S3.12.p3.13.m13.1.1.cmml"><mi id="S3.12.p3.13.m13.1.1.2" xref="S3.12.p3.13.m13.1.1.2.cmml">T</mi><mi id="S3.12.p3.13.m13.1.1.3" xref="S3.12.p3.13.m13.1.1.3.cmml">Y</mi></msub><annotation-xml encoding="MathML-Content" id="S3.12.p3.13.m13.1b"><apply id="S3.12.p3.13.m13.1.1.cmml" xref="S3.12.p3.13.m13.1.1"><csymbol cd="ambiguous" id="S3.12.p3.13.m13.1.1.1.cmml" xref="S3.12.p3.13.m13.1.1">subscript</csymbol><ci id="S3.12.p3.13.m13.1.1.2.cmml" xref="S3.12.p3.13.m13.1.1.2">𝑇</ci><ci id="S3.12.p3.13.m13.1.1.3.cmml" xref="S3.12.p3.13.m13.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.13.m13.1c">T_{Y}</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.13.m13.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="T_{X}" class="ltx_Math" display="inline" id="S3.12.p3.14.m14.1"><semantics id="S3.12.p3.14.m14.1a"><msub id="S3.12.p3.14.m14.1.1" xref="S3.12.p3.14.m14.1.1.cmml"><mi id="S3.12.p3.14.m14.1.1.2" xref="S3.12.p3.14.m14.1.1.2.cmml">T</mi><mi id="S3.12.p3.14.m14.1.1.3" xref="S3.12.p3.14.m14.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S3.12.p3.14.m14.1b"><apply id="S3.12.p3.14.m14.1.1.cmml" xref="S3.12.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S3.12.p3.14.m14.1.1.1.cmml" xref="S3.12.p3.14.m14.1.1">subscript</csymbol><ci id="S3.12.p3.14.m14.1.1.2.cmml" xref="S3.12.p3.14.m14.1.1.2">𝑇</ci><ci id="S3.12.p3.14.m14.1.1.3.cmml" xref="S3.12.p3.14.m14.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.14.m14.1c">T_{X}</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.14.m14.1d">italic_T start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> using the edge <math alttext="xy" class="ltx_Math" display="inline" id="S3.12.p3.15.m15.1"><semantics id="S3.12.p3.15.m15.1a"><mrow id="S3.12.p3.15.m15.1.1" xref="S3.12.p3.15.m15.1.1.cmml"><mi id="S3.12.p3.15.m15.1.1.2" xref="S3.12.p3.15.m15.1.1.2.cmml">x</mi><mo id="S3.12.p3.15.m15.1.1.1" xref="S3.12.p3.15.m15.1.1.1.cmml">⁢</mo><mi id="S3.12.p3.15.m15.1.1.3" xref="S3.12.p3.15.m15.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.15.m15.1b"><apply id="S3.12.p3.15.m15.1.1.cmml" xref="S3.12.p3.15.m15.1.1"><times id="S3.12.p3.15.m15.1.1.1.cmml" xref="S3.12.p3.15.m15.1.1.1"></times><ci id="S3.12.p3.15.m15.1.1.2.cmml" xref="S3.12.p3.15.m15.1.1.2">𝑥</ci><ci id="S3.12.p3.15.m15.1.1.3.cmml" xref="S3.12.p3.15.m15.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.15.m15.1c">xy</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.15.m15.1d">italic_x italic_y</annotation></semantics></math> gives the desired tree decomposition <math alttext="\mathcal{T}:=(B_{x}:x\in V(T))" class="ltx_math_unparsed" display="inline" id="S3.12.p3.16.m16.1"><semantics id="S3.12.p3.16.m16.1a"><mrow id="S3.12.p3.16.m16.1b"><mi class="ltx_font_mathcaligraphic" id="S3.12.p3.16.m16.1.1">𝒯</mi><mo id="S3.12.p3.16.m16.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S3.12.p3.16.m16.1.3"><mo id="S3.12.p3.16.m16.1.3.1" stretchy="false">(</mo><msub id="S3.12.p3.16.m16.1.3.2"><mi id="S3.12.p3.16.m16.1.3.2.2">B</mi><mi id="S3.12.p3.16.m16.1.3.2.3">x</mi></msub><mo id="S3.12.p3.16.m16.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.12.p3.16.m16.1.3.4">x</mi><mo id="S3.12.p3.16.m16.1.3.5">∈</mo><mi id="S3.12.p3.16.m16.1.3.6">V</mi><mrow id="S3.12.p3.16.m16.1.3.7"><mo id="S3.12.p3.16.m16.1.3.7.1" stretchy="false">(</mo><mi id="S3.12.p3.16.m16.1.3.7.2">T</mi><mo id="S3.12.p3.16.m16.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S3.12.p3.16.m16.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.12.p3.16.m16.1c">\mathcal{T}:=(B_{x}:x\in V(T))</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.16.m16.1d">caligraphic_T := ( italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT : italic_x ∈ italic_V ( italic_T ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.13.p4"> <p class="ltx_p" id="S3.13.p4.8">If <math alttext="\ell=0" class="ltx_Math" display="inline" id="S3.13.p4.1.m1.1"><semantics id="S3.13.p4.1.m1.1a"><mrow id="S3.13.p4.1.m1.1.1" xref="S3.13.p4.1.m1.1.1.cmml"><mi id="S3.13.p4.1.m1.1.1.2" mathvariant="normal" xref="S3.13.p4.1.m1.1.1.2.cmml">ℓ</mi><mo id="S3.13.p4.1.m1.1.1.1" xref="S3.13.p4.1.m1.1.1.1.cmml">=</mo><mn id="S3.13.p4.1.m1.1.1.3" xref="S3.13.p4.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.13.p4.1.m1.1b"><apply id="S3.13.p4.1.m1.1.1.cmml" xref="S3.13.p4.1.m1.1.1"><eq id="S3.13.p4.1.m1.1.1.1.cmml" xref="S3.13.p4.1.m1.1.1.1"></eq><ci id="S3.13.p4.1.m1.1.1.2.cmml" xref="S3.13.p4.1.m1.1.1.2">ℓ</ci><cn id="S3.13.p4.1.m1.1.1.3.cmml" type="integer" xref="S3.13.p4.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.p4.1.m1.1c">\ell=0</annotation><annotation encoding="application/x-llamapun" id="S3.13.p4.1.m1.1d">roman_ℓ = 0</annotation></semantics></math> then we use the trivial tree decomposition in which <math alttext="T_{Y}" class="ltx_Math" display="inline" id="S3.13.p4.2.m2.1"><semantics id="S3.13.p4.2.m2.1a"><msub id="S3.13.p4.2.m2.1.1" xref="S3.13.p4.2.m2.1.1.cmml"><mi id="S3.13.p4.2.m2.1.1.2" xref="S3.13.p4.2.m2.1.1.2.cmml">T</mi><mi id="S3.13.p4.2.m2.1.1.3" xref="S3.13.p4.2.m2.1.1.3.cmml">Y</mi></msub><annotation-xml encoding="MathML-Content" id="S3.13.p4.2.m2.1b"><apply id="S3.13.p4.2.m2.1.1.cmml" xref="S3.13.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.13.p4.2.m2.1.1.1.cmml" xref="S3.13.p4.2.m2.1.1">subscript</csymbol><ci id="S3.13.p4.2.m2.1.1.2.cmml" xref="S3.13.p4.2.m2.1.1.2">𝑇</ci><ci id="S3.13.p4.2.m2.1.1.3.cmml" xref="S3.13.p4.2.m2.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.p4.2.m2.1c">T_{Y}</annotation><annotation encoding="application/x-llamapun" id="S3.13.p4.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math> has a single node <math alttext="y" class="ltx_Math" display="inline" id="S3.13.p4.3.m3.1"><semantics id="S3.13.p4.3.m3.1a"><mi id="S3.13.p4.3.m3.1.1" xref="S3.13.p4.3.m3.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.13.p4.3.m3.1b"><ci id="S3.13.p4.3.m3.1.1.cmml" xref="S3.13.p4.3.m3.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.13.p4.3.m3.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.13.p4.3.m3.1d">italic_y</annotation></semantics></math> where <math alttext="B_{y}:=W\cup Z" class="ltx_Math" display="inline" id="S3.13.p4.4.m4.1"><semantics id="S3.13.p4.4.m4.1a"><mrow id="S3.13.p4.4.m4.1.1" xref="S3.13.p4.4.m4.1.1.cmml"><msub id="S3.13.p4.4.m4.1.1.2" xref="S3.13.p4.4.m4.1.1.2.cmml"><mi id="S3.13.p4.4.m4.1.1.2.2" xref="S3.13.p4.4.m4.1.1.2.2.cmml">B</mi><mi id="S3.13.p4.4.m4.1.1.2.3" xref="S3.13.p4.4.m4.1.1.2.3.cmml">y</mi></msub><mo id="S3.13.p4.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.13.p4.4.m4.1.1.1.cmml">:=</mo><mrow id="S3.13.p4.4.m4.1.1.3" xref="S3.13.p4.4.m4.1.1.3.cmml"><mi id="S3.13.p4.4.m4.1.1.3.2" xref="S3.13.p4.4.m4.1.1.3.2.cmml">W</mi><mo id="S3.13.p4.4.m4.1.1.3.1" xref="S3.13.p4.4.m4.1.1.3.1.cmml">∪</mo><mi id="S3.13.p4.4.m4.1.1.3.3" xref="S3.13.p4.4.m4.1.1.3.3.cmml">Z</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.13.p4.4.m4.1b"><apply id="S3.13.p4.4.m4.1.1.cmml" xref="S3.13.p4.4.m4.1.1"><csymbol cd="latexml" id="S3.13.p4.4.m4.1.1.1.cmml" xref="S3.13.p4.4.m4.1.1.1">assign</csymbol><apply id="S3.13.p4.4.m4.1.1.2.cmml" xref="S3.13.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.13.p4.4.m4.1.1.2.1.cmml" xref="S3.13.p4.4.m4.1.1.2">subscript</csymbol><ci id="S3.13.p4.4.m4.1.1.2.2.cmml" xref="S3.13.p4.4.m4.1.1.2.2">𝐵</ci><ci id="S3.13.p4.4.m4.1.1.2.3.cmml" xref="S3.13.p4.4.m4.1.1.2.3">𝑦</ci></apply><apply id="S3.13.p4.4.m4.1.1.3.cmml" xref="S3.13.p4.4.m4.1.1.3"><union id="S3.13.p4.4.m4.1.1.3.1.cmml" xref="S3.13.p4.4.m4.1.1.3.1"></union><ci id="S3.13.p4.4.m4.1.1.3.2.cmml" xref="S3.13.p4.4.m4.1.1.3.2">𝑊</ci><ci id="S3.13.p4.4.m4.1.1.3.3.cmml" xref="S3.13.p4.4.m4.1.1.3.3">𝑍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.p4.4.m4.1c">B_{y}:=W\cup Z</annotation><annotation encoding="application/x-llamapun" id="S3.13.p4.4.m4.1d">italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT := italic_W ∪ italic_Z</annotation></semantics></math>. Since <math alttext="W\cup Z\subseteq Y" class="ltx_Math" display="inline" id="S3.13.p4.5.m5.1"><semantics id="S3.13.p4.5.m5.1a"><mrow id="S3.13.p4.5.m5.1.1" xref="S3.13.p4.5.m5.1.1.cmml"><mrow id="S3.13.p4.5.m5.1.1.2" xref="S3.13.p4.5.m5.1.1.2.cmml"><mi id="S3.13.p4.5.m5.1.1.2.2" xref="S3.13.p4.5.m5.1.1.2.2.cmml">W</mi><mo id="S3.13.p4.5.m5.1.1.2.1" xref="S3.13.p4.5.m5.1.1.2.1.cmml">∪</mo><mi id="S3.13.p4.5.m5.1.1.2.3" xref="S3.13.p4.5.m5.1.1.2.3.cmml">Z</mi></mrow><mo id="S3.13.p4.5.m5.1.1.1" xref="S3.13.p4.5.m5.1.1.1.cmml">⊆</mo><mi id="S3.13.p4.5.m5.1.1.3" xref="S3.13.p4.5.m5.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.13.p4.5.m5.1b"><apply id="S3.13.p4.5.m5.1.1.cmml" xref="S3.13.p4.5.m5.1.1"><subset id="S3.13.p4.5.m5.1.1.1.cmml" xref="S3.13.p4.5.m5.1.1.1"></subset><apply id="S3.13.p4.5.m5.1.1.2.cmml" xref="S3.13.p4.5.m5.1.1.2"><union id="S3.13.p4.5.m5.1.1.2.1.cmml" xref="S3.13.p4.5.m5.1.1.2.1"></union><ci id="S3.13.p4.5.m5.1.1.2.2.cmml" xref="S3.13.p4.5.m5.1.1.2.2">𝑊</ci><ci id="S3.13.p4.5.m5.1.1.2.3.cmml" xref="S3.13.p4.5.m5.1.1.2.3">𝑍</ci></apply><ci id="S3.13.p4.5.m5.1.1.3.cmml" xref="S3.13.p4.5.m5.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.p4.5.m5.1c">W\cup Z\subseteq Y</annotation><annotation encoding="application/x-llamapun" id="S3.13.p4.5.m5.1d">italic_W ∪ italic_Z ⊆ italic_Y</annotation></semantics></math> and <math alttext="|Y|=|W_{1}|=|W|+|Z|<2|W|<(2t+1)a" class="ltx_Math" display="inline" id="S3.13.p4.6.m6.7"><semantics id="S3.13.p4.6.m6.7a"><mrow id="S3.13.p4.6.m6.7.7" xref="S3.13.p4.6.m6.7.7.cmml"><mrow id="S3.13.p4.6.m6.7.7.4.2" xref="S3.13.p4.6.m6.7.7.4.1.cmml"><mo id="S3.13.p4.6.m6.7.7.4.2.1" stretchy="false" xref="S3.13.p4.6.m6.7.7.4.1.1.cmml">|</mo><mi id="S3.13.p4.6.m6.1.1" xref="S3.13.p4.6.m6.1.1.cmml">Y</mi><mo id="S3.13.p4.6.m6.7.7.4.2.2" stretchy="false" xref="S3.13.p4.6.m6.7.7.4.1.1.cmml">|</mo></mrow><mo id="S3.13.p4.6.m6.7.7.5" xref="S3.13.p4.6.m6.7.7.5.cmml">=</mo><mrow id="S3.13.p4.6.m6.6.6.1.1" xref="S3.13.p4.6.m6.6.6.1.2.cmml"><mo id="S3.13.p4.6.m6.6.6.1.1.2" stretchy="false" xref="S3.13.p4.6.m6.6.6.1.2.1.cmml">|</mo><msub id="S3.13.p4.6.m6.6.6.1.1.1" xref="S3.13.p4.6.m6.6.6.1.1.1.cmml"><mi id="S3.13.p4.6.m6.6.6.1.1.1.2" xref="S3.13.p4.6.m6.6.6.1.1.1.2.cmml">W</mi><mn id="S3.13.p4.6.m6.6.6.1.1.1.3" xref="S3.13.p4.6.m6.6.6.1.1.1.3.cmml">1</mn></msub><mo id="S3.13.p4.6.m6.6.6.1.1.3" stretchy="false" xref="S3.13.p4.6.m6.6.6.1.2.1.cmml">|</mo></mrow><mo id="S3.13.p4.6.m6.7.7.6" xref="S3.13.p4.6.m6.7.7.6.cmml">=</mo><mrow id="S3.13.p4.6.m6.7.7.7" xref="S3.13.p4.6.m6.7.7.7.cmml"><mrow id="S3.13.p4.6.m6.7.7.7.2.2" xref="S3.13.p4.6.m6.7.7.7.2.1.cmml"><mo id="S3.13.p4.6.m6.7.7.7.2.2.1" stretchy="false" xref="S3.13.p4.6.m6.7.7.7.2.1.1.cmml">|</mo><mi id="S3.13.p4.6.m6.2.2" xref="S3.13.p4.6.m6.2.2.cmml">W</mi><mo id="S3.13.p4.6.m6.7.7.7.2.2.2" stretchy="false" xref="S3.13.p4.6.m6.7.7.7.2.1.1.cmml">|</mo></mrow><mo id="S3.13.p4.6.m6.7.7.7.1" xref="S3.13.p4.6.m6.7.7.7.1.cmml">+</mo><mrow id="S3.13.p4.6.m6.7.7.7.3.2" xref="S3.13.p4.6.m6.7.7.7.3.1.cmml"><mo id="S3.13.p4.6.m6.7.7.7.3.2.1" stretchy="false" xref="S3.13.p4.6.m6.7.7.7.3.1.1.cmml">|</mo><mi id="S3.13.p4.6.m6.4.4" xref="S3.13.p4.6.m6.4.4.cmml">Z</mi><mo id="S3.13.p4.6.m6.7.7.7.3.2.2" stretchy="false" xref="S3.13.p4.6.m6.7.7.7.3.1.1.cmml">|</mo></mrow></mrow><mo id="S3.13.p4.6.m6.7.7.8" xref="S3.13.p4.6.m6.7.7.8.cmml"><</mo><mrow id="S3.13.p4.6.m6.7.7.9" xref="S3.13.p4.6.m6.7.7.9.cmml"><mn id="S3.13.p4.6.m6.3.3" xref="S3.13.p4.6.m6.3.3.cmml">2</mn><mo id="S3.13.p4.6.m6.7.7.9.1" xref="S3.13.p4.6.m6.7.7.9.1.cmml">⁢</mo><mrow id="S3.13.p4.6.m6.7.7.9.2.2" xref="S3.13.p4.6.m6.7.7.9.2.1.cmml"><mo id="S3.13.p4.6.m6.7.7.9.2.2.1" stretchy="false" xref="S3.13.p4.6.m6.7.7.9.2.1.1.cmml">|</mo><mi id="S3.13.p4.6.m6.5.5" xref="S3.13.p4.6.m6.5.5.cmml">W</mi><mo id="S3.13.p4.6.m6.7.7.9.2.2.2" stretchy="false" xref="S3.13.p4.6.m6.7.7.9.2.1.1.cmml">|</mo></mrow></mrow><mo id="S3.13.p4.6.m6.7.7.10" xref="S3.13.p4.6.m6.7.7.10.cmml"><</mo><mrow id="S3.13.p4.6.m6.7.7.2" xref="S3.13.p4.6.m6.7.7.2.cmml"><mrow id="S3.13.p4.6.m6.7.7.2.1.1" xref="S3.13.p4.6.m6.7.7.2.1.1.1.cmml"><mo id="S3.13.p4.6.m6.7.7.2.1.1.2" stretchy="false" xref="S3.13.p4.6.m6.7.7.2.1.1.1.cmml">(</mo><mrow id="S3.13.p4.6.m6.7.7.2.1.1.1" xref="S3.13.p4.6.m6.7.7.2.1.1.1.cmml"><mrow id="S3.13.p4.6.m6.7.7.2.1.1.1.2" xref="S3.13.p4.6.m6.7.7.2.1.1.1.2.cmml"><mn id="S3.13.p4.6.m6.7.7.2.1.1.1.2.2" xref="S3.13.p4.6.m6.7.7.2.1.1.1.2.2.cmml">2</mn><mo id="S3.13.p4.6.m6.7.7.2.1.1.1.2.1" xref="S3.13.p4.6.m6.7.7.2.1.1.1.2.1.cmml">⁢</mo><mi id="S3.13.p4.6.m6.7.7.2.1.1.1.2.3" xref="S3.13.p4.6.m6.7.7.2.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.13.p4.6.m6.7.7.2.1.1.1.1" xref="S3.13.p4.6.m6.7.7.2.1.1.1.1.cmml">+</mo><mn id="S3.13.p4.6.m6.7.7.2.1.1.1.3" xref="S3.13.p4.6.m6.7.7.2.1.1.1.3.cmml">1</mn></mrow><mo id="S3.13.p4.6.m6.7.7.2.1.1.3" stretchy="false" xref="S3.13.p4.6.m6.7.7.2.1.1.1.cmml">)</mo></mrow><mo id="S3.13.p4.6.m6.7.7.2.2" xref="S3.13.p4.6.m6.7.7.2.2.cmml">⁢</mo><mi id="S3.13.p4.6.m6.7.7.2.3" xref="S3.13.p4.6.m6.7.7.2.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.13.p4.6.m6.7b"><apply id="S3.13.p4.6.m6.7.7.cmml" xref="S3.13.p4.6.m6.7.7"><and id="S3.13.p4.6.m6.7.7a.cmml" xref="S3.13.p4.6.m6.7.7"></and><apply id="S3.13.p4.6.m6.7.7b.cmml" xref="S3.13.p4.6.m6.7.7"><eq id="S3.13.p4.6.m6.7.7.5.cmml" xref="S3.13.p4.6.m6.7.7.5"></eq><apply id="S3.13.p4.6.m6.7.7.4.1.cmml" xref="S3.13.p4.6.m6.7.7.4.2"><abs id="S3.13.p4.6.m6.7.7.4.1.1.cmml" xref="S3.13.p4.6.m6.7.7.4.2.1"></abs><ci id="S3.13.p4.6.m6.1.1.cmml" xref="S3.13.p4.6.m6.1.1">𝑌</ci></apply><apply id="S3.13.p4.6.m6.6.6.1.2.cmml" xref="S3.13.p4.6.m6.6.6.1.1"><abs id="S3.13.p4.6.m6.6.6.1.2.1.cmml" xref="S3.13.p4.6.m6.6.6.1.1.2"></abs><apply id="S3.13.p4.6.m6.6.6.1.1.1.cmml" xref="S3.13.p4.6.m6.6.6.1.1.1"><csymbol cd="ambiguous" id="S3.13.p4.6.m6.6.6.1.1.1.1.cmml" xref="S3.13.p4.6.m6.6.6.1.1.1">subscript</csymbol><ci id="S3.13.p4.6.m6.6.6.1.1.1.2.cmml" xref="S3.13.p4.6.m6.6.6.1.1.1.2">𝑊</ci><cn id="S3.13.p4.6.m6.6.6.1.1.1.3.cmml" type="integer" xref="S3.13.p4.6.m6.6.6.1.1.1.3">1</cn></apply></apply></apply><apply id="S3.13.p4.6.m6.7.7c.cmml" xref="S3.13.p4.6.m6.7.7"><eq id="S3.13.p4.6.m6.7.7.6.cmml" xref="S3.13.p4.6.m6.7.7.6"></eq><share href="https://arxiv.org/html/2503.17112v1#S3.13.p4.6.m6.6.6.1.cmml" id="S3.13.p4.6.m6.7.7d.cmml" xref="S3.13.p4.6.m6.7.7"></share><apply id="S3.13.p4.6.m6.7.7.7.cmml" xref="S3.13.p4.6.m6.7.7.7"><plus id="S3.13.p4.6.m6.7.7.7.1.cmml" xref="S3.13.p4.6.m6.7.7.7.1"></plus><apply id="S3.13.p4.6.m6.7.7.7.2.1.cmml" xref="S3.13.p4.6.m6.7.7.7.2.2"><abs id="S3.13.p4.6.m6.7.7.7.2.1.1.cmml" xref="S3.13.p4.6.m6.7.7.7.2.2.1"></abs><ci id="S3.13.p4.6.m6.2.2.cmml" xref="S3.13.p4.6.m6.2.2">𝑊</ci></apply><apply id="S3.13.p4.6.m6.7.7.7.3.1.cmml" xref="S3.13.p4.6.m6.7.7.7.3.2"><abs id="S3.13.p4.6.m6.7.7.7.3.1.1.cmml" xref="S3.13.p4.6.m6.7.7.7.3.2.1"></abs><ci id="S3.13.p4.6.m6.4.4.cmml" xref="S3.13.p4.6.m6.4.4">𝑍</ci></apply></apply></apply><apply id="S3.13.p4.6.m6.7.7e.cmml" xref="S3.13.p4.6.m6.7.7"><lt id="S3.13.p4.6.m6.7.7.8.cmml" xref="S3.13.p4.6.m6.7.7.8"></lt><share href="https://arxiv.org/html/2503.17112v1#S3.13.p4.6.m6.7.7.7.cmml" id="S3.13.p4.6.m6.7.7f.cmml" xref="S3.13.p4.6.m6.7.7"></share><apply id="S3.13.p4.6.m6.7.7.9.cmml" xref="S3.13.p4.6.m6.7.7.9"><times id="S3.13.p4.6.m6.7.7.9.1.cmml" xref="S3.13.p4.6.m6.7.7.9.1"></times><cn id="S3.13.p4.6.m6.3.3.cmml" type="integer" xref="S3.13.p4.6.m6.3.3">2</cn><apply id="S3.13.p4.6.m6.7.7.9.2.1.cmml" xref="S3.13.p4.6.m6.7.7.9.2.2"><abs id="S3.13.p4.6.m6.7.7.9.2.1.1.cmml" xref="S3.13.p4.6.m6.7.7.9.2.2.1"></abs><ci id="S3.13.p4.6.m6.5.5.cmml" xref="S3.13.p4.6.m6.5.5">𝑊</ci></apply></apply></apply><apply id="S3.13.p4.6.m6.7.7g.cmml" xref="S3.13.p4.6.m6.7.7"><lt id="S3.13.p4.6.m6.7.7.10.cmml" xref="S3.13.p4.6.m6.7.7.10"></lt><share href="https://arxiv.org/html/2503.17112v1#S3.13.p4.6.m6.7.7.9.cmml" id="S3.13.p4.6.m6.7.7h.cmml" xref="S3.13.p4.6.m6.7.7"></share><apply id="S3.13.p4.6.m6.7.7.2.cmml" xref="S3.13.p4.6.m6.7.7.2"><times id="S3.13.p4.6.m6.7.7.2.2.cmml" xref="S3.13.p4.6.m6.7.7.2.2"></times><apply id="S3.13.p4.6.m6.7.7.2.1.1.1.cmml" xref="S3.13.p4.6.m6.7.7.2.1.1"><plus id="S3.13.p4.6.m6.7.7.2.1.1.1.1.cmml" xref="S3.13.p4.6.m6.7.7.2.1.1.1.1"></plus><apply id="S3.13.p4.6.m6.7.7.2.1.1.1.2.cmml" xref="S3.13.p4.6.m6.7.7.2.1.1.1.2"><times id="S3.13.p4.6.m6.7.7.2.1.1.1.2.1.cmml" xref="S3.13.p4.6.m6.7.7.2.1.1.1.2.1"></times><cn id="S3.13.p4.6.m6.7.7.2.1.1.1.2.2.cmml" type="integer" xref="S3.13.p4.6.m6.7.7.2.1.1.1.2.2">2</cn><ci id="S3.13.p4.6.m6.7.7.2.1.1.1.2.3.cmml" xref="S3.13.p4.6.m6.7.7.2.1.1.1.2.3">𝑡</ci></apply><cn id="S3.13.p4.6.m6.7.7.2.1.1.1.3.cmml" type="integer" xref="S3.13.p4.6.m6.7.7.2.1.1.1.3">1</cn></apply><ci id="S3.13.p4.6.m6.7.7.2.3.cmml" xref="S3.13.p4.6.m6.7.7.2.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.p4.6.m6.7c">|Y|=|W_{1}|=|W|+|Z|<2|W|<(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.13.p4.6.m6.7d">| italic_Y | = | italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = | italic_W | + | italic_Z | < 2 | italic_W | < ( 2 italic_t + 1 ) italic_a</annotation></semantics></math>, this decomposition has width less than <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="S3.13.p4.7.m7.1"><semantics id="S3.13.p4.7.m7.1a"><mrow id="S3.13.p4.7.m7.1.1" xref="S3.13.p4.7.m7.1.1.cmml"><mrow id="S3.13.p4.7.m7.1.1.1.1" xref="S3.13.p4.7.m7.1.1.1.1.1.cmml"><mo id="S3.13.p4.7.m7.1.1.1.1.2" stretchy="false" xref="S3.13.p4.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S3.13.p4.7.m7.1.1.1.1.1" xref="S3.13.p4.7.m7.1.1.1.1.1.cmml"><mrow id="S3.13.p4.7.m7.1.1.1.1.1.2" xref="S3.13.p4.7.m7.1.1.1.1.1.2.cmml"><mn id="S3.13.p4.7.m7.1.1.1.1.1.2.2" xref="S3.13.p4.7.m7.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.13.p4.7.m7.1.1.1.1.1.2.1" xref="S3.13.p4.7.m7.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.13.p4.7.m7.1.1.1.1.1.2.3" xref="S3.13.p4.7.m7.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.13.p4.7.m7.1.1.1.1.1.1" xref="S3.13.p4.7.m7.1.1.1.1.1.1.cmml">+</mo><mn id="S3.13.p4.7.m7.1.1.1.1.1.3" xref="S3.13.p4.7.m7.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.13.p4.7.m7.1.1.1.1.3" stretchy="false" xref="S3.13.p4.7.m7.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.13.p4.7.m7.1.1.2" xref="S3.13.p4.7.m7.1.1.2.cmml">⁢</mo><mi id="S3.13.p4.7.m7.1.1.3" xref="S3.13.p4.7.m7.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.13.p4.7.m7.1b"><apply id="S3.13.p4.7.m7.1.1.cmml" xref="S3.13.p4.7.m7.1.1"><times id="S3.13.p4.7.m7.1.1.2.cmml" xref="S3.13.p4.7.m7.1.1.2"></times><apply id="S3.13.p4.7.m7.1.1.1.1.1.cmml" xref="S3.13.p4.7.m7.1.1.1.1"><plus id="S3.13.p4.7.m7.1.1.1.1.1.1.cmml" xref="S3.13.p4.7.m7.1.1.1.1.1.1"></plus><apply id="S3.13.p4.7.m7.1.1.1.1.1.2.cmml" xref="S3.13.p4.7.m7.1.1.1.1.1.2"><times id="S3.13.p4.7.m7.1.1.1.1.1.2.1.cmml" xref="S3.13.p4.7.m7.1.1.1.1.1.2.1"></times><cn id="S3.13.p4.7.m7.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.13.p4.7.m7.1.1.1.1.1.2.2">2</cn><ci id="S3.13.p4.7.m7.1.1.1.1.1.2.3.cmml" xref="S3.13.p4.7.m7.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.13.p4.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S3.13.p4.7.m7.1.1.1.1.1.3">1</cn></apply><ci id="S3.13.p4.7.m7.1.1.3.cmml" xref="S3.13.p4.7.m7.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.p4.7.m7.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.13.p4.7.m7.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math>. We now assume that <math alttext="\ell\geq 1" class="ltx_Math" display="inline" id="S3.13.p4.8.m8.1"><semantics id="S3.13.p4.8.m8.1a"><mrow id="S3.13.p4.8.m8.1.1" xref="S3.13.p4.8.m8.1.1.cmml"><mi id="S3.13.p4.8.m8.1.1.2" mathvariant="normal" xref="S3.13.p4.8.m8.1.1.2.cmml">ℓ</mi><mo id="S3.13.p4.8.m8.1.1.1" xref="S3.13.p4.8.m8.1.1.1.cmml">≥</mo><mn id="S3.13.p4.8.m8.1.1.3" xref="S3.13.p4.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.13.p4.8.m8.1b"><apply id="S3.13.p4.8.m8.1.1.cmml" xref="S3.13.p4.8.m8.1.1"><geq id="S3.13.p4.8.m8.1.1.1.cmml" xref="S3.13.p4.8.m8.1.1.1"></geq><ci id="S3.13.p4.8.m8.1.1.2.cmml" xref="S3.13.p4.8.m8.1.1.2">ℓ</ci><cn id="S3.13.p4.8.m8.1.1.3.cmml" type="integer" xref="S3.13.p4.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.p4.8.m8.1c">\ell\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.13.p4.8.m8.1d">roman_ℓ ≥ 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.14.p5"> <p class="ltx_p" id="S3.14.p5.7">Since <math alttext="\ell\geq 1" class="ltx_Math" display="inline" id="S3.14.p5.1.m1.1"><semantics id="S3.14.p5.1.m1.1a"><mrow id="S3.14.p5.1.m1.1.1" xref="S3.14.p5.1.m1.1.1.cmml"><mi id="S3.14.p5.1.m1.1.1.2" mathvariant="normal" xref="S3.14.p5.1.m1.1.1.2.cmml">ℓ</mi><mo id="S3.14.p5.1.m1.1.1.1" xref="S3.14.p5.1.m1.1.1.1.cmml">≥</mo><mn id="S3.14.p5.1.m1.1.1.3" xref="S3.14.p5.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.14.p5.1.m1.1b"><apply id="S3.14.p5.1.m1.1.1.cmml" xref="S3.14.p5.1.m1.1.1"><geq id="S3.14.p5.1.m1.1.1.1.cmml" xref="S3.14.p5.1.m1.1.1.1"></geq><ci id="S3.14.p5.1.m1.1.1.2.cmml" xref="S3.14.p5.1.m1.1.1.2">ℓ</ci><cn id="S3.14.p5.1.m1.1.1.3.cmml" type="integer" xref="S3.14.p5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.p5.1.m1.1c">\ell\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.14.p5.1.m1.1d">roman_ℓ ≥ 1</annotation></semantics></math>, <math alttext="W_{0},\ldots,W_{\ell+1}" class="ltx_Math" display="inline" id="S3.14.p5.2.m2.3"><semantics id="S3.14.p5.2.m2.3a"><mrow id="S3.14.p5.2.m2.3.3.2" xref="S3.14.p5.2.m2.3.3.3.cmml"><msub id="S3.14.p5.2.m2.2.2.1.1" xref="S3.14.p5.2.m2.2.2.1.1.cmml"><mi id="S3.14.p5.2.m2.2.2.1.1.2" xref="S3.14.p5.2.m2.2.2.1.1.2.cmml">W</mi><mn id="S3.14.p5.2.m2.2.2.1.1.3" xref="S3.14.p5.2.m2.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.14.p5.2.m2.3.3.2.3" xref="S3.14.p5.2.m2.3.3.3.cmml">,</mo><mi id="S3.14.p5.2.m2.1.1" mathvariant="normal" xref="S3.14.p5.2.m2.1.1.cmml">…</mi><mo id="S3.14.p5.2.m2.3.3.2.4" xref="S3.14.p5.2.m2.3.3.3.cmml">,</mo><msub id="S3.14.p5.2.m2.3.3.2.2" xref="S3.14.p5.2.m2.3.3.2.2.cmml"><mi id="S3.14.p5.2.m2.3.3.2.2.2" xref="S3.14.p5.2.m2.3.3.2.2.2.cmml">W</mi><mrow id="S3.14.p5.2.m2.3.3.2.2.3" xref="S3.14.p5.2.m2.3.3.2.2.3.cmml"><mi id="S3.14.p5.2.m2.3.3.2.2.3.2" mathvariant="normal" xref="S3.14.p5.2.m2.3.3.2.2.3.2.cmml">ℓ</mi><mo id="S3.14.p5.2.m2.3.3.2.2.3.1" xref="S3.14.p5.2.m2.3.3.2.2.3.1.cmml">+</mo><mn id="S3.14.p5.2.m2.3.3.2.2.3.3" xref="S3.14.p5.2.m2.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.14.p5.2.m2.3b"><list id="S3.14.p5.2.m2.3.3.3.cmml" xref="S3.14.p5.2.m2.3.3.2"><apply id="S3.14.p5.2.m2.2.2.1.1.cmml" xref="S3.14.p5.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3.14.p5.2.m2.2.2.1.1.1.cmml" xref="S3.14.p5.2.m2.2.2.1.1">subscript</csymbol><ci id="S3.14.p5.2.m2.2.2.1.1.2.cmml" xref="S3.14.p5.2.m2.2.2.1.1.2">𝑊</ci><cn id="S3.14.p5.2.m2.2.2.1.1.3.cmml" type="integer" xref="S3.14.p5.2.m2.2.2.1.1.3">0</cn></apply><ci id="S3.14.p5.2.m2.1.1.cmml" xref="S3.14.p5.2.m2.1.1">…</ci><apply id="S3.14.p5.2.m2.3.3.2.2.cmml" xref="S3.14.p5.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S3.14.p5.2.m2.3.3.2.2.1.cmml" xref="S3.14.p5.2.m2.3.3.2.2">subscript</csymbol><ci id="S3.14.p5.2.m2.3.3.2.2.2.cmml" xref="S3.14.p5.2.m2.3.3.2.2.2">𝑊</ci><apply id="S3.14.p5.2.m2.3.3.2.2.3.cmml" xref="S3.14.p5.2.m2.3.3.2.2.3"><plus id="S3.14.p5.2.m2.3.3.2.2.3.1.cmml" xref="S3.14.p5.2.m2.3.3.2.2.3.1"></plus><ci id="S3.14.p5.2.m2.3.3.2.2.3.2.cmml" xref="S3.14.p5.2.m2.3.3.2.2.3.2">ℓ</ci><cn id="S3.14.p5.2.m2.3.3.2.2.3.3.cmml" type="integer" xref="S3.14.p5.2.m2.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.14.p5.2.m2.3c">W_{0},\ldots,W_{\ell+1}</annotation><annotation encoding="application/x-llamapun" id="S3.14.p5.2.m2.3d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT</annotation></semantics></math> satisfies the conditions of <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm7" title="Lemma 7. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">7</span></a>. Let <math alttext="\mathcal{T}^{\prime}_{Y}:=(B^{\prime}_{y}:y\in V(T^{\prime}_{Y}))" class="ltx_math_unparsed" display="inline" id="S3.14.p5.3.m3.1"><semantics id="S3.14.p5.3.m3.1a"><mrow id="S3.14.p5.3.m3.1b"><msubsup id="S3.14.p5.3.m3.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.14.p5.3.m3.1.1.2.2">𝒯</mi><mi id="S3.14.p5.3.m3.1.1.3">Y</mi><mo id="S3.14.p5.3.m3.1.1.2.3">′</mo></msubsup><mo id="S3.14.p5.3.m3.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S3.14.p5.3.m3.1.3"><mo id="S3.14.p5.3.m3.1.3.1" stretchy="false">(</mo><msubsup id="S3.14.p5.3.m3.1.3.2"><mi id="S3.14.p5.3.m3.1.3.2.2.2">B</mi><mi id="S3.14.p5.3.m3.1.3.2.3">y</mi><mo id="S3.14.p5.3.m3.1.3.2.2.3">′</mo></msubsup><mo id="S3.14.p5.3.m3.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.14.p5.3.m3.1.3.4">y</mi><mo id="S3.14.p5.3.m3.1.3.5">∈</mo><mi id="S3.14.p5.3.m3.1.3.6">V</mi><mrow id="S3.14.p5.3.m3.1.3.7"><mo id="S3.14.p5.3.m3.1.3.7.1" stretchy="false">(</mo><msubsup id="S3.14.p5.3.m3.1.3.7.2"><mi id="S3.14.p5.3.m3.1.3.7.2.2.2">T</mi><mi id="S3.14.p5.3.m3.1.3.7.2.3">Y</mi><mo id="S3.14.p5.3.m3.1.3.7.2.2.3">′</mo></msubsup><mo id="S3.14.p5.3.m3.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S3.14.p5.3.m3.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.14.p5.3.m3.1c">\mathcal{T}^{\prime}_{Y}:=(B^{\prime}_{y}:y\in V(T^{\prime}_{Y}))</annotation><annotation encoding="application/x-llamapun" id="S3.14.p5.3.m3.1d">caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT := ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT : italic_y ∈ italic_V ( italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) )</annotation></semantics></math> be the tree decomposition of <math alttext="G[W_{\ell+1}]" class="ltx_Math" display="inline" id="S3.14.p5.4.m4.1"><semantics id="S3.14.p5.4.m4.1a"><mrow id="S3.14.p5.4.m4.1.1" xref="S3.14.p5.4.m4.1.1.cmml"><mi id="S3.14.p5.4.m4.1.1.3" xref="S3.14.p5.4.m4.1.1.3.cmml">G</mi><mo id="S3.14.p5.4.m4.1.1.2" xref="S3.14.p5.4.m4.1.1.2.cmml">⁢</mo><mrow id="S3.14.p5.4.m4.1.1.1.1" xref="S3.14.p5.4.m4.1.1.1.2.cmml"><mo id="S3.14.p5.4.m4.1.1.1.1.2" stretchy="false" xref="S3.14.p5.4.m4.1.1.1.2.1.cmml">[</mo><msub id="S3.14.p5.4.m4.1.1.1.1.1" xref="S3.14.p5.4.m4.1.1.1.1.1.cmml"><mi id="S3.14.p5.4.m4.1.1.1.1.1.2" xref="S3.14.p5.4.m4.1.1.1.1.1.2.cmml">W</mi><mrow id="S3.14.p5.4.m4.1.1.1.1.1.3" xref="S3.14.p5.4.m4.1.1.1.1.1.3.cmml"><mi id="S3.14.p5.4.m4.1.1.1.1.1.3.2" mathvariant="normal" xref="S3.14.p5.4.m4.1.1.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.14.p5.4.m4.1.1.1.1.1.3.1" xref="S3.14.p5.4.m4.1.1.1.1.1.3.1.cmml">+</mo><mn id="S3.14.p5.4.m4.1.1.1.1.1.3.3" xref="S3.14.p5.4.m4.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.14.p5.4.m4.1.1.1.1.3" stretchy="false" xref="S3.14.p5.4.m4.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.14.p5.4.m4.1b"><apply id="S3.14.p5.4.m4.1.1.cmml" xref="S3.14.p5.4.m4.1.1"><times id="S3.14.p5.4.m4.1.1.2.cmml" xref="S3.14.p5.4.m4.1.1.2"></times><ci id="S3.14.p5.4.m4.1.1.3.cmml" xref="S3.14.p5.4.m4.1.1.3">𝐺</ci><apply id="S3.14.p5.4.m4.1.1.1.2.cmml" xref="S3.14.p5.4.m4.1.1.1.1"><csymbol cd="latexml" id="S3.14.p5.4.m4.1.1.1.2.1.cmml" xref="S3.14.p5.4.m4.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.14.p5.4.m4.1.1.1.1.1.cmml" xref="S3.14.p5.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.14.p5.4.m4.1.1.1.1.1.1.cmml" xref="S3.14.p5.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S3.14.p5.4.m4.1.1.1.1.1.2.cmml" xref="S3.14.p5.4.m4.1.1.1.1.1.2">𝑊</ci><apply id="S3.14.p5.4.m4.1.1.1.1.1.3.cmml" xref="S3.14.p5.4.m4.1.1.1.1.1.3"><plus id="S3.14.p5.4.m4.1.1.1.1.1.3.1.cmml" xref="S3.14.p5.4.m4.1.1.1.1.1.3.1"></plus><ci id="S3.14.p5.4.m4.1.1.1.1.1.3.2.cmml" xref="S3.14.p5.4.m4.1.1.1.1.1.3.2">ℓ</ci><cn id="S3.14.p5.4.m4.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.14.p5.4.m4.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.p5.4.m4.1c">G[W_{\ell+1}]</annotation><annotation encoding="application/x-llamapun" id="S3.14.p5.4.m4.1d">italic_G [ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math> obtained by applying <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm5" title="Lemma 5. ‣ 2 Preliminaries ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5</span></a> to <math alttext="G[W_{\ell+1}]" class="ltx_Math" display="inline" id="S3.14.p5.5.m5.1"><semantics id="S3.14.p5.5.m5.1a"><mrow id="S3.14.p5.5.m5.1.1" xref="S3.14.p5.5.m5.1.1.cmml"><mi id="S3.14.p5.5.m5.1.1.3" xref="S3.14.p5.5.m5.1.1.3.cmml">G</mi><mo id="S3.14.p5.5.m5.1.1.2" xref="S3.14.p5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S3.14.p5.5.m5.1.1.1.1" xref="S3.14.p5.5.m5.1.1.1.2.cmml"><mo id="S3.14.p5.5.m5.1.1.1.1.2" stretchy="false" xref="S3.14.p5.5.m5.1.1.1.2.1.cmml">[</mo><msub id="S3.14.p5.5.m5.1.1.1.1.1" xref="S3.14.p5.5.m5.1.1.1.1.1.cmml"><mi id="S3.14.p5.5.m5.1.1.1.1.1.2" xref="S3.14.p5.5.m5.1.1.1.1.1.2.cmml">W</mi><mrow id="S3.14.p5.5.m5.1.1.1.1.1.3" xref="S3.14.p5.5.m5.1.1.1.1.1.3.cmml"><mi id="S3.14.p5.5.m5.1.1.1.1.1.3.2" mathvariant="normal" xref="S3.14.p5.5.m5.1.1.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.14.p5.5.m5.1.1.1.1.1.3.1" xref="S3.14.p5.5.m5.1.1.1.1.1.3.1.cmml">+</mo><mn id="S3.14.p5.5.m5.1.1.1.1.1.3.3" xref="S3.14.p5.5.m5.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.14.p5.5.m5.1.1.1.1.3" stretchy="false" xref="S3.14.p5.5.m5.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.14.p5.5.m5.1b"><apply id="S3.14.p5.5.m5.1.1.cmml" xref="S3.14.p5.5.m5.1.1"><times id="S3.14.p5.5.m5.1.1.2.cmml" xref="S3.14.p5.5.m5.1.1.2"></times><ci id="S3.14.p5.5.m5.1.1.3.cmml" xref="S3.14.p5.5.m5.1.1.3">𝐺</ci><apply id="S3.14.p5.5.m5.1.1.1.2.cmml" xref="S3.14.p5.5.m5.1.1.1.1"><csymbol cd="latexml" id="S3.14.p5.5.m5.1.1.1.2.1.cmml" xref="S3.14.p5.5.m5.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.14.p5.5.m5.1.1.1.1.1.cmml" xref="S3.14.p5.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.14.p5.5.m5.1.1.1.1.1.1.cmml" xref="S3.14.p5.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.14.p5.5.m5.1.1.1.1.1.2.cmml" xref="S3.14.p5.5.m5.1.1.1.1.1.2">𝑊</ci><apply id="S3.14.p5.5.m5.1.1.1.1.1.3.cmml" xref="S3.14.p5.5.m5.1.1.1.1.1.3"><plus id="S3.14.p5.5.m5.1.1.1.1.1.3.1.cmml" xref="S3.14.p5.5.m5.1.1.1.1.1.3.1"></plus><ci id="S3.14.p5.5.m5.1.1.1.1.1.3.2.cmml" xref="S3.14.p5.5.m5.1.1.1.1.1.3.2">ℓ</ci><cn id="S3.14.p5.5.m5.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.14.p5.5.m5.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.p5.5.m5.1c">G[W_{\ell+1}]</annotation><annotation encoding="application/x-llamapun" id="S3.14.p5.5.m5.1d">italic_G [ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math> with the height <math alttext="h" class="ltx_Math" display="inline" id="S3.14.p5.6.m6.1"><semantics id="S3.14.p5.6.m6.1a"><mi id="S3.14.p5.6.m6.1.1" xref="S3.14.p5.6.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.14.p5.6.m6.1b"><ci id="S3.14.p5.6.m6.1.1.cmml" xref="S3.14.p5.6.m6.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.14.p5.6.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.14.p5.6.m6.1d">italic_h</annotation></semantics></math> defined above. The following claim will be used to bound the width of a tree decomposition that we derive from <math alttext="\mathcal{T}^{\prime}_{Y}" class="ltx_Math" display="inline" id="S3.14.p5.7.m7.1"><semantics id="S3.14.p5.7.m7.1a"><msubsup id="S3.14.p5.7.m7.1.1" xref="S3.14.p5.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.14.p5.7.m7.1.1.2.2" xref="S3.14.p5.7.m7.1.1.2.2.cmml">𝒯</mi><mi id="S3.14.p5.7.m7.1.1.3" xref="S3.14.p5.7.m7.1.1.3.cmml">Y</mi><mo id="S3.14.p5.7.m7.1.1.2.3" xref="S3.14.p5.7.m7.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.14.p5.7.m7.1b"><apply id="S3.14.p5.7.m7.1.1.cmml" xref="S3.14.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S3.14.p5.7.m7.1.1.1.cmml" xref="S3.14.p5.7.m7.1.1">subscript</csymbol><apply id="S3.14.p5.7.m7.1.1.2.cmml" xref="S3.14.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S3.14.p5.7.m7.1.1.2.1.cmml" xref="S3.14.p5.7.m7.1.1">superscript</csymbol><ci id="S3.14.p5.7.m7.1.1.2.2.cmml" xref="S3.14.p5.7.m7.1.1.2.2">𝒯</ci><ci id="S3.14.p5.7.m7.1.1.2.3.cmml" xref="S3.14.p5.7.m7.1.1.2.3">′</ci></apply><ci id="S3.14.p5.7.m7.1.1.3.cmml" xref="S3.14.p5.7.m7.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.p5.7.m7.1c">\mathcal{T}^{\prime}_{Y}</annotation><annotation encoding="application/x-llamapun" id="S3.14.p5.7.m7.1d">caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_clm" id="Thmthm9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm9.1.1.1">Claim 9</span></span><span class="ltx_text ltx_font_bold" id="Thmthm9.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm9.p1"> <p class="ltx_p" id="Thmthm9.p1.4"><span class="ltx_text ltx_font_italic" id="Thmthm9.p1.4.4">For each <math alttext="d\in\{0,\ldots,h\}" class="ltx_Math" display="inline" id="Thmthm9.p1.1.1.m1.3"><semantics id="Thmthm9.p1.1.1.m1.3a"><mrow id="Thmthm9.p1.1.1.m1.3.4" xref="Thmthm9.p1.1.1.m1.3.4.cmml"><mi id="Thmthm9.p1.1.1.m1.3.4.2" xref="Thmthm9.p1.1.1.m1.3.4.2.cmml">d</mi><mo id="Thmthm9.p1.1.1.m1.3.4.1" xref="Thmthm9.p1.1.1.m1.3.4.1.cmml">∈</mo><mrow id="Thmthm9.p1.1.1.m1.3.4.3.2" xref="Thmthm9.p1.1.1.m1.3.4.3.1.cmml"><mo id="Thmthm9.p1.1.1.m1.3.4.3.2.1" stretchy="false" xref="Thmthm9.p1.1.1.m1.3.4.3.1.cmml">{</mo><mn id="Thmthm9.p1.1.1.m1.1.1" xref="Thmthm9.p1.1.1.m1.1.1.cmml">0</mn><mo id="Thmthm9.p1.1.1.m1.3.4.3.2.2" xref="Thmthm9.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="Thmthm9.p1.1.1.m1.2.2" mathvariant="normal" xref="Thmthm9.p1.1.1.m1.2.2.cmml">…</mi><mo id="Thmthm9.p1.1.1.m1.3.4.3.2.3" xref="Thmthm9.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="Thmthm9.p1.1.1.m1.3.3" xref="Thmthm9.p1.1.1.m1.3.3.cmml">h</mi><mo id="Thmthm9.p1.1.1.m1.3.4.3.2.4" stretchy="false" xref="Thmthm9.p1.1.1.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.1.1.m1.3b"><apply id="Thmthm9.p1.1.1.m1.3.4.cmml" xref="Thmthm9.p1.1.1.m1.3.4"><in id="Thmthm9.p1.1.1.m1.3.4.1.cmml" xref="Thmthm9.p1.1.1.m1.3.4.1"></in><ci id="Thmthm9.p1.1.1.m1.3.4.2.cmml" xref="Thmthm9.p1.1.1.m1.3.4.2">𝑑</ci><set id="Thmthm9.p1.1.1.m1.3.4.3.1.cmml" xref="Thmthm9.p1.1.1.m1.3.4.3.2"><cn id="Thmthm9.p1.1.1.m1.1.1.cmml" type="integer" xref="Thmthm9.p1.1.1.m1.1.1">0</cn><ci id="Thmthm9.p1.1.1.m1.2.2.cmml" xref="Thmthm9.p1.1.1.m1.2.2">…</ci><ci id="Thmthm9.p1.1.1.m1.3.3.cmml" xref="Thmthm9.p1.1.1.m1.3.3">ℎ</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.1.1.m1.3c">d\in\{0,\ldots,h\}</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.1.1.m1.3d">italic_d ∈ { 0 , … , italic_h }</annotation></semantics></math> and each node <math alttext="y" class="ltx_Math" display="inline" id="Thmthm9.p1.2.2.m2.1"><semantics id="Thmthm9.p1.2.2.m2.1a"><mi id="Thmthm9.p1.2.2.m2.1.1" xref="Thmthm9.p1.2.2.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.2.2.m2.1b"><ci id="Thmthm9.p1.2.2.m2.1.1.cmml" xref="Thmthm9.p1.2.2.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.2.2.m2.1c">y</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.2.2.m2.1d">italic_y</annotation></semantics></math> of <math alttext="T^{\prime}_{y}" class="ltx_Math" display="inline" id="Thmthm9.p1.3.3.m3.1"><semantics id="Thmthm9.p1.3.3.m3.1a"><msubsup id="Thmthm9.p1.3.3.m3.1.1" xref="Thmthm9.p1.3.3.m3.1.1.cmml"><mi id="Thmthm9.p1.3.3.m3.1.1.2.2" xref="Thmthm9.p1.3.3.m3.1.1.2.2.cmml">T</mi><mi id="Thmthm9.p1.3.3.m3.1.1.3" xref="Thmthm9.p1.3.3.m3.1.1.3.cmml">y</mi><mo id="Thmthm9.p1.3.3.m3.1.1.2.3" xref="Thmthm9.p1.3.3.m3.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.3.3.m3.1b"><apply id="Thmthm9.p1.3.3.m3.1.1.cmml" xref="Thmthm9.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="Thmthm9.p1.3.3.m3.1.1.1.cmml" xref="Thmthm9.p1.3.3.m3.1.1">subscript</csymbol><apply id="Thmthm9.p1.3.3.m3.1.1.2.cmml" xref="Thmthm9.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="Thmthm9.p1.3.3.m3.1.1.2.1.cmml" xref="Thmthm9.p1.3.3.m3.1.1">superscript</csymbol><ci id="Thmthm9.p1.3.3.m3.1.1.2.2.cmml" xref="Thmthm9.p1.3.3.m3.1.1.2.2">𝑇</ci><ci id="Thmthm9.p1.3.3.m3.1.1.2.3.cmml" xref="Thmthm9.p1.3.3.m3.1.1.2.3">′</ci></apply><ci id="Thmthm9.p1.3.3.m3.1.1.3.cmml" xref="Thmthm9.p1.3.3.m3.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.3.3.m3.1c">T^{\prime}_{y}</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.3.3.m3.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="|\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)|\leq(\tfrac{2}{3})^{d}\cdot|% W_{\ell+1}|" class="ltx_Math" display="inline" 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xref="Thmthm9.p1.4.4.m4.3.3.1.1.1.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="Thmthm9.p1.4.4.m4.3.3.1.1.1.1a" xref="Thmthm9.p1.4.4.m4.3.3.1.1.1.2.cmml">⁡</mo><mrow id="Thmthm9.p1.4.4.m4.3.3.1.1.1.1.2" xref="Thmthm9.p1.4.4.m4.3.3.1.1.1.2.cmml"><mo id="Thmthm9.p1.4.4.m4.3.3.1.1.1.1.2.1" stretchy="false" xref="Thmthm9.p1.4.4.m4.3.3.1.1.1.2.cmml">(</mo><mi id="Thmthm9.p1.4.4.m4.1.1" xref="Thmthm9.p1.4.4.m4.1.1.cmml">y</mi><mo id="Thmthm9.p1.4.4.m4.3.3.1.1.1.1.2.2" stretchy="false" xref="Thmthm9.p1.4.4.m4.3.3.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmthm9.p1.4.4.m4.3.3.1.1.3" stretchy="false" xref="Thmthm9.p1.4.4.m4.3.3.1.2.1.cmml">|</mo></mrow><mo id="Thmthm9.p1.4.4.m4.4.4.3" xref="Thmthm9.p1.4.4.m4.4.4.3.cmml">≤</mo><mrow id="Thmthm9.p1.4.4.m4.4.4.2" xref="Thmthm9.p1.4.4.m4.4.4.2.cmml"><msup id="Thmthm9.p1.4.4.m4.4.4.2.3" xref="Thmthm9.p1.4.4.m4.4.4.2.3.cmml"><mrow id="Thmthm9.p1.4.4.m4.4.4.2.3.2.2" xref="Thmthm9.p1.4.4.m4.2.2.cmml"><mo id="Thmthm9.p1.4.4.m4.4.4.2.3.2.2.1" 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xref="Thmthm9.p1.4.4.m4.4.4.2.1.1"><abs id="Thmthm9.p1.4.4.m4.4.4.2.1.2.1.cmml" xref="Thmthm9.p1.4.4.m4.4.4.2.1.1.2"></abs><apply id="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.cmml" xref="Thmthm9.p1.4.4.m4.4.4.2.1.1.1"><csymbol cd="ambiguous" id="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.1.cmml" xref="Thmthm9.p1.4.4.m4.4.4.2.1.1.1">subscript</csymbol><ci id="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.2.cmml" xref="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.2">𝑊</ci><apply id="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.3.cmml" xref="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.3"><plus id="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.3.1.cmml" xref="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.3.1"></plus><ci id="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.3.2.cmml" xref="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.3.2">ℓ</ci><cn id="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.3.3.cmml" type="integer" xref="Thmthm9.p1.4.4.m4.4.4.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.4.4.m4.4c">|\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)|\leq(\tfrac{2}{3})^{d}\cdot|% W_{\ell+1}|</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.4.4.m4.4d">| roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) | ≤ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ⋅ | italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT |</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)\cap(W\cup Z)|\leq(2+\tfrac{1% }{6})ta\cdot(\tfrac{2}{3})^{d}+3da" class="ltx_Math" display="block" id="S3.E9.m1.4"><semantics 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caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∩ ( italic_W ∪ italic_Z ) | ≤ ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) italic_t italic_a ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT + 3 italic_d italic_a</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_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">(9)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S3.16.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Claim:.</h6> <div class="ltx_para" id="S3.15.1.p1"> <p class="ltx_p" id="S3.15.1.p1.3">Consider the separation</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) , italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ∖ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.15.1.p1.2">of <math alttext="G[W_{\ell+1}]" class="ltx_Math" display="inline" id="S3.15.1.p1.1.m1.1"><semantics id="S3.15.1.p1.1.m1.1a"><mrow id="S3.15.1.p1.1.m1.1.1" xref="S3.15.1.p1.1.m1.1.1.cmml"><mi id="S3.15.1.p1.1.m1.1.1.3" xref="S3.15.1.p1.1.m1.1.1.3.cmml">G</mi><mo id="S3.15.1.p1.1.m1.1.1.2" xref="S3.15.1.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.15.1.p1.1.m1.1.1.1.1" xref="S3.15.1.p1.1.m1.1.1.1.2.cmml"><mo id="S3.15.1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S3.15.1.p1.1.m1.1.1.1.2.1.cmml">[</mo><msub id="S3.15.1.p1.1.m1.1.1.1.1.1" 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id="S3.15.1.p1.1.m1.1.1.1.2.1.cmml" xref="S3.15.1.p1.1.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.15.1.p1.1.m1.1.1.1.1.1.cmml" xref="S3.15.1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.15.1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.15.1.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.15.1.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.15.1.p1.1.m1.1.1.1.1.1.2">𝑊</ci><apply id="S3.15.1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.15.1.p1.1.m1.1.1.1.1.1.3"><plus id="S3.15.1.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.15.1.p1.1.m1.1.1.1.1.1.3.1"></plus><ci id="S3.15.1.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S3.15.1.p1.1.m1.1.1.1.1.1.3.2">ℓ</ci><cn id="S3.15.1.p1.1.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.15.1.p1.1.m1.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.1.p1.1.m1.1c">G[W_{\ell+1}]</annotation><annotation encoding="application/x-llamapun" id="S3.15.1.p1.1.m1.1d">italic_G [ italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math> and observe that <math alttext="A\setminus B:=\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)" class="ltx_Math" display="inline" id="S3.15.1.p1.2.m2.2"><semantics id="S3.15.1.p1.2.m2.2a"><mrow id="S3.15.1.p1.2.m2.2.2" xref="S3.15.1.p1.2.m2.2.2.cmml"><mrow id="S3.15.1.p1.2.m2.2.2.3" xref="S3.15.1.p1.2.m2.2.2.3.cmml"><mi id="S3.15.1.p1.2.m2.2.2.3.2" xref="S3.15.1.p1.2.m2.2.2.3.2.cmml">A</mi><mo id="S3.15.1.p1.2.m2.2.2.3.1" xref="S3.15.1.p1.2.m2.2.2.3.1.cmml">∖</mo><mi id="S3.15.1.p1.2.m2.2.2.3.3" xref="S3.15.1.p1.2.m2.2.2.3.3.cmml">B</mi></mrow><mo id="S3.15.1.p1.2.m2.2.2.2" lspace="0.278em" rspace="0.278em" xref="S3.15.1.p1.2.m2.2.2.2.cmml">:=</mo><mrow id="S3.15.1.p1.2.m2.2.2.1.1" xref="S3.15.1.p1.2.m2.2.2.1.2.cmml"><msub id="S3.15.1.p1.2.m2.2.2.1.1.1" xref="S3.15.1.p1.2.m2.2.2.1.1.1.cmml"><mi id="S3.15.1.p1.2.m2.2.2.1.1.1.2" xref="S3.15.1.p1.2.m2.2.2.1.1.1.2.cmml">int</mi><msubsup id="S3.15.1.p1.2.m2.2.2.1.1.1.3" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.2" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.2.cmml">𝒯</mi><mi id="S3.15.1.p1.2.m2.2.2.1.1.1.3.3" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3.3.cmml">Y</mi><mo id="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.3" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.15.1.p1.2.m2.2.2.1.1a" xref="S3.15.1.p1.2.m2.2.2.1.2.cmml">⁡</mo><mrow id="S3.15.1.p1.2.m2.2.2.1.1.2" xref="S3.15.1.p1.2.m2.2.2.1.2.cmml"><mo id="S3.15.1.p1.2.m2.2.2.1.1.2.1" stretchy="false" xref="S3.15.1.p1.2.m2.2.2.1.2.cmml">(</mo><mi id="S3.15.1.p1.2.m2.1.1" xref="S3.15.1.p1.2.m2.1.1.cmml">y</mi><mo id="S3.15.1.p1.2.m2.2.2.1.1.2.2" stretchy="false" xref="S3.15.1.p1.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.15.1.p1.2.m2.2b"><apply id="S3.15.1.p1.2.m2.2.2.cmml" xref="S3.15.1.p1.2.m2.2.2"><csymbol cd="latexml" id="S3.15.1.p1.2.m2.2.2.2.cmml" xref="S3.15.1.p1.2.m2.2.2.2">assign</csymbol><apply id="S3.15.1.p1.2.m2.2.2.3.cmml" xref="S3.15.1.p1.2.m2.2.2.3"><setdiff id="S3.15.1.p1.2.m2.2.2.3.1.cmml" xref="S3.15.1.p1.2.m2.2.2.3.1"></setdiff><ci id="S3.15.1.p1.2.m2.2.2.3.2.cmml" xref="S3.15.1.p1.2.m2.2.2.3.2">𝐴</ci><ci id="S3.15.1.p1.2.m2.2.2.3.3.cmml" xref="S3.15.1.p1.2.m2.2.2.3.3">𝐵</ci></apply><apply id="S3.15.1.p1.2.m2.2.2.1.2.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1"><apply id="S3.15.1.p1.2.m2.2.2.1.1.1.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.15.1.p1.2.m2.2.2.1.1.1.1.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S3.15.1.p1.2.m2.2.2.1.1.1.2.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1.2">int</ci><apply id="S3.15.1.p1.2.m2.2.2.1.1.1.3.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.15.1.p1.2.m2.2.2.1.1.1.3.1.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3">subscript</csymbol><apply id="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.1.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3">superscript</csymbol><ci id="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.2.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.2">𝒯</ci><ci id="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.3.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3.2.3">′</ci></apply><ci id="S3.15.1.p1.2.m2.2.2.1.1.1.3.3.cmml" xref="S3.15.1.p1.2.m2.2.2.1.1.1.3.3">𝑌</ci></apply></apply><ci id="S3.15.1.p1.2.m2.1.1.cmml" xref="S3.15.1.p1.2.m2.1.1">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.1.p1.2.m2.2c">A\setminus B:=\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)</annotation><annotation encoding="application/x-llamapun" id="S3.15.1.p1.2.m2.2d">italic_A ∖ italic_B := roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y )</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm7" title="Lemma 7. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">7</span></a>,</p> </div> <div class="ltx_para" id="S3.16.2.p2"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx5"> <tbody id="S3.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle|\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)\cap(W\cup Z)|" class="ltx_Math" display="inline" id="S3.Ex15.m1.2"><semantics id="S3.Ex15.m1.2a"><mrow id="S3.Ex15.m1.2.2.1" xref="S3.Ex15.m1.2.2.2.cmml"><mo id="S3.Ex15.m1.2.2.1.2" stretchy="false" xref="S3.Ex15.m1.2.2.2.1.cmml">|</mo><mrow id="S3.Ex15.m1.2.2.1.1" xref="S3.Ex15.m1.2.2.1.1.cmml"><mrow id="S3.Ex15.m1.2.2.1.1.1.1" xref="S3.Ex15.m1.2.2.1.1.1.2.cmml"><msub id="S3.Ex15.m1.2.2.1.1.1.1.1" xref="S3.Ex15.m1.2.2.1.1.1.1.1.cmml"><mi id="S3.Ex15.m1.2.2.1.1.1.1.1.2" xref="S3.Ex15.m1.2.2.1.1.1.1.1.2.cmml">int</mi><msubsup id="S3.Ex15.m1.2.2.1.1.1.1.1.3" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.2" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.2.cmml">𝒯</mi><mi id="S3.Ex15.m1.2.2.1.1.1.1.1.3.3" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3.3.cmml">Y</mi><mo id="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.3" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.Ex15.m1.2.2.1.1.1.1a" xref="S3.Ex15.m1.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S3.Ex15.m1.2.2.1.1.1.1.2" xref="S3.Ex15.m1.2.2.1.1.1.2.cmml"><mo id="S3.Ex15.m1.2.2.1.1.1.1.2.1" stretchy="false" xref="S3.Ex15.m1.2.2.1.1.1.2.cmml">(</mo><mi id="S3.Ex15.m1.1.1" xref="S3.Ex15.m1.1.1.cmml">y</mi><mo id="S3.Ex15.m1.2.2.1.1.1.1.2.2" stretchy="false" xref="S3.Ex15.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex15.m1.2.2.1.1.3" xref="S3.Ex15.m1.2.2.1.1.3.cmml">∩</mo><mrow id="S3.Ex15.m1.2.2.1.1.2.1" xref="S3.Ex15.m1.2.2.1.1.2.1.1.cmml"><mo id="S3.Ex15.m1.2.2.1.1.2.1.2" stretchy="false" xref="S3.Ex15.m1.2.2.1.1.2.1.1.cmml">(</mo><mrow id="S3.Ex15.m1.2.2.1.1.2.1.1" xref="S3.Ex15.m1.2.2.1.1.2.1.1.cmml"><mi id="S3.Ex15.m1.2.2.1.1.2.1.1.2" xref="S3.Ex15.m1.2.2.1.1.2.1.1.2.cmml">W</mi><mo id="S3.Ex15.m1.2.2.1.1.2.1.1.1" xref="S3.Ex15.m1.2.2.1.1.2.1.1.1.cmml">∪</mo><mi id="S3.Ex15.m1.2.2.1.1.2.1.1.3" xref="S3.Ex15.m1.2.2.1.1.2.1.1.3.cmml">Z</mi></mrow><mo id="S3.Ex15.m1.2.2.1.1.2.1.3" stretchy="false" xref="S3.Ex15.m1.2.2.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex15.m1.2.2.1.3" stretchy="false" xref="S3.Ex15.m1.2.2.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex15.m1.2b"><apply id="S3.Ex15.m1.2.2.2.cmml" xref="S3.Ex15.m1.2.2.1"><abs id="S3.Ex15.m1.2.2.2.1.cmml" xref="S3.Ex15.m1.2.2.1.2"></abs><apply id="S3.Ex15.m1.2.2.1.1.cmml" xref="S3.Ex15.m1.2.2.1.1"><intersect id="S3.Ex15.m1.2.2.1.1.3.cmml" xref="S3.Ex15.m1.2.2.1.1.3"></intersect><apply id="S3.Ex15.m1.2.2.1.1.1.2.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1"><apply id="S3.Ex15.m1.2.2.1.1.1.1.1.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex15.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex15.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1.2">int</ci><apply id="S3.Ex15.m1.2.2.1.1.1.1.1.3.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex15.m1.2.2.1.1.1.1.1.3.1.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3">subscript</csymbol><apply id="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.1.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3">superscript</csymbol><ci id="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.2.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.2">𝒯</ci><ci id="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.3.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3.2.3">′</ci></apply><ci id="S3.Ex15.m1.2.2.1.1.1.1.1.3.3.cmml" xref="S3.Ex15.m1.2.2.1.1.1.1.1.3.3">𝑌</ci></apply></apply><ci id="S3.Ex15.m1.1.1.cmml" xref="S3.Ex15.m1.1.1">𝑦</ci></apply><apply id="S3.Ex15.m1.2.2.1.1.2.1.1.cmml" xref="S3.Ex15.m1.2.2.1.1.2.1"><union id="S3.Ex15.m1.2.2.1.1.2.1.1.1.cmml" xref="S3.Ex15.m1.2.2.1.1.2.1.1.1"></union><ci id="S3.Ex15.m1.2.2.1.1.2.1.1.2.cmml" xref="S3.Ex15.m1.2.2.1.1.2.1.1.2">𝑊</ci><ci id="S3.Ex15.m1.2.2.1.1.2.1.1.3.cmml" xref="S3.Ex15.m1.2.2.1.1.2.1.1.3">𝑍</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex15.m1.2c">\displaystyle|\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)\cap(W\cup Z)|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex15.m1.2d">| roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∩ ( italic_W ∪ italic_Z ) |</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=|(A\setminus B)\cap(W\cup Z)|" class="ltx_Math" display="inline" id="S3.Ex15.m2.1"><semantics id="S3.Ex15.m2.1a"><mrow id="S3.Ex15.m2.1.1" xref="S3.Ex15.m2.1.1.cmml"><mi id="S3.Ex15.m2.1.1.3" xref="S3.Ex15.m2.1.1.3.cmml"></mi><mo id="S3.Ex15.m2.1.1.2" xref="S3.Ex15.m2.1.1.2.cmml">=</mo><mrow id="S3.Ex15.m2.1.1.1.1" xref="S3.Ex15.m2.1.1.1.2.cmml"><mo id="S3.Ex15.m2.1.1.1.1.2" stretchy="false" xref="S3.Ex15.m2.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex15.m2.1.1.1.1.1" xref="S3.Ex15.m2.1.1.1.1.1.cmml"><mrow id="S3.Ex15.m2.1.1.1.1.1.1.1" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex15.m2.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex15.m2.1.1.1.1.1.1.1.1" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex15.m2.1.1.1.1.1.1.1.1.2" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.Ex15.m2.1.1.1.1.1.1.1.1.1" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex15.m2.1.1.1.1.1.1.1.1.3" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex15.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex15.m2.1.1.1.1.1.3" xref="S3.Ex15.m2.1.1.1.1.1.3.cmml">∩</mo><mrow id="S3.Ex15.m2.1.1.1.1.1.2.1" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.cmml"><mo id="S3.Ex15.m2.1.1.1.1.1.2.1.2" stretchy="false" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S3.Ex15.m2.1.1.1.1.1.2.1.1" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.cmml"><mi id="S3.Ex15.m2.1.1.1.1.1.2.1.1.2" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.2.cmml">W</mi><mo id="S3.Ex15.m2.1.1.1.1.1.2.1.1.1" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.1.cmml">∪</mo><mi id="S3.Ex15.m2.1.1.1.1.1.2.1.1.3" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.3.cmml">Z</mi></mrow><mo id="S3.Ex15.m2.1.1.1.1.1.2.1.3" stretchy="false" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex15.m2.1.1.1.1.3" stretchy="false" xref="S3.Ex15.m2.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex15.m2.1b"><apply id="S3.Ex15.m2.1.1.cmml" xref="S3.Ex15.m2.1.1"><eq id="S3.Ex15.m2.1.1.2.cmml" xref="S3.Ex15.m2.1.1.2"></eq><csymbol cd="latexml" id="S3.Ex15.m2.1.1.3.cmml" xref="S3.Ex15.m2.1.1.3">absent</csymbol><apply id="S3.Ex15.m2.1.1.1.2.cmml" xref="S3.Ex15.m2.1.1.1.1"><abs id="S3.Ex15.m2.1.1.1.2.1.cmml" xref="S3.Ex15.m2.1.1.1.1.2"></abs><apply id="S3.Ex15.m2.1.1.1.1.1.cmml" xref="S3.Ex15.m2.1.1.1.1.1"><intersect id="S3.Ex15.m2.1.1.1.1.1.3.cmml" xref="S3.Ex15.m2.1.1.1.1.1.3"></intersect><apply id="S3.Ex15.m2.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex15.m2.1.1.1.1.1.1.1"><setdiff id="S3.Ex15.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.1"></setdiff><ci id="S3.Ex15.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.Ex15.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex15.m2.1.1.1.1.1.1.1.1.3">𝐵</ci></apply><apply id="S3.Ex15.m2.1.1.1.1.1.2.1.1.cmml" xref="S3.Ex15.m2.1.1.1.1.1.2.1"><union id="S3.Ex15.m2.1.1.1.1.1.2.1.1.1.cmml" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.1"></union><ci id="S3.Ex15.m2.1.1.1.1.1.2.1.1.2.cmml" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.2">𝑊</ci><ci id="S3.Ex15.m2.1.1.1.1.1.2.1.1.3.cmml" xref="S3.Ex15.m2.1.1.1.1.1.2.1.1.3">𝑍</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex15.m2.1c">\displaystyle=|(A\setminus B)\cap(W\cup Z)|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex15.m2.1d">= | ( italic_A ∖ italic_B ) ∩ ( italic_W ∪ italic_Z ) |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq|W\setminus B|+|Z\setminus B|" class="ltx_Math" display="inline" id="S3.Ex16.m1.2"><semantics id="S3.Ex16.m1.2a"><mrow id="S3.Ex16.m1.2.2" xref="S3.Ex16.m1.2.2.cmml"><mi id="S3.Ex16.m1.2.2.4" xref="S3.Ex16.m1.2.2.4.cmml"></mi><mo id="S3.Ex16.m1.2.2.3" xref="S3.Ex16.m1.2.2.3.cmml">≤</mo><mrow id="S3.Ex16.m1.2.2.2" xref="S3.Ex16.m1.2.2.2.cmml"><mrow id="S3.Ex16.m1.1.1.1.1.1" xref="S3.Ex16.m1.1.1.1.1.2.cmml"><mo id="S3.Ex16.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex16.m1.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex16.m1.1.1.1.1.1.1" xref="S3.Ex16.m1.1.1.1.1.1.1.cmml"><mi id="S3.Ex16.m1.1.1.1.1.1.1.2" xref="S3.Ex16.m1.1.1.1.1.1.1.2.cmml">W</mi><mo id="S3.Ex16.m1.1.1.1.1.1.1.1" xref="S3.Ex16.m1.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex16.m1.1.1.1.1.1.1.3" xref="S3.Ex16.m1.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex16.m1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex16.m1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex16.m1.2.2.2.3" xref="S3.Ex16.m1.2.2.2.3.cmml">+</mo><mrow id="S3.Ex16.m1.2.2.2.2.1" xref="S3.Ex16.m1.2.2.2.2.2.cmml"><mo id="S3.Ex16.m1.2.2.2.2.1.2" stretchy="false" xref="S3.Ex16.m1.2.2.2.2.2.1.cmml">|</mo><mrow id="S3.Ex16.m1.2.2.2.2.1.1" xref="S3.Ex16.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex16.m1.2.2.2.2.1.1.2" xref="S3.Ex16.m1.2.2.2.2.1.1.2.cmml">Z</mi><mo id="S3.Ex16.m1.2.2.2.2.1.1.1" xref="S3.Ex16.m1.2.2.2.2.1.1.1.cmml">∖</mo><mi id="S3.Ex16.m1.2.2.2.2.1.1.3" xref="S3.Ex16.m1.2.2.2.2.1.1.3.cmml">B</mi></mrow><mo id="S3.Ex16.m1.2.2.2.2.1.3" stretchy="false" xref="S3.Ex16.m1.2.2.2.2.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex16.m1.2b"><apply id="S3.Ex16.m1.2.2.cmml" xref="S3.Ex16.m1.2.2"><leq id="S3.Ex16.m1.2.2.3.cmml" xref="S3.Ex16.m1.2.2.3"></leq><csymbol cd="latexml" id="S3.Ex16.m1.2.2.4.cmml" xref="S3.Ex16.m1.2.2.4">absent</csymbol><apply id="S3.Ex16.m1.2.2.2.cmml" xref="S3.Ex16.m1.2.2.2"><plus id="S3.Ex16.m1.2.2.2.3.cmml" xref="S3.Ex16.m1.2.2.2.3"></plus><apply id="S3.Ex16.m1.1.1.1.1.2.cmml" xref="S3.Ex16.m1.1.1.1.1.1"><abs id="S3.Ex16.m1.1.1.1.1.2.1.cmml" xref="S3.Ex16.m1.1.1.1.1.1.2"></abs><apply id="S3.Ex16.m1.1.1.1.1.1.1.cmml" xref="S3.Ex16.m1.1.1.1.1.1.1"><setdiff id="S3.Ex16.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex16.m1.1.1.1.1.1.1.1"></setdiff><ci id="S3.Ex16.m1.1.1.1.1.1.1.2.cmml" xref="S3.Ex16.m1.1.1.1.1.1.1.2">𝑊</ci><ci id="S3.Ex16.m1.1.1.1.1.1.1.3.cmml" xref="S3.Ex16.m1.1.1.1.1.1.1.3">𝐵</ci></apply></apply><apply id="S3.Ex16.m1.2.2.2.2.2.cmml" xref="S3.Ex16.m1.2.2.2.2.1"><abs id="S3.Ex16.m1.2.2.2.2.2.1.cmml" xref="S3.Ex16.m1.2.2.2.2.1.2"></abs><apply id="S3.Ex16.m1.2.2.2.2.1.1.cmml" xref="S3.Ex16.m1.2.2.2.2.1.1"><setdiff id="S3.Ex16.m1.2.2.2.2.1.1.1.cmml" xref="S3.Ex16.m1.2.2.2.2.1.1.1"></setdiff><ci id="S3.Ex16.m1.2.2.2.2.1.1.2.cmml" xref="S3.Ex16.m1.2.2.2.2.1.1.2">𝑍</ci><ci id="S3.Ex16.m1.2.2.2.2.1.1.3.cmml" xref="S3.Ex16.m1.2.2.2.2.1.1.3">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex16.m1.2c">\displaystyle\leq|W\setminus B|+|Z\setminus B|</annotation><annotation 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id="S3.Ex17.m1.3.3.1.1.1.1.1.cmml" xref="S3.Ex17.m1.3.3.1.1.1.1.1"><intersect id="S3.Ex17.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.Ex17.m1.3.3.1.1.1.1.1.1"></intersect><ci id="S3.Ex17.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.Ex17.m1.3.3.1.1.1.1.1.2">𝐴</ci><ci id="S3.Ex17.m1.3.3.1.1.1.1.1.3.cmml" xref="S3.Ex17.m1.3.3.1.1.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex17.m1.3c">\displaystyle\leq\frac{(2+\tfrac{1}{6})\,|A\setminus B|}{\ell+2}+3\,|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex17.m1.3d">≤ divide start_ARG ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) | italic_A ∖ italic_B | end_ARG start_ARG roman_ℓ + 2 end_ARG + 3 | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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xref="S3.Ex18.m1.3.3.5.3">2</cn></apply></apply><apply id="S3.Ex18.m1.4.4.1.1.cmml" xref="S3.Ex18.m1.4.4.1.1"><times id="S3.Ex18.m1.4.4.1.1.2.cmml" xref="S3.Ex18.m1.4.4.1.1.2"></times><cn id="S3.Ex18.m1.4.4.1.1.3.cmml" type="integer" xref="S3.Ex18.m1.4.4.1.1.3">3</cn><apply id="S3.Ex18.m1.4.4.1.1.1.2.cmml" xref="S3.Ex18.m1.4.4.1.1.1.1"><abs id="S3.Ex18.m1.4.4.1.1.1.2.1.cmml" xref="S3.Ex18.m1.4.4.1.1.1.1.2"></abs><apply id="S3.Ex18.m1.4.4.1.1.1.1.1.cmml" xref="S3.Ex18.m1.4.4.1.1.1.1.1"><intersect id="S3.Ex18.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.Ex18.m1.4.4.1.1.1.1.1.1"></intersect><ci id="S3.Ex18.m1.4.4.1.1.1.1.1.2.cmml" xref="S3.Ex18.m1.4.4.1.1.1.1.1.2">𝐴</ci><ci id="S3.Ex18.m1.4.4.1.1.1.1.1.3.cmml" xref="S3.Ex18.m1.4.4.1.1.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex18.m1.4c">\displaystyle\leq\frac{(2+\tfrac{1}{6})\,(\tfrac{2}{3})^{d}|W_{\ell+1}|}{\ell+% 2}+3\,|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex18.m1.4d">≤ divide start_ARG ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT | end_ARG start_ARG roman_ℓ + 2 end_ARG + 3 | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{(2+\tfrac{1}{6})\,(\tfrac{2}{3})^{d}((\ell+1)|W|+|\Delta_{% \ell+1}|)}{\ell+2}+3\,|A\cap B|" class="ltx_Math" display="inline" id="S3.Ex19.m1.5"><semantics id="S3.Ex19.m1.5a"><mrow id="S3.Ex19.m1.5.5" xref="S3.Ex19.m1.5.5.cmml"><mi id="S3.Ex19.m1.5.5.3" 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xref="S3.Ex19.m1.5.5.1.1.1.1.1"><intersect id="S3.Ex19.m1.5.5.1.1.1.1.1.1.cmml" xref="S3.Ex19.m1.5.5.1.1.1.1.1.1"></intersect><ci id="S3.Ex19.m1.5.5.1.1.1.1.1.2.cmml" xref="S3.Ex19.m1.5.5.1.1.1.1.1.2">𝐴</ci><ci id="S3.Ex19.m1.5.5.1.1.1.1.1.3.cmml" xref="S3.Ex19.m1.5.5.1.1.1.1.1.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex19.m1.5c">\displaystyle=\frac{(2+\tfrac{1}{6})\,(\tfrac{2}{3})^{d}((\ell+1)|W|+|\Delta_{% \ell+1}|)}{\ell+2}+3\,|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex19.m1.5d">= divide start_ARG ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ( ( roman_ℓ + 1 ) | italic_W | + | roman_Δ start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT | ) end_ARG start_ARG roman_ℓ + 2 end_ARG + 3 | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle<\frac{(2+\tfrac{1}{6})\,(\tfrac{2}{3})^{d}(\ell+2)|W|}{\ell+2}+3% \,|A\cap B|" class="ltx_Math" display="inline" id="S3.Ex20.m1.5"><semantics id="S3.Ex20.m1.5a"><mrow id="S3.Ex20.m1.5.5" xref="S3.Ex20.m1.5.5.cmml"><mi id="S3.Ex20.m1.5.5.3" xref="S3.Ex20.m1.5.5.3.cmml"></mi><mo id="S3.Ex20.m1.5.5.2" xref="S3.Ex20.m1.5.5.2.cmml"><</mo><mrow id="S3.Ex20.m1.5.5.1" xref="S3.Ex20.m1.5.5.1.cmml"><mstyle displaystyle="true" id="S3.Ex20.m1.4.4" xref="S3.Ex20.m1.4.4.cmml"><mfrac id="S3.Ex20.m1.4.4a" xref="S3.Ex20.m1.4.4.cmml"><mrow id="S3.Ex20.m1.4.4.4" xref="S3.Ex20.m1.4.4.4.cmml"><mrow id="S3.Ex20.m1.3.3.3.3.1" xref="S3.Ex20.m1.3.3.3.3.1.1.cmml"><mo id="S3.Ex20.m1.3.3.3.3.1.2" stretchy="false" 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encoding="application/x-tex" id="S3.Ex20.m1.5c">\displaystyle<\frac{(2+\tfrac{1}{6})\,(\tfrac{2}{3})^{d}(\ell+2)|W|}{\ell+2}+3% \,|A\cap B|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex20.m1.5d">< divide start_ARG ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ( roman_ℓ + 2 ) | italic_W | end_ARG start_ARG roman_ℓ + 2 end_ARG + 3 | italic_A ∩ italic_B |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq(2+\tfrac{1}{6})ta\cdot(\tfrac{2}{3})^{d}+3da\enspace.\qed" class="ltx_Math" display="inline" id="S3.Ex21.m1.2"><semantics id="S3.Ex21.m1.2a"><mrow 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id="S3.Ex21.m1.2c">\displaystyle\leq(2+\tfrac{1}{6})ta\cdot(\tfrac{2}{3})^{d}+3da\enspace.\qed</annotation><annotation encoding="application/x-llamapun" id="S3.Ex21.m1.2d">≤ ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) italic_t italic_a ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT + 3 italic_d italic_a . italic_∎</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.17.p6"> <p class="ltx_p" id="S3.17.p6.8">For each node <math alttext="y" class="ltx_Math" display="inline" id="S3.17.p6.1.m1.1"><semantics id="S3.17.p6.1.m1.1a"><mi id="S3.17.p6.1.m1.1.1" xref="S3.17.p6.1.m1.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.17.p6.1.m1.1b"><ci id="S3.17.p6.1.m1.1.1.cmml" xref="S3.17.p6.1.m1.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p6.1.m1.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.17.p6.1.m1.1d">italic_y</annotation></semantics></math> of <math alttext="T^{\prime}_{y}" class="ltx_Math" display="inline" id="S3.17.p6.2.m2.1"><semantics id="S3.17.p6.2.m2.1a"><msubsup id="S3.17.p6.2.m2.1.1" xref="S3.17.p6.2.m2.1.1.cmml"><mi id="S3.17.p6.2.m2.1.1.2.2" xref="S3.17.p6.2.m2.1.1.2.2.cmml">T</mi><mi id="S3.17.p6.2.m2.1.1.3" xref="S3.17.p6.2.m2.1.1.3.cmml">y</mi><mo id="S3.17.p6.2.m2.1.1.2.3" xref="S3.17.p6.2.m2.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.17.p6.2.m2.1b"><apply id="S3.17.p6.2.m2.1.1.cmml" xref="S3.17.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.17.p6.2.m2.1.1.1.cmml" xref="S3.17.p6.2.m2.1.1">subscript</csymbol><apply id="S3.17.p6.2.m2.1.1.2.cmml" xref="S3.17.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.17.p6.2.m2.1.1.2.1.cmml" xref="S3.17.p6.2.m2.1.1">superscript</csymbol><ci id="S3.17.p6.2.m2.1.1.2.2.cmml" xref="S3.17.p6.2.m2.1.1.2.2">𝑇</ci><ci id="S3.17.p6.2.m2.1.1.2.3.cmml" xref="S3.17.p6.2.m2.1.1.2.3">′</ci></apply><ci id="S3.17.p6.2.m2.1.1.3.cmml" xref="S3.17.p6.2.m2.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p6.2.m2.1c">T^{\prime}_{y}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p6.2.m2.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>, define <math alttext="B_{y}:=(B^{\prime}_{y}\cap Y)\cup(\operatorname{int}_{\mathcal{T}^{\prime}_{Y}% }(y)\cap(W\cup Z))" class="ltx_Math" display="inline" id="S3.17.p6.3.m3.3"><semantics id="S3.17.p6.3.m3.3a"><mrow id="S3.17.p6.3.m3.3.3" xref="S3.17.p6.3.m3.3.3.cmml"><msub id="S3.17.p6.3.m3.3.3.4" xref="S3.17.p6.3.m3.3.3.4.cmml"><mi id="S3.17.p6.3.m3.3.3.4.2" xref="S3.17.p6.3.m3.3.3.4.2.cmml">B</mi><mi id="S3.17.p6.3.m3.3.3.4.3" xref="S3.17.p6.3.m3.3.3.4.3.cmml">y</mi></msub><mo id="S3.17.p6.3.m3.3.3.3" lspace="0.278em" rspace="0.278em" xref="S3.17.p6.3.m3.3.3.3.cmml">:=</mo><mrow id="S3.17.p6.3.m3.3.3.2" xref="S3.17.p6.3.m3.3.3.2.cmml"><mrow id="S3.17.p6.3.m3.2.2.1.1.1" xref="S3.17.p6.3.m3.2.2.1.1.1.1.cmml"><mo id="S3.17.p6.3.m3.2.2.1.1.1.2" stretchy="false" xref="S3.17.p6.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.17.p6.3.m3.2.2.1.1.1.1" xref="S3.17.p6.3.m3.2.2.1.1.1.1.cmml"><msubsup id="S3.17.p6.3.m3.2.2.1.1.1.1.2" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2.cmml"><mi id="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.2" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.2.cmml">B</mi><mi id="S3.17.p6.3.m3.2.2.1.1.1.1.2.3" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2.3.cmml">y</mi><mo id="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.3" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S3.17.p6.3.m3.2.2.1.1.1.1.1" xref="S3.17.p6.3.m3.2.2.1.1.1.1.1.cmml">∩</mo><mi id="S3.17.p6.3.m3.2.2.1.1.1.1.3" xref="S3.17.p6.3.m3.2.2.1.1.1.1.3.cmml">Y</mi></mrow><mo id="S3.17.p6.3.m3.2.2.1.1.1.3" stretchy="false" xref="S3.17.p6.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S3.17.p6.3.m3.3.3.2.3" xref="S3.17.p6.3.m3.3.3.2.3.cmml">∪</mo><mrow id="S3.17.p6.3.m3.3.3.2.2.1" xref="S3.17.p6.3.m3.3.3.2.2.1.1.cmml"><mo id="S3.17.p6.3.m3.3.3.2.2.1.2" stretchy="false" xref="S3.17.p6.3.m3.3.3.2.2.1.1.cmml">(</mo><mrow id="S3.17.p6.3.m3.3.3.2.2.1.1" xref="S3.17.p6.3.m3.3.3.2.2.1.1.cmml"><mrow id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.2.cmml"><msub id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.cmml"><mi id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.2" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.2.cmml">int</mi><msubsup id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.2" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.2.cmml">𝒯</mi><mi id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.3" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.3.cmml">Y</mi><mo id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.3" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.3.cmml">′</mo></msubsup></msub><mo id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1a" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.2" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.2.cmml"><mo id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.2.1" stretchy="false" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.2.cmml">(</mo><mi id="S3.17.p6.3.m3.1.1" xref="S3.17.p6.3.m3.1.1.cmml">y</mi><mo id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.2.2" stretchy="false" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.17.p6.3.m3.3.3.2.2.1.1.3" xref="S3.17.p6.3.m3.3.3.2.2.1.1.3.cmml">∩</mo><mrow id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.cmml"><mo id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.2" stretchy="false" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.cmml">(</mo><mrow id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.cmml"><mi id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.2" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.2.cmml">W</mi><mo id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.1" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.1.cmml">∪</mo><mi id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.3" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.3.cmml">Z</mi></mrow><mo id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.3" stretchy="false" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.17.p6.3.m3.3.3.2.2.1.3" stretchy="false" xref="S3.17.p6.3.m3.3.3.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p6.3.m3.3b"><apply id="S3.17.p6.3.m3.3.3.cmml" xref="S3.17.p6.3.m3.3.3"><csymbol cd="latexml" id="S3.17.p6.3.m3.3.3.3.cmml" xref="S3.17.p6.3.m3.3.3.3">assign</csymbol><apply id="S3.17.p6.3.m3.3.3.4.cmml" xref="S3.17.p6.3.m3.3.3.4"><csymbol cd="ambiguous" id="S3.17.p6.3.m3.3.3.4.1.cmml" xref="S3.17.p6.3.m3.3.3.4">subscript</csymbol><ci id="S3.17.p6.3.m3.3.3.4.2.cmml" xref="S3.17.p6.3.m3.3.3.4.2">𝐵</ci><ci id="S3.17.p6.3.m3.3.3.4.3.cmml" xref="S3.17.p6.3.m3.3.3.4.3">𝑦</ci></apply><apply id="S3.17.p6.3.m3.3.3.2.cmml" xref="S3.17.p6.3.m3.3.3.2"><union id="S3.17.p6.3.m3.3.3.2.3.cmml" xref="S3.17.p6.3.m3.3.3.2.3"></union><apply id="S3.17.p6.3.m3.2.2.1.1.1.1.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1"><intersect id="S3.17.p6.3.m3.2.2.1.1.1.1.1.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.1"></intersect><apply id="S3.17.p6.3.m3.2.2.1.1.1.1.2.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.17.p6.3.m3.2.2.1.1.1.1.2.1.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2">subscript</csymbol><apply id="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.1.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2">superscript</csymbol><ci id="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.2.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.2">𝐵</ci><ci id="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.3.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2.2.3">′</ci></apply><ci id="S3.17.p6.3.m3.2.2.1.1.1.1.2.3.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.2.3">𝑦</ci></apply><ci id="S3.17.p6.3.m3.2.2.1.1.1.1.3.cmml" xref="S3.17.p6.3.m3.2.2.1.1.1.1.3">𝑌</ci></apply><apply id="S3.17.p6.3.m3.3.3.2.2.1.1.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1"><intersect id="S3.17.p6.3.m3.3.3.2.2.1.1.3.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.3"></intersect><apply id="S3.17.p6.3.m3.3.3.2.2.1.1.1.2.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1"><apply id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.1.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.2.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.2">int</ci><apply id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.1.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3">subscript</csymbol><apply id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.1.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3">superscript</csymbol><ci id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.2.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.2">𝒯</ci><ci id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.3.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.2.3">′</ci></apply><ci id="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.3.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.1.1.1.3.3">𝑌</ci></apply></apply><ci id="S3.17.p6.3.m3.1.1.cmml" xref="S3.17.p6.3.m3.1.1">𝑦</ci></apply><apply id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1"><union id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.1.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.1"></union><ci id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.2.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.2">𝑊</ci><ci id="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.3.cmml" xref="S3.17.p6.3.m3.3.3.2.2.1.1.2.1.1.3">𝑍</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p6.3.m3.3c">B_{y}:=(B^{\prime}_{y}\cap Y)\cup(\operatorname{int}_{\mathcal{T}^{\prime}_{Y}% }(y)\cap(W\cup Z))</annotation><annotation encoding="application/x-llamapun" id="S3.17.p6.3.m3.3d">italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT := ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∩ italic_Y ) ∪ ( roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∩ ( italic_W ∪ italic_Z ) )</annotation></semantics></math> and let <math alttext="\mathcal{T}^{\prime\prime}_{Y}:=(B_{y}:y\in V(T^{\prime}_{y}))" class="ltx_math_unparsed" display="inline" id="S3.17.p6.4.m4.1"><semantics id="S3.17.p6.4.m4.1a"><mrow id="S3.17.p6.4.m4.1b"><msubsup id="S3.17.p6.4.m4.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.17.p6.4.m4.1.1.2.2">𝒯</mi><mi id="S3.17.p6.4.m4.1.1.3">Y</mi><mo id="S3.17.p6.4.m4.1.1.2.3">′′</mo></msubsup><mo id="S3.17.p6.4.m4.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S3.17.p6.4.m4.1.3"><mo id="S3.17.p6.4.m4.1.3.1" stretchy="false">(</mo><msub id="S3.17.p6.4.m4.1.3.2"><mi id="S3.17.p6.4.m4.1.3.2.2">B</mi><mi id="S3.17.p6.4.m4.1.3.2.3">y</mi></msub><mo id="S3.17.p6.4.m4.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.17.p6.4.m4.1.3.4">y</mi><mo id="S3.17.p6.4.m4.1.3.5">∈</mo><mi id="S3.17.p6.4.m4.1.3.6">V</mi><mrow id="S3.17.p6.4.m4.1.3.7"><mo id="S3.17.p6.4.m4.1.3.7.1" stretchy="false">(</mo><msubsup id="S3.17.p6.4.m4.1.3.7.2"><mi id="S3.17.p6.4.m4.1.3.7.2.2.2">T</mi><mi id="S3.17.p6.4.m4.1.3.7.2.3">y</mi><mo id="S3.17.p6.4.m4.1.3.7.2.2.3">′</mo></msubsup><mo id="S3.17.p6.4.m4.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S3.17.p6.4.m4.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.17.p6.4.m4.1c">\mathcal{T}^{\prime\prime}_{Y}:=(B_{y}:y\in V(T^{\prime}_{y}))</annotation><annotation encoding="application/x-llamapun" id="S3.17.p6.4.m4.1d">caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT := ( italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT : italic_y ∈ italic_V ( italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) )</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm4" title="Lemma 4. ‣ 2 Preliminaries ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4</span></a> (applied with the separation <math alttext="(X\cup W,Y\cup W)" class="ltx_Math" display="inline" id="S3.17.p6.5.m5.2"><semantics id="S3.17.p6.5.m5.2a"><mrow id="S3.17.p6.5.m5.2.2.2" xref="S3.17.p6.5.m5.2.2.3.cmml"><mo id="S3.17.p6.5.m5.2.2.2.3" stretchy="false" xref="S3.17.p6.5.m5.2.2.3.cmml">(</mo><mrow id="S3.17.p6.5.m5.1.1.1.1" xref="S3.17.p6.5.m5.1.1.1.1.cmml"><mi id="S3.17.p6.5.m5.1.1.1.1.2" xref="S3.17.p6.5.m5.1.1.1.1.2.cmml">X</mi><mo id="S3.17.p6.5.m5.1.1.1.1.1" xref="S3.17.p6.5.m5.1.1.1.1.1.cmml">∪</mo><mi id="S3.17.p6.5.m5.1.1.1.1.3" xref="S3.17.p6.5.m5.1.1.1.1.3.cmml">W</mi></mrow><mo id="S3.17.p6.5.m5.2.2.2.4" xref="S3.17.p6.5.m5.2.2.3.cmml">,</mo><mrow id="S3.17.p6.5.m5.2.2.2.2" xref="S3.17.p6.5.m5.2.2.2.2.cmml"><mi id="S3.17.p6.5.m5.2.2.2.2.2" xref="S3.17.p6.5.m5.2.2.2.2.2.cmml">Y</mi><mo id="S3.17.p6.5.m5.2.2.2.2.1" xref="S3.17.p6.5.m5.2.2.2.2.1.cmml">∪</mo><mi id="S3.17.p6.5.m5.2.2.2.2.3" xref="S3.17.p6.5.m5.2.2.2.2.3.cmml">W</mi></mrow><mo id="S3.17.p6.5.m5.2.2.2.5" stretchy="false" xref="S3.17.p6.5.m5.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p6.5.m5.2b"><interval closure="open" id="S3.17.p6.5.m5.2.2.3.cmml" xref="S3.17.p6.5.m5.2.2.2"><apply id="S3.17.p6.5.m5.1.1.1.1.cmml" xref="S3.17.p6.5.m5.1.1.1.1"><union id="S3.17.p6.5.m5.1.1.1.1.1.cmml" xref="S3.17.p6.5.m5.1.1.1.1.1"></union><ci id="S3.17.p6.5.m5.1.1.1.1.2.cmml" xref="S3.17.p6.5.m5.1.1.1.1.2">𝑋</ci><ci id="S3.17.p6.5.m5.1.1.1.1.3.cmml" xref="S3.17.p6.5.m5.1.1.1.1.3">𝑊</ci></apply><apply id="S3.17.p6.5.m5.2.2.2.2.cmml" xref="S3.17.p6.5.m5.2.2.2.2"><union id="S3.17.p6.5.m5.2.2.2.2.1.cmml" xref="S3.17.p6.5.m5.2.2.2.2.1"></union><ci id="S3.17.p6.5.m5.2.2.2.2.2.cmml" xref="S3.17.p6.5.m5.2.2.2.2.2">𝑌</ci><ci id="S3.17.p6.5.m5.2.2.2.2.3.cmml" xref="S3.17.p6.5.m5.2.2.2.2.3">𝑊</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p6.5.m5.2c">(X\cup W,Y\cup W)</annotation><annotation encoding="application/x-llamapun" id="S3.17.p6.5.m5.2d">( italic_X ∪ italic_W , italic_Y ∪ italic_W )</annotation></semantics></math>), <math alttext="\mathcal{T}^{\prime\prime}_{Y}" class="ltx_Math" display="inline" id="S3.17.p6.6.m6.1"><semantics id="S3.17.p6.6.m6.1a"><msubsup id="S3.17.p6.6.m6.1.1" xref="S3.17.p6.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.17.p6.6.m6.1.1.2.2" xref="S3.17.p6.6.m6.1.1.2.2.cmml">𝒯</mi><mi id="S3.17.p6.6.m6.1.1.3" xref="S3.17.p6.6.m6.1.1.3.cmml">Y</mi><mo id="S3.17.p6.6.m6.1.1.2.3" xref="S3.17.p6.6.m6.1.1.2.3.cmml">′′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.17.p6.6.m6.1b"><apply id="S3.17.p6.6.m6.1.1.cmml" xref="S3.17.p6.6.m6.1.1"><csymbol cd="ambiguous" id="S3.17.p6.6.m6.1.1.1.cmml" xref="S3.17.p6.6.m6.1.1">subscript</csymbol><apply id="S3.17.p6.6.m6.1.1.2.cmml" xref="S3.17.p6.6.m6.1.1"><csymbol cd="ambiguous" id="S3.17.p6.6.m6.1.1.2.1.cmml" xref="S3.17.p6.6.m6.1.1">superscript</csymbol><ci id="S3.17.p6.6.m6.1.1.2.2.cmml" xref="S3.17.p6.6.m6.1.1.2.2">𝒯</ci><ci id="S3.17.p6.6.m6.1.1.2.3.cmml" xref="S3.17.p6.6.m6.1.1.2.3">′′</ci></apply><ci id="S3.17.p6.6.m6.1.1.3.cmml" xref="S3.17.p6.6.m6.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p6.6.m6.1c">\mathcal{T}^{\prime\prime}_{Y}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p6.6.m6.1d">caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math> is a tree decomposition of <math alttext="G[Y]" class="ltx_Math" display="inline" id="S3.17.p6.7.m7.1"><semantics id="S3.17.p6.7.m7.1a"><mrow id="S3.17.p6.7.m7.1.2" xref="S3.17.p6.7.m7.1.2.cmml"><mi id="S3.17.p6.7.m7.1.2.2" xref="S3.17.p6.7.m7.1.2.2.cmml">G</mi><mo id="S3.17.p6.7.m7.1.2.1" xref="S3.17.p6.7.m7.1.2.1.cmml">⁢</mo><mrow id="S3.17.p6.7.m7.1.2.3.2" xref="S3.17.p6.7.m7.1.2.3.1.cmml"><mo id="S3.17.p6.7.m7.1.2.3.2.1" stretchy="false" xref="S3.17.p6.7.m7.1.2.3.1.1.cmml">[</mo><mi id="S3.17.p6.7.m7.1.1" xref="S3.17.p6.7.m7.1.1.cmml">Y</mi><mo id="S3.17.p6.7.m7.1.2.3.2.2" stretchy="false" xref="S3.17.p6.7.m7.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p6.7.m7.1b"><apply id="S3.17.p6.7.m7.1.2.cmml" xref="S3.17.p6.7.m7.1.2"><times id="S3.17.p6.7.m7.1.2.1.cmml" xref="S3.17.p6.7.m7.1.2.1"></times><ci id="S3.17.p6.7.m7.1.2.2.cmml" xref="S3.17.p6.7.m7.1.2.2">𝐺</ci><apply id="S3.17.p6.7.m7.1.2.3.1.cmml" xref="S3.17.p6.7.m7.1.2.3.2"><csymbol cd="latexml" id="S3.17.p6.7.m7.1.2.3.1.1.cmml" xref="S3.17.p6.7.m7.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.17.p6.7.m7.1.1.cmml" xref="S3.17.p6.7.m7.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p6.7.m7.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="S3.17.p6.7.m7.1d">italic_G [ italic_Y ]</annotation></semantics></math> in which the root bag contains <math alttext="(X\cup W)\cap(Y\cup W)=Z\cup W" class="ltx_Math" display="inline" id="S3.17.p6.8.m8.2"><semantics id="S3.17.p6.8.m8.2a"><mrow id="S3.17.p6.8.m8.2.2" xref="S3.17.p6.8.m8.2.2.cmml"><mrow id="S3.17.p6.8.m8.2.2.2" xref="S3.17.p6.8.m8.2.2.2.cmml"><mrow id="S3.17.p6.8.m8.1.1.1.1.1" xref="S3.17.p6.8.m8.1.1.1.1.1.1.cmml"><mo id="S3.17.p6.8.m8.1.1.1.1.1.2" stretchy="false" xref="S3.17.p6.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.17.p6.8.m8.1.1.1.1.1.1" xref="S3.17.p6.8.m8.1.1.1.1.1.1.cmml"><mi id="S3.17.p6.8.m8.1.1.1.1.1.1.2" xref="S3.17.p6.8.m8.1.1.1.1.1.1.2.cmml">X</mi><mo id="S3.17.p6.8.m8.1.1.1.1.1.1.1" xref="S3.17.p6.8.m8.1.1.1.1.1.1.1.cmml">∪</mo><mi id="S3.17.p6.8.m8.1.1.1.1.1.1.3" xref="S3.17.p6.8.m8.1.1.1.1.1.1.3.cmml">W</mi></mrow><mo id="S3.17.p6.8.m8.1.1.1.1.1.3" stretchy="false" xref="S3.17.p6.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.17.p6.8.m8.2.2.2.3" xref="S3.17.p6.8.m8.2.2.2.3.cmml">∩</mo><mrow id="S3.17.p6.8.m8.2.2.2.2.1" xref="S3.17.p6.8.m8.2.2.2.2.1.1.cmml"><mo id="S3.17.p6.8.m8.2.2.2.2.1.2" stretchy="false" xref="S3.17.p6.8.m8.2.2.2.2.1.1.cmml">(</mo><mrow id="S3.17.p6.8.m8.2.2.2.2.1.1" xref="S3.17.p6.8.m8.2.2.2.2.1.1.cmml"><mi id="S3.17.p6.8.m8.2.2.2.2.1.1.2" xref="S3.17.p6.8.m8.2.2.2.2.1.1.2.cmml">Y</mi><mo id="S3.17.p6.8.m8.2.2.2.2.1.1.1" xref="S3.17.p6.8.m8.2.2.2.2.1.1.1.cmml">∪</mo><mi id="S3.17.p6.8.m8.2.2.2.2.1.1.3" xref="S3.17.p6.8.m8.2.2.2.2.1.1.3.cmml">W</mi></mrow><mo id="S3.17.p6.8.m8.2.2.2.2.1.3" stretchy="false" xref="S3.17.p6.8.m8.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.17.p6.8.m8.2.2.3" xref="S3.17.p6.8.m8.2.2.3.cmml">=</mo><mrow id="S3.17.p6.8.m8.2.2.4" xref="S3.17.p6.8.m8.2.2.4.cmml"><mi id="S3.17.p6.8.m8.2.2.4.2" xref="S3.17.p6.8.m8.2.2.4.2.cmml">Z</mi><mo id="S3.17.p6.8.m8.2.2.4.1" xref="S3.17.p6.8.m8.2.2.4.1.cmml">∪</mo><mi id="S3.17.p6.8.m8.2.2.4.3" xref="S3.17.p6.8.m8.2.2.4.3.cmml">W</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p6.8.m8.2b"><apply id="S3.17.p6.8.m8.2.2.cmml" xref="S3.17.p6.8.m8.2.2"><eq id="S3.17.p6.8.m8.2.2.3.cmml" xref="S3.17.p6.8.m8.2.2.3"></eq><apply id="S3.17.p6.8.m8.2.2.2.cmml" xref="S3.17.p6.8.m8.2.2.2"><intersect id="S3.17.p6.8.m8.2.2.2.3.cmml" xref="S3.17.p6.8.m8.2.2.2.3"></intersect><apply id="S3.17.p6.8.m8.1.1.1.1.1.1.cmml" xref="S3.17.p6.8.m8.1.1.1.1.1"><union id="S3.17.p6.8.m8.1.1.1.1.1.1.1.cmml" xref="S3.17.p6.8.m8.1.1.1.1.1.1.1"></union><ci id="S3.17.p6.8.m8.1.1.1.1.1.1.2.cmml" xref="S3.17.p6.8.m8.1.1.1.1.1.1.2">𝑋</ci><ci id="S3.17.p6.8.m8.1.1.1.1.1.1.3.cmml" xref="S3.17.p6.8.m8.1.1.1.1.1.1.3">𝑊</ci></apply><apply id="S3.17.p6.8.m8.2.2.2.2.1.1.cmml" xref="S3.17.p6.8.m8.2.2.2.2.1"><union id="S3.17.p6.8.m8.2.2.2.2.1.1.1.cmml" xref="S3.17.p6.8.m8.2.2.2.2.1.1.1"></union><ci id="S3.17.p6.8.m8.2.2.2.2.1.1.2.cmml" xref="S3.17.p6.8.m8.2.2.2.2.1.1.2">𝑌</ci><ci id="S3.17.p6.8.m8.2.2.2.2.1.1.3.cmml" xref="S3.17.p6.8.m8.2.2.2.2.1.1.3">𝑊</ci></apply></apply><apply id="S3.17.p6.8.m8.2.2.4.cmml" xref="S3.17.p6.8.m8.2.2.4"><union id="S3.17.p6.8.m8.2.2.4.1.cmml" xref="S3.17.p6.8.m8.2.2.4.1"></union><ci id="S3.17.p6.8.m8.2.2.4.2.cmml" xref="S3.17.p6.8.m8.2.2.4.2">𝑍</ci><ci id="S3.17.p6.8.m8.2.2.4.3.cmml" xref="S3.17.p6.8.m8.2.2.4.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p6.8.m8.2c">(X\cup W)\cap(Y\cup W)=Z\cup W</annotation><annotation encoding="application/x-llamapun" id="S3.17.p6.8.m8.2d">( italic_X ∪ italic_W ) ∩ ( italic_Y ∪ italic_W ) = italic_Z ∪ italic_W</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_clm" id="Thmthm10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm10.1.1.1">Claim 10</span></span><span class="ltx_text ltx_font_bold" id="Thmthm10.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm10.p1"> <p class="ltx_p" id="Thmthm10.p1.3"><span class="ltx_text ltx_font_italic" id="Thmthm10.p1.3.3">For each leaf <math alttext="y" class="ltx_Math" display="inline" id="Thmthm10.p1.1.1.m1.1"><semantics id="Thmthm10.p1.1.1.m1.1a"><mi id="Thmthm10.p1.1.1.m1.1.1" xref="Thmthm10.p1.1.1.m1.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.1.1.m1.1b"><ci id="Thmthm10.p1.1.1.m1.1.1.cmml" xref="Thmthm10.p1.1.1.m1.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.1.1.m1.1c">y</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.1.1.m1.1d">italic_y</annotation></semantics></math> of <math alttext="T^{\prime}_{Y}" class="ltx_Math" display="inline" id="Thmthm10.p1.2.2.m2.1"><semantics id="Thmthm10.p1.2.2.m2.1a"><msubsup id="Thmthm10.p1.2.2.m2.1.1" xref="Thmthm10.p1.2.2.m2.1.1.cmml"><mi id="Thmthm10.p1.2.2.m2.1.1.2.2" xref="Thmthm10.p1.2.2.m2.1.1.2.2.cmml">T</mi><mi id="Thmthm10.p1.2.2.m2.1.1.3" xref="Thmthm10.p1.2.2.m2.1.1.3.cmml">Y</mi><mo id="Thmthm10.p1.2.2.m2.1.1.2.3" xref="Thmthm10.p1.2.2.m2.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.2.2.m2.1b"><apply id="Thmthm10.p1.2.2.m2.1.1.cmml" xref="Thmthm10.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmthm10.p1.2.2.m2.1.1.1.cmml" xref="Thmthm10.p1.2.2.m2.1.1">subscript</csymbol><apply id="Thmthm10.p1.2.2.m2.1.1.2.cmml" xref="Thmthm10.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmthm10.p1.2.2.m2.1.1.2.1.cmml" xref="Thmthm10.p1.2.2.m2.1.1">superscript</csymbol><ci id="Thmthm10.p1.2.2.m2.1.1.2.2.cmml" xref="Thmthm10.p1.2.2.m2.1.1.2.2">𝑇</ci><ci id="Thmthm10.p1.2.2.m2.1.1.2.3.cmml" xref="Thmthm10.p1.2.2.m2.1.1.2.3">′</ci></apply><ci id="Thmthm10.p1.2.2.m2.1.1.3.cmml" xref="Thmthm10.p1.2.2.m2.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.2.2.m2.1c">T^{\prime}_{Y}</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.2.2.m2.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="|\operatorname{\partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)|\leq ta" class="ltx_Math" display="inline" id="Thmthm10.p1.3.3.m3.2"><semantics id="Thmthm10.p1.3.3.m3.2a"><mrow id="Thmthm10.p1.3.3.m3.2.2" xref="Thmthm10.p1.3.3.m3.2.2.cmml"><mrow id="Thmthm10.p1.3.3.m3.2.2.1.1" xref="Thmthm10.p1.3.3.m3.2.2.1.2.cmml"><mo id="Thmthm10.p1.3.3.m3.2.2.1.1.2" stretchy="false" xref="Thmthm10.p1.3.3.m3.2.2.1.2.1.cmml">|</mo><mrow id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.2.cmml"><msub id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.cmml"><mi id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.2" mathvariant="normal" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.2.cmml">∂</mi><msubsup id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.2" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.2.cmml">𝒯</mi><mi id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.3" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.3.cmml">Y</mi><mo id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.3" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.3.cmml">′′</mo></msubsup></msub><mo id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1a" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.2.cmml">⁡</mo><mrow id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.2" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.2.cmml"><mo id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.2.1" stretchy="false" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.2.cmml">(</mo><mi id="Thmthm10.p1.3.3.m3.1.1" xref="Thmthm10.p1.3.3.m3.1.1.cmml">y</mi><mo id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.2.2" stretchy="false" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="Thmthm10.p1.3.3.m3.2.2.1.1.3" stretchy="false" xref="Thmthm10.p1.3.3.m3.2.2.1.2.1.cmml">|</mo></mrow><mo id="Thmthm10.p1.3.3.m3.2.2.2" xref="Thmthm10.p1.3.3.m3.2.2.2.cmml">≤</mo><mrow id="Thmthm10.p1.3.3.m3.2.2.3" xref="Thmthm10.p1.3.3.m3.2.2.3.cmml"><mi id="Thmthm10.p1.3.3.m3.2.2.3.2" xref="Thmthm10.p1.3.3.m3.2.2.3.2.cmml">t</mi><mo id="Thmthm10.p1.3.3.m3.2.2.3.1" xref="Thmthm10.p1.3.3.m3.2.2.3.1.cmml">⁢</mo><mi id="Thmthm10.p1.3.3.m3.2.2.3.3" xref="Thmthm10.p1.3.3.m3.2.2.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.3.3.m3.2b"><apply id="Thmthm10.p1.3.3.m3.2.2.cmml" xref="Thmthm10.p1.3.3.m3.2.2"><leq id="Thmthm10.p1.3.3.m3.2.2.2.cmml" xref="Thmthm10.p1.3.3.m3.2.2.2"></leq><apply id="Thmthm10.p1.3.3.m3.2.2.1.2.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1"><abs id="Thmthm10.p1.3.3.m3.2.2.1.2.1.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.2"></abs><apply id="Thmthm10.p1.3.3.m3.2.2.1.1.1.2.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1"><apply id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.1.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1">subscript</csymbol><partialdiff id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.2.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.2"></partialdiff><apply id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.1.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3">subscript</csymbol><apply id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.1.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3">superscript</csymbol><ci id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.2.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.2">𝒯</ci><ci id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.3.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.2.3">′′</ci></apply><ci id="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.3.cmml" xref="Thmthm10.p1.3.3.m3.2.2.1.1.1.1.1.3.3">𝑌</ci></apply></apply><ci id="Thmthm10.p1.3.3.m3.1.1.cmml" xref="Thmthm10.p1.3.3.m3.1.1">𝑦</ci></apply></apply><apply id="Thmthm10.p1.3.3.m3.2.2.3.cmml" xref="Thmthm10.p1.3.3.m3.2.2.3"><times id="Thmthm10.p1.3.3.m3.2.2.3.1.cmml" xref="Thmthm10.p1.3.3.m3.2.2.3.1"></times><ci id="Thmthm10.p1.3.3.m3.2.2.3.2.cmml" xref="Thmthm10.p1.3.3.m3.2.2.3.2">𝑡</ci><ci id="Thmthm10.p1.3.3.m3.2.2.3.3.cmml" xref="Thmthm10.p1.3.3.m3.2.2.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.3.3.m3.2c">|\operatorname{\partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)|\leq ta</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.3.3.m3.2d">| ∂ start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) | ≤ italic_t italic_a</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.18.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.18.3.p1"> <p class="ltx_p" id="S3.18.3.p1.6">Let <math alttext="y" class="ltx_Math" display="inline" id="S3.18.3.p1.1.m1.1"><semantics id="S3.18.3.p1.1.m1.1a"><mi id="S3.18.3.p1.1.m1.1.1" xref="S3.18.3.p1.1.m1.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.18.3.p1.1.m1.1b"><ci id="S3.18.3.p1.1.m1.1.1.cmml" xref="S3.18.3.p1.1.m1.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.18.3.p1.1.m1.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.1.m1.1d">italic_y</annotation></semantics></math> be a leaf of <math alttext="T^{\prime}_{Y}" class="ltx_Math" display="inline" id="S3.18.3.p1.2.m2.1"><semantics id="S3.18.3.p1.2.m2.1a"><msubsup id="S3.18.3.p1.2.m2.1.1" xref="S3.18.3.p1.2.m2.1.1.cmml"><mi id="S3.18.3.p1.2.m2.1.1.2.2" xref="S3.18.3.p1.2.m2.1.1.2.2.cmml">T</mi><mi id="S3.18.3.p1.2.m2.1.1.3" xref="S3.18.3.p1.2.m2.1.1.3.cmml">Y</mi><mo id="S3.18.3.p1.2.m2.1.1.2.3" xref="S3.18.3.p1.2.m2.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.18.3.p1.2.m2.1b"><apply id="S3.18.3.p1.2.m2.1.1.cmml" xref="S3.18.3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.18.3.p1.2.m2.1.1.1.cmml" xref="S3.18.3.p1.2.m2.1.1">subscript</csymbol><apply id="S3.18.3.p1.2.m2.1.1.2.cmml" xref="S3.18.3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.18.3.p1.2.m2.1.1.2.1.cmml" xref="S3.18.3.p1.2.m2.1.1">superscript</csymbol><ci id="S3.18.3.p1.2.m2.1.1.2.2.cmml" xref="S3.18.3.p1.2.m2.1.1.2.2">𝑇</ci><ci id="S3.18.3.p1.2.m2.1.1.2.3.cmml" xref="S3.18.3.p1.2.m2.1.1.2.3">′</ci></apply><ci id="S3.18.3.p1.2.m2.1.1.3.cmml" xref="S3.18.3.p1.2.m2.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.18.3.p1.2.m2.1c">T^{\prime}_{Y}</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.2.m2.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math> and let <math alttext="z" class="ltx_Math" display="inline" id="S3.18.3.p1.3.m3.1"><semantics id="S3.18.3.p1.3.m3.1a"><mi id="S3.18.3.p1.3.m3.1.1" xref="S3.18.3.p1.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.18.3.p1.3.m3.1b"><ci id="S3.18.3.p1.3.m3.1.1.cmml" xref="S3.18.3.p1.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.18.3.p1.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.3.m3.1d">italic_z</annotation></semantics></math> be the parent of <math alttext="y" class="ltx_Math" display="inline" id="S3.18.3.p1.4.m4.1"><semantics id="S3.18.3.p1.4.m4.1a"><mi id="S3.18.3.p1.4.m4.1.1" xref="S3.18.3.p1.4.m4.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.18.3.p1.4.m4.1b"><ci id="S3.18.3.p1.4.m4.1.1.cmml" xref="S3.18.3.p1.4.m4.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.18.3.p1.4.m4.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.4.m4.1d">italic_y</annotation></semantics></math>. Then, by <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm5" title="Lemma 5. ‣ 2 Preliminaries ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5</span></a>, <math alttext="|\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)|\leq(\tfrac{2}{3})^{h}\cdot|% W_{\ell+1}|" class="ltx_Math" display="inline" id="S3.18.3.p1.5.m5.4"><semantics id="S3.18.3.p1.5.m5.4a"><mrow id="S3.18.3.p1.5.m5.4.4" xref="S3.18.3.p1.5.m5.4.4.cmml"><mrow id="S3.18.3.p1.5.m5.3.3.1.1" xref="S3.18.3.p1.5.m5.3.3.1.2.cmml"><mo id="S3.18.3.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S3.18.3.p1.5.m5.3.3.1.2.1.cmml">|</mo><mrow id="S3.18.3.p1.5.m5.3.3.1.1.1.1" xref="S3.18.3.p1.5.m5.3.3.1.1.1.2.cmml"><msub id="S3.18.3.p1.5.m5.3.3.1.1.1.1.1" xref="S3.18.3.p1.5.m5.3.3.1.1.1.1.1.cmml"><mi id="S3.18.3.p1.5.m5.3.3.1.1.1.1.1.2" xref="S3.18.3.p1.5.m5.3.3.1.1.1.1.1.2.cmml">int</mi><msubsup 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xref="S3.18.3.p1.5.m5.2.2.3">3</cn></apply><ci id="S3.18.3.p1.5.m5.4.4.2.3.3.cmml" xref="S3.18.3.p1.5.m5.4.4.2.3.3">ℎ</ci></apply><apply id="S3.18.3.p1.5.m5.4.4.2.1.2.cmml" xref="S3.18.3.p1.5.m5.4.4.2.1.1"><abs id="S3.18.3.p1.5.m5.4.4.2.1.2.1.cmml" xref="S3.18.3.p1.5.m5.4.4.2.1.1.2"></abs><apply id="S3.18.3.p1.5.m5.4.4.2.1.1.1.cmml" xref="S3.18.3.p1.5.m5.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S3.18.3.p1.5.m5.4.4.2.1.1.1.1.cmml" xref="S3.18.3.p1.5.m5.4.4.2.1.1.1">subscript</csymbol><ci id="S3.18.3.p1.5.m5.4.4.2.1.1.1.2.cmml" xref="S3.18.3.p1.5.m5.4.4.2.1.1.1.2">𝑊</ci><apply id="S3.18.3.p1.5.m5.4.4.2.1.1.1.3.cmml" xref="S3.18.3.p1.5.m5.4.4.2.1.1.1.3"><plus id="S3.18.3.p1.5.m5.4.4.2.1.1.1.3.1.cmml" xref="S3.18.3.p1.5.m5.4.4.2.1.1.1.3.1"></plus><ci id="S3.18.3.p1.5.m5.4.4.2.1.1.1.3.2.cmml" xref="S3.18.3.p1.5.m5.4.4.2.1.1.1.3.2">ℓ</ci><cn id="S3.18.3.p1.5.m5.4.4.2.1.1.1.3.3.cmml" type="integer" xref="S3.18.3.p1.5.m5.4.4.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.18.3.p1.5.m5.4c">|\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)|\leq(\tfrac{2}{3})^{h}\cdot|% W_{\ell+1}|</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.5.m5.4d">| roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) | ≤ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ⋅ | italic_W start_POSTSUBSCRIPT roman_ℓ + 1 end_POSTSUBSCRIPT |</annotation></semantics></math> and <math alttext="\operatorname{\partial}_{\mathcal{T}_{Y}^{\prime\prime}}(x)=B_{y}\cap B_{z}% \subseteq(B^{\prime}_{y}\cap B^{\prime}_{z})\cup(\operatorname{int}_{\mathcal{% T}^{\prime}_{Y}}(y)\cap(W\cup Z))=\operatorname{\partial}_{\mathcal{T}_{Y}^{% \prime}}(y)\cup(\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)\cap(W\cup Z))" class="ltx_Math" display="inline" id="S3.18.3.p1.6.m6.9"><semantics id="S3.18.3.p1.6.m6.9a"><mrow id="S3.18.3.p1.6.m6.9.9" xref="S3.18.3.p1.6.m6.9.9.cmml"><mrow id="S3.18.3.p1.6.m6.5.5.1.1" xref="S3.18.3.p1.6.m6.5.5.1.2.cmml"><msub id="S3.18.3.p1.6.m6.5.5.1.1.1" xref="S3.18.3.p1.6.m6.5.5.1.1.1.cmml"><mi id="S3.18.3.p1.6.m6.5.5.1.1.1.2" mathvariant="normal" xref="S3.18.3.p1.6.m6.5.5.1.1.1.2.cmml">∂</mi><msubsup id="S3.18.3.p1.6.m6.5.5.1.1.1.3" xref="S3.18.3.p1.6.m6.5.5.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.18.3.p1.6.m6.5.5.1.1.1.3.2.2" xref="S3.18.3.p1.6.m6.5.5.1.1.1.3.2.2.cmml">𝒯</mi><mi id="S3.18.3.p1.6.m6.5.5.1.1.1.3.2.3" xref="S3.18.3.p1.6.m6.5.5.1.1.1.3.2.3.cmml">Y</mi><mo id="S3.18.3.p1.6.m6.5.5.1.1.1.3.3" xref="S3.18.3.p1.6.m6.5.5.1.1.1.3.3.cmml">′′</mo></msubsup></msub><mo id="S3.18.3.p1.6.m6.5.5.1.1a" xref="S3.18.3.p1.6.m6.5.5.1.2.cmml">⁡</mo><mrow id="S3.18.3.p1.6.m6.5.5.1.1.2" 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B^{\prime}_{z})\cup(\operatorname{int}_{\mathcal{% T}^{\prime}_{Y}}(y)\cap(W\cup Z))=\operatorname{\partial}_{\mathcal{T}_{Y}^{% \prime}}(y)\cup(\operatorname{int}_{\mathcal{T}^{\prime}_{Y}}(y)\cap(W\cup Z))</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.6.m6.9d">∂ start_POSTSUBSCRIPT caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) = italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∩ italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⊆ ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∩ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) ∪ ( roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∩ ( italic_W ∪ italic_Z ) ) = ∂ start_POSTSUBSCRIPT caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∪ ( roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∩ ( italic_W ∪ italic_Z ) )</annotation></semantics></math>. Therefore, by <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm5" title="Lemma 5. ‣ 2 Preliminaries ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm9" title="Claim 9. ‣ Proof of Theorem 1. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Claim</span> <span class="ltx_text ltx_ref_tag">9</span></a></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex22"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|\operatorname{\partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)|\leq|% \operatorname{\partial}_{\mathcal{T}^{\prime}_{Y}}(y)|+|\operatorname{int}_{% \mathcal{T}^{\prime}_{Y}}(y)\cap(W\cup Z)|\leq ha+(2+\tfrac{1}{6})ta\cdot(% \tfrac{2}{3})^{h}+3ha=(2+\tfrac{1}{6})ta\cdot(\tfrac{2}{3})^{h}+4ha\enspace." class="ltx_Math" display="block" id="S3.Ex22.m1.6"><semantics id="S3.Ex22.m1.6a"><mrow id="S3.Ex22.m1.6.6.1" xref="S3.Ex22.m1.6.6.1.1.cmml"><mrow id="S3.Ex22.m1.6.6.1.1" xref="S3.Ex22.m1.6.6.1.1.cmml"><mrow id="S3.Ex22.m1.6.6.1.1.1.1" xref="S3.Ex22.m1.6.6.1.1.1.2.cmml"><mo id="S3.Ex22.m1.6.6.1.1.1.1.2" stretchy="false" xref="S3.Ex22.m1.6.6.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex22.m1.6.6.1.1.1.1.1.1" xref="S3.Ex22.m1.6.6.1.1.1.1.1.2.cmml"><msub id="S3.Ex22.m1.6.6.1.1.1.1.1.1.1" xref="S3.Ex22.m1.6.6.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex22.m1.6.6.1.1.1.1.1.1.1.2" mathvariant="normal" xref="S3.Ex22.m1.6.6.1.1.1.1.1.1.1.2.cmml">∂</mi><msubsup id="S3.Ex22.m1.6.6.1.1.1.1.1.1.1.3" xref="S3.Ex22.m1.6.6.1.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex22.m1.6.6.1.1.1.1.1.1.1.3.2.2" 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′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) | ≤ | ∂ start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) | + | roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∩ ( italic_W ∪ italic_Z ) | ≤ italic_h italic_a + ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) italic_t italic_a ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT + 3 italic_h italic_a = ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) italic_t italic_a ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT + 4 italic_h italic_a .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.18.3.p1.9">The values of <math alttext="h" class="ltx_Math" display="inline" id="S3.18.3.p1.7.m1.1"><semantics id="S3.18.3.p1.7.m1.1a"><mi id="S3.18.3.p1.7.m1.1.1" xref="S3.18.3.p1.7.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.18.3.p1.7.m1.1b"><ci id="S3.18.3.p1.7.m1.1.1.cmml" xref="S3.18.3.p1.7.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.18.3.p1.7.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.7.m1.1d">italic_h</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" id="S3.18.3.p1.8.m2.1"><semantics id="S3.18.3.p1.8.m2.1a"><mi id="S3.18.3.p1.8.m2.1.1" xref="S3.18.3.p1.8.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.18.3.p1.8.m2.1b"><ci id="S3.18.3.p1.8.m2.1.1.cmml" xref="S3.18.3.p1.8.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.18.3.p1.8.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.8.m2.1d">italic_t</annotation></semantics></math>, defined above, are chosen so that the right hand side of this inequality is equal to <math alttext="ta" class="ltx_Math" display="inline" id="S3.18.3.p1.9.m3.1"><semantics id="S3.18.3.p1.9.m3.1a"><mrow id="S3.18.3.p1.9.m3.1.1" xref="S3.18.3.p1.9.m3.1.1.cmml"><mi id="S3.18.3.p1.9.m3.1.1.2" xref="S3.18.3.p1.9.m3.1.1.2.cmml">t</mi><mo id="S3.18.3.p1.9.m3.1.1.1" xref="S3.18.3.p1.9.m3.1.1.1.cmml">⁢</mo><mi id="S3.18.3.p1.9.m3.1.1.3" xref="S3.18.3.p1.9.m3.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.18.3.p1.9.m3.1b"><apply id="S3.18.3.p1.9.m3.1.1.cmml" xref="S3.18.3.p1.9.m3.1.1"><times id="S3.18.3.p1.9.m3.1.1.1.cmml" xref="S3.18.3.p1.9.m3.1.1.1"></times><ci id="S3.18.3.p1.9.m3.1.1.2.cmml" xref="S3.18.3.p1.9.m3.1.1.2">𝑡</ci><ci id="S3.18.3.p1.9.m3.1.1.3.cmml" xref="S3.18.3.p1.9.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.18.3.p1.9.m3.1c">ta</annotation><annotation encoding="application/x-llamapun" id="S3.18.3.p1.9.m3.1d">italic_t italic_a</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S3.19.p7"> <p class="ltx_p" id="S3.19.p7.18">By <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm10" title="Claim 10. ‣ Proof of Theorem 1. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Claim</span> <span class="ltx_text ltx_ref_tag">10</span></a>, <math alttext="|\operatorname{\partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)|\leq ta" class="ltx_Math" display="inline" id="S3.19.p7.1.m1.2"><semantics id="S3.19.p7.1.m1.2a"><mrow id="S3.19.p7.1.m1.2.2" xref="S3.19.p7.1.m1.2.2.cmml"><mrow id="S3.19.p7.1.m1.2.2.1.1" xref="S3.19.p7.1.m1.2.2.1.2.cmml"><mo id="S3.19.p7.1.m1.2.2.1.1.2" stretchy="false" xref="S3.19.p7.1.m1.2.2.1.2.1.cmml">|</mo><mrow id="S3.19.p7.1.m1.2.2.1.1.1.1" xref="S3.19.p7.1.m1.2.2.1.1.1.2.cmml"><msub id="S3.19.p7.1.m1.2.2.1.1.1.1.1" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.cmml"><mi id="S3.19.p7.1.m1.2.2.1.1.1.1.1.2" mathvariant="normal" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.2.cmml">∂</mi><msubsup id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.2" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.2.cmml">𝒯</mi><mi id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.3" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.3.cmml">Y</mi><mo id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.3" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.3.cmml">′′</mo></msubsup></msub><mo id="S3.19.p7.1.m1.2.2.1.1.1.1a" xref="S3.19.p7.1.m1.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S3.19.p7.1.m1.2.2.1.1.1.1.2" xref="S3.19.p7.1.m1.2.2.1.1.1.2.cmml"><mo id="S3.19.p7.1.m1.2.2.1.1.1.1.2.1" stretchy="false" xref="S3.19.p7.1.m1.2.2.1.1.1.2.cmml">(</mo><mi id="S3.19.p7.1.m1.1.1" xref="S3.19.p7.1.m1.1.1.cmml">y</mi><mo id="S3.19.p7.1.m1.2.2.1.1.1.1.2.2" stretchy="false" xref="S3.19.p7.1.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.19.p7.1.m1.2.2.1.1.3" stretchy="false" xref="S3.19.p7.1.m1.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.19.p7.1.m1.2.2.2" xref="S3.19.p7.1.m1.2.2.2.cmml">≤</mo><mrow id="S3.19.p7.1.m1.2.2.3" xref="S3.19.p7.1.m1.2.2.3.cmml"><mi id="S3.19.p7.1.m1.2.2.3.2" xref="S3.19.p7.1.m1.2.2.3.2.cmml">t</mi><mo id="S3.19.p7.1.m1.2.2.3.1" xref="S3.19.p7.1.m1.2.2.3.1.cmml">⁢</mo><mi id="S3.19.p7.1.m1.2.2.3.3" xref="S3.19.p7.1.m1.2.2.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.19.p7.1.m1.2b"><apply id="S3.19.p7.1.m1.2.2.cmml" xref="S3.19.p7.1.m1.2.2"><leq id="S3.19.p7.1.m1.2.2.2.cmml" xref="S3.19.p7.1.m1.2.2.2"></leq><apply id="S3.19.p7.1.m1.2.2.1.2.cmml" xref="S3.19.p7.1.m1.2.2.1.1"><abs id="S3.19.p7.1.m1.2.2.1.2.1.cmml" xref="S3.19.p7.1.m1.2.2.1.1.2"></abs><apply id="S3.19.p7.1.m1.2.2.1.1.1.2.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1"><apply id="S3.19.p7.1.m1.2.2.1.1.1.1.1.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.19.p7.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1">subscript</csymbol><partialdiff id="S3.19.p7.1.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.2"></partialdiff><apply id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.1.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3">subscript</csymbol><apply id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.1.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3">superscript</csymbol><ci id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.2.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.2">𝒯</ci><ci id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.3.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.2.3">′′</ci></apply><ci id="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.3.cmml" xref="S3.19.p7.1.m1.2.2.1.1.1.1.1.3.3">𝑌</ci></apply></apply><ci id="S3.19.p7.1.m1.1.1.cmml" xref="S3.19.p7.1.m1.1.1">𝑦</ci></apply></apply><apply id="S3.19.p7.1.m1.2.2.3.cmml" xref="S3.19.p7.1.m1.2.2.3"><times id="S3.19.p7.1.m1.2.2.3.1.cmml" xref="S3.19.p7.1.m1.2.2.3.1"></times><ci id="S3.19.p7.1.m1.2.2.3.2.cmml" xref="S3.19.p7.1.m1.2.2.3.2">𝑡</ci><ci id="S3.19.p7.1.m1.2.2.3.3.cmml" xref="S3.19.p7.1.m1.2.2.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.1.m1.2c">|\operatorname{\partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)|\leq ta</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.1.m1.2d">| ∂ start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) | ≤ italic_t italic_a</annotation></semantics></math> for each leaf <math alttext="y" class="ltx_Math" display="inline" id="S3.19.p7.2.m2.1"><semantics id="S3.19.p7.2.m2.1a"><mi id="S3.19.p7.2.m2.1.1" xref="S3.19.p7.2.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.19.p7.2.m2.1b"><ci id="S3.19.p7.2.m2.1.1.cmml" xref="S3.19.p7.2.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.2.m2.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.2.m2.1d">italic_y</annotation></semantics></math> of <math alttext="T_{Y}^{\prime}" class="ltx_Math" display="inline" id="S3.19.p7.3.m3.1"><semantics id="S3.19.p7.3.m3.1a"><msubsup id="S3.19.p7.3.m3.1.1" xref="S3.19.p7.3.m3.1.1.cmml"><mi id="S3.19.p7.3.m3.1.1.2.2" xref="S3.19.p7.3.m3.1.1.2.2.cmml">T</mi><mi id="S3.19.p7.3.m3.1.1.2.3" xref="S3.19.p7.3.m3.1.1.2.3.cmml">Y</mi><mo id="S3.19.p7.3.m3.1.1.3" xref="S3.19.p7.3.m3.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.19.p7.3.m3.1b"><apply id="S3.19.p7.3.m3.1.1.cmml" xref="S3.19.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S3.19.p7.3.m3.1.1.1.cmml" xref="S3.19.p7.3.m3.1.1">superscript</csymbol><apply id="S3.19.p7.3.m3.1.1.2.cmml" xref="S3.19.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S3.19.p7.3.m3.1.1.2.1.cmml" xref="S3.19.p7.3.m3.1.1">subscript</csymbol><ci id="S3.19.p7.3.m3.1.1.2.2.cmml" xref="S3.19.p7.3.m3.1.1.2.2">𝑇</ci><ci id="S3.19.p7.3.m3.1.1.2.3.cmml" xref="S3.19.p7.3.m3.1.1.2.3">𝑌</ci></apply><ci id="S3.19.p7.3.m3.1.1.3.cmml" xref="S3.19.p7.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.3.m3.1c">T_{Y}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Therefore, by the inductive hypothesis, <math alttext="G[\operatorname{int}_{\mathcal{T}^{\prime\prime}_{Y}}(y)\cup\operatorname{% \partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)]" class="ltx_Math" display="inline" id="S3.19.p7.4.m4.3"><semantics id="S3.19.p7.4.m4.3a"><mrow id="S3.19.p7.4.m4.3.3" xref="S3.19.p7.4.m4.3.3.cmml"><mi id="S3.19.p7.4.m4.3.3.3" xref="S3.19.p7.4.m4.3.3.3.cmml">G</mi><mo id="S3.19.p7.4.m4.3.3.2" xref="S3.19.p7.4.m4.3.3.2.cmml">⁢</mo><mrow id="S3.19.p7.4.m4.3.3.1.1" xref="S3.19.p7.4.m4.3.3.1.2.cmml"><mo id="S3.19.p7.4.m4.3.3.1.1.2" stretchy="false" xref="S3.19.p7.4.m4.3.3.1.2.1.cmml">[</mo><mrow id="S3.19.p7.4.m4.3.3.1.1.1" xref="S3.19.p7.4.m4.3.3.1.1.1.cmml"><mrow id="S3.19.p7.4.m4.3.3.1.1.1.1.1" xref="S3.19.p7.4.m4.3.3.1.1.1.1.2.cmml"><msub id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.cmml"><mi id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.2" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.2.cmml">int</mi><msubsup id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3" 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xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.1.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.2.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.2">int</ci><apply id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.1.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3">subscript</csymbol><apply id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.2.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.2.1.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.2.2.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.2.2">𝒯</ci><ci id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.2.3.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.2.3">′′</ci></apply><ci id="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.3.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.1.1.1.3.3">𝑌</ci></apply></apply><ci id="S3.19.p7.4.m4.1.1.cmml" xref="S3.19.p7.4.m4.1.1">𝑦</ci></apply><apply id="S3.19.p7.4.m4.3.3.1.1.1.2.2.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1"><apply id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.1.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1">subscript</csymbol><partialdiff id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.2.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.2"></partialdiff><apply id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.1.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3">subscript</csymbol><apply id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.2.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.2.1.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3">superscript</csymbol><ci id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.2.2.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.2.2">𝒯</ci><ci id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.2.3.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.2.3">′′</ci></apply><ci id="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.3.cmml" xref="S3.19.p7.4.m4.3.3.1.1.1.2.1.1.3.3">𝑌</ci></apply></apply><ci id="S3.19.p7.4.m4.2.2.cmml" xref="S3.19.p7.4.m4.2.2">𝑦</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.4.m4.3c">G[\operatorname{int}_{\mathcal{T}^{\prime\prime}_{Y}}(y)\cup\operatorname{% \partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)]</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.4.m4.3d">italic_G [ roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∪ ∂ start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) ]</annotation></semantics></math> has a tree decomposition <math alttext="\mathcal{T}^{y}:=(B_{y}:y\in V(T^{y}))" class="ltx_math_unparsed" display="inline" id="S3.19.p7.5.m5.1"><semantics id="S3.19.p7.5.m5.1a"><mrow id="S3.19.p7.5.m5.1b"><msup id="S3.19.p7.5.m5.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.19.p7.5.m5.1.1.2">𝒯</mi><mi id="S3.19.p7.5.m5.1.1.3">y</mi></msup><mo id="S3.19.p7.5.m5.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S3.19.p7.5.m5.1.3"><mo id="S3.19.p7.5.m5.1.3.1" stretchy="false">(</mo><msub id="S3.19.p7.5.m5.1.3.2"><mi id="S3.19.p7.5.m5.1.3.2.2">B</mi><mi id="S3.19.p7.5.m5.1.3.2.3">y</mi></msub><mo id="S3.19.p7.5.m5.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.19.p7.5.m5.1.3.4">y</mi><mo id="S3.19.p7.5.m5.1.3.5">∈</mo><mi id="S3.19.p7.5.m5.1.3.6">V</mi><mrow id="S3.19.p7.5.m5.1.3.7"><mo id="S3.19.p7.5.m5.1.3.7.1" stretchy="false">(</mo><msup id="S3.19.p7.5.m5.1.3.7.2"><mi id="S3.19.p7.5.m5.1.3.7.2.2">T</mi><mi id="S3.19.p7.5.m5.1.3.7.2.3">y</mi></msup><mo id="S3.19.p7.5.m5.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S3.19.p7.5.m5.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.19.p7.5.m5.1c">\mathcal{T}^{y}:=(B_{y}:y\in V(T^{y}))</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.5.m5.1d">caligraphic_T start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT := ( italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT : italic_y ∈ italic_V ( italic_T start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ) )</annotation></semantics></math> of width less than <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="S3.19.p7.6.m6.1"><semantics id="S3.19.p7.6.m6.1a"><mrow id="S3.19.p7.6.m6.1.1" xref="S3.19.p7.6.m6.1.1.cmml"><mrow id="S3.19.p7.6.m6.1.1.1.1" xref="S3.19.p7.6.m6.1.1.1.1.1.cmml"><mo id="S3.19.p7.6.m6.1.1.1.1.2" stretchy="false" xref="S3.19.p7.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S3.19.p7.6.m6.1.1.1.1.1" xref="S3.19.p7.6.m6.1.1.1.1.1.cmml"><mrow id="S3.19.p7.6.m6.1.1.1.1.1.2" xref="S3.19.p7.6.m6.1.1.1.1.1.2.cmml"><mn id="S3.19.p7.6.m6.1.1.1.1.1.2.2" xref="S3.19.p7.6.m6.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.19.p7.6.m6.1.1.1.1.1.2.1" xref="S3.19.p7.6.m6.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.19.p7.6.m6.1.1.1.1.1.2.3" xref="S3.19.p7.6.m6.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.19.p7.6.m6.1.1.1.1.1.1" xref="S3.19.p7.6.m6.1.1.1.1.1.1.cmml">+</mo><mn id="S3.19.p7.6.m6.1.1.1.1.1.3" xref="S3.19.p7.6.m6.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.19.p7.6.m6.1.1.1.1.3" stretchy="false" xref="S3.19.p7.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.19.p7.6.m6.1.1.2" xref="S3.19.p7.6.m6.1.1.2.cmml">⁢</mo><mi id="S3.19.p7.6.m6.1.1.3" xref="S3.19.p7.6.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.19.p7.6.m6.1b"><apply id="S3.19.p7.6.m6.1.1.cmml" xref="S3.19.p7.6.m6.1.1"><times id="S3.19.p7.6.m6.1.1.2.cmml" xref="S3.19.p7.6.m6.1.1.2"></times><apply id="S3.19.p7.6.m6.1.1.1.1.1.cmml" xref="S3.19.p7.6.m6.1.1.1.1"><plus id="S3.19.p7.6.m6.1.1.1.1.1.1.cmml" xref="S3.19.p7.6.m6.1.1.1.1.1.1"></plus><apply id="S3.19.p7.6.m6.1.1.1.1.1.2.cmml" xref="S3.19.p7.6.m6.1.1.1.1.1.2"><times id="S3.19.p7.6.m6.1.1.1.1.1.2.1.cmml" xref="S3.19.p7.6.m6.1.1.1.1.1.2.1"></times><cn id="S3.19.p7.6.m6.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.19.p7.6.m6.1.1.1.1.1.2.2">2</cn><ci id="S3.19.p7.6.m6.1.1.1.1.1.2.3.cmml" xref="S3.19.p7.6.m6.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.19.p7.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S3.19.p7.6.m6.1.1.1.1.1.3">1</cn></apply><ci id="S3.19.p7.6.m6.1.1.3.cmml" xref="S3.19.p7.6.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.6.m6.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.6.m6.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math> in which some bag <math alttext="B_{y_{0}}" class="ltx_Math" display="inline" id="S3.19.p7.7.m7.1"><semantics id="S3.19.p7.7.m7.1a"><msub id="S3.19.p7.7.m7.1.1" xref="S3.19.p7.7.m7.1.1.cmml"><mi id="S3.19.p7.7.m7.1.1.2" xref="S3.19.p7.7.m7.1.1.2.cmml">B</mi><msub id="S3.19.p7.7.m7.1.1.3" xref="S3.19.p7.7.m7.1.1.3.cmml"><mi id="S3.19.p7.7.m7.1.1.3.2" xref="S3.19.p7.7.m7.1.1.3.2.cmml">y</mi><mn id="S3.19.p7.7.m7.1.1.3.3" xref="S3.19.p7.7.m7.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.19.p7.7.m7.1b"><apply id="S3.19.p7.7.m7.1.1.cmml" xref="S3.19.p7.7.m7.1.1"><csymbol cd="ambiguous" id="S3.19.p7.7.m7.1.1.1.cmml" xref="S3.19.p7.7.m7.1.1">subscript</csymbol><ci id="S3.19.p7.7.m7.1.1.2.cmml" xref="S3.19.p7.7.m7.1.1.2">𝐵</ci><apply id="S3.19.p7.7.m7.1.1.3.cmml" xref="S3.19.p7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.7.m7.1.1.3.1.cmml" xref="S3.19.p7.7.m7.1.1.3">subscript</csymbol><ci id="S3.19.p7.7.m7.1.1.3.2.cmml" xref="S3.19.p7.7.m7.1.1.3.2">𝑦</ci><cn id="S3.19.p7.7.m7.1.1.3.3.cmml" type="integer" xref="S3.19.p7.7.m7.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.7.m7.1c">B_{y_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.7.m7.1d">italic_B start_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> contains <math alttext="\operatorname{\partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)" class="ltx_Math" display="inline" id="S3.19.p7.8.m8.2"><semantics id="S3.19.p7.8.m8.2a"><mrow id="S3.19.p7.8.m8.2.2.1" xref="S3.19.p7.8.m8.2.2.2.cmml"><msub id="S3.19.p7.8.m8.2.2.1.1" xref="S3.19.p7.8.m8.2.2.1.1.cmml"><mi id="S3.19.p7.8.m8.2.2.1.1.2" mathvariant="normal" xref="S3.19.p7.8.m8.2.2.1.1.2.cmml">∂</mi><msubsup id="S3.19.p7.8.m8.2.2.1.1.3" xref="S3.19.p7.8.m8.2.2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.19.p7.8.m8.2.2.1.1.3.2.2" xref="S3.19.p7.8.m8.2.2.1.1.3.2.2.cmml">𝒯</mi><mi id="S3.19.p7.8.m8.2.2.1.1.3.3" xref="S3.19.p7.8.m8.2.2.1.1.3.3.cmml">Y</mi><mo id="S3.19.p7.8.m8.2.2.1.1.3.2.3" xref="S3.19.p7.8.m8.2.2.1.1.3.2.3.cmml">′′</mo></msubsup></msub><mo id="S3.19.p7.8.m8.2.2.1a" xref="S3.19.p7.8.m8.2.2.2.cmml">⁡</mo><mrow id="S3.19.p7.8.m8.2.2.1.2" xref="S3.19.p7.8.m8.2.2.2.cmml"><mo id="S3.19.p7.8.m8.2.2.1.2.1" stretchy="false" xref="S3.19.p7.8.m8.2.2.2.cmml">(</mo><mi id="S3.19.p7.8.m8.1.1" xref="S3.19.p7.8.m8.1.1.cmml">y</mi><mo id="S3.19.p7.8.m8.2.2.1.2.2" stretchy="false" xref="S3.19.p7.8.m8.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.19.p7.8.m8.2b"><apply id="S3.19.p7.8.m8.2.2.2.cmml" xref="S3.19.p7.8.m8.2.2.1"><apply id="S3.19.p7.8.m8.2.2.1.1.cmml" xref="S3.19.p7.8.m8.2.2.1.1"><csymbol cd="ambiguous" id="S3.19.p7.8.m8.2.2.1.1.1.cmml" xref="S3.19.p7.8.m8.2.2.1.1">subscript</csymbol><partialdiff id="S3.19.p7.8.m8.2.2.1.1.2.cmml" xref="S3.19.p7.8.m8.2.2.1.1.2"></partialdiff><apply id="S3.19.p7.8.m8.2.2.1.1.3.cmml" xref="S3.19.p7.8.m8.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.8.m8.2.2.1.1.3.1.cmml" xref="S3.19.p7.8.m8.2.2.1.1.3">subscript</csymbol><apply id="S3.19.p7.8.m8.2.2.1.1.3.2.cmml" xref="S3.19.p7.8.m8.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.19.p7.8.m8.2.2.1.1.3.2.1.cmml" xref="S3.19.p7.8.m8.2.2.1.1.3">superscript</csymbol><ci id="S3.19.p7.8.m8.2.2.1.1.3.2.2.cmml" xref="S3.19.p7.8.m8.2.2.1.1.3.2.2">𝒯</ci><ci id="S3.19.p7.8.m8.2.2.1.1.3.2.3.cmml" xref="S3.19.p7.8.m8.2.2.1.1.3.2.3">′′</ci></apply><ci id="S3.19.p7.8.m8.2.2.1.1.3.3.cmml" xref="S3.19.p7.8.m8.2.2.1.1.3.3">𝑌</ci></apply></apply><ci id="S3.19.p7.8.m8.1.1.cmml" xref="S3.19.p7.8.m8.1.1">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.8.m8.2c">\operatorname{\partial}_{\mathcal{T}^{\prime\prime}_{Y}}(y)</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.8.m8.2d">∂ start_POSTSUBSCRIPT caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y )</annotation></semantics></math>, for each leaf <math alttext="y" class="ltx_Math" display="inline" id="S3.19.p7.9.m9.1"><semantics id="S3.19.p7.9.m9.1a"><mi id="S3.19.p7.9.m9.1.1" xref="S3.19.p7.9.m9.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.19.p7.9.m9.1b"><ci id="S3.19.p7.9.m9.1.1.cmml" xref="S3.19.p7.9.m9.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.9.m9.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.9.m9.1d">italic_y</annotation></semantics></math> of <math alttext="T_{Y}^{\prime}" class="ltx_Math" display="inline" id="S3.19.p7.10.m10.1"><semantics id="S3.19.p7.10.m10.1a"><msubsup id="S3.19.p7.10.m10.1.1" xref="S3.19.p7.10.m10.1.1.cmml"><mi id="S3.19.p7.10.m10.1.1.2.2" xref="S3.19.p7.10.m10.1.1.2.2.cmml">T</mi><mi id="S3.19.p7.10.m10.1.1.2.3" xref="S3.19.p7.10.m10.1.1.2.3.cmml">Y</mi><mo id="S3.19.p7.10.m10.1.1.3" xref="S3.19.p7.10.m10.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.19.p7.10.m10.1b"><apply id="S3.19.p7.10.m10.1.1.cmml" xref="S3.19.p7.10.m10.1.1"><csymbol cd="ambiguous" id="S3.19.p7.10.m10.1.1.1.cmml" xref="S3.19.p7.10.m10.1.1">superscript</csymbol><apply id="S3.19.p7.10.m10.1.1.2.cmml" xref="S3.19.p7.10.m10.1.1"><csymbol cd="ambiguous" id="S3.19.p7.10.m10.1.1.2.1.cmml" xref="S3.19.p7.10.m10.1.1">subscript</csymbol><ci id="S3.19.p7.10.m10.1.1.2.2.cmml" xref="S3.19.p7.10.m10.1.1.2.2">𝑇</ci><ci id="S3.19.p7.10.m10.1.1.2.3.cmml" xref="S3.19.p7.10.m10.1.1.2.3">𝑌</ci></apply><ci id="S3.19.p7.10.m10.1.1.3.cmml" xref="S3.19.p7.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.10.m10.1c">T_{Y}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.10.m10.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Create a new tree <math alttext="T_{Y}" class="ltx_Math" display="inline" id="S3.19.p7.11.m11.1"><semantics id="S3.19.p7.11.m11.1a"><msub id="S3.19.p7.11.m11.1.1" xref="S3.19.p7.11.m11.1.1.cmml"><mi id="S3.19.p7.11.m11.1.1.2" xref="S3.19.p7.11.m11.1.1.2.cmml">T</mi><mi id="S3.19.p7.11.m11.1.1.3" xref="S3.19.p7.11.m11.1.1.3.cmml">Y</mi></msub><annotation-xml encoding="MathML-Content" id="S3.19.p7.11.m11.1b"><apply id="S3.19.p7.11.m11.1.1.cmml" xref="S3.19.p7.11.m11.1.1"><csymbol cd="ambiguous" id="S3.19.p7.11.m11.1.1.1.cmml" xref="S3.19.p7.11.m11.1.1">subscript</csymbol><ci id="S3.19.p7.11.m11.1.1.2.cmml" xref="S3.19.p7.11.m11.1.1.2">𝑇</ci><ci id="S3.19.p7.11.m11.1.1.3.cmml" xref="S3.19.p7.11.m11.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.11.m11.1c">T_{Y}</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.11.m11.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="T_{Y}^{\prime}" class="ltx_Math" display="inline" id="S3.19.p7.12.m12.1"><semantics id="S3.19.p7.12.m12.1a"><msubsup id="S3.19.p7.12.m12.1.1" xref="S3.19.p7.12.m12.1.1.cmml"><mi id="S3.19.p7.12.m12.1.1.2.2" xref="S3.19.p7.12.m12.1.1.2.2.cmml">T</mi><mi id="S3.19.p7.12.m12.1.1.2.3" xref="S3.19.p7.12.m12.1.1.2.3.cmml">Y</mi><mo id="S3.19.p7.12.m12.1.1.3" xref="S3.19.p7.12.m12.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.19.p7.12.m12.1b"><apply id="S3.19.p7.12.m12.1.1.cmml" xref="S3.19.p7.12.m12.1.1"><csymbol cd="ambiguous" id="S3.19.p7.12.m12.1.1.1.cmml" xref="S3.19.p7.12.m12.1.1">superscript</csymbol><apply id="S3.19.p7.12.m12.1.1.2.cmml" xref="S3.19.p7.12.m12.1.1"><csymbol cd="ambiguous" id="S3.19.p7.12.m12.1.1.2.1.cmml" xref="S3.19.p7.12.m12.1.1">subscript</csymbol><ci id="S3.19.p7.12.m12.1.1.2.2.cmml" xref="S3.19.p7.12.m12.1.1.2.2">𝑇</ci><ci id="S3.19.p7.12.m12.1.1.2.3.cmml" xref="S3.19.p7.12.m12.1.1.2.3">𝑌</ci></apply><ci id="S3.19.p7.12.m12.1.1.3.cmml" xref="S3.19.p7.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.12.m12.1c">T_{Y}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.12.m12.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> by replacing each leaf <math alttext="y" class="ltx_Math" display="inline" id="S3.19.p7.13.m13.1"><semantics id="S3.19.p7.13.m13.1a"><mi id="S3.19.p7.13.m13.1.1" xref="S3.19.p7.13.m13.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.19.p7.13.m13.1b"><ci id="S3.19.p7.13.m13.1.1.cmml" xref="S3.19.p7.13.m13.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.13.m13.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.13.m13.1d">italic_y</annotation></semantics></math> of <math alttext="T_{Y}^{\prime}" class="ltx_Math" display="inline" id="S3.19.p7.14.m14.1"><semantics id="S3.19.p7.14.m14.1a"><msubsup id="S3.19.p7.14.m14.1.1" xref="S3.19.p7.14.m14.1.1.cmml"><mi id="S3.19.p7.14.m14.1.1.2.2" xref="S3.19.p7.14.m14.1.1.2.2.cmml">T</mi><mi id="S3.19.p7.14.m14.1.1.2.3" xref="S3.19.p7.14.m14.1.1.2.3.cmml">Y</mi><mo id="S3.19.p7.14.m14.1.1.3" xref="S3.19.p7.14.m14.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.19.p7.14.m14.1b"><apply id="S3.19.p7.14.m14.1.1.cmml" xref="S3.19.p7.14.m14.1.1"><csymbol cd="ambiguous" id="S3.19.p7.14.m14.1.1.1.cmml" xref="S3.19.p7.14.m14.1.1">superscript</csymbol><apply id="S3.19.p7.14.m14.1.1.2.cmml" xref="S3.19.p7.14.m14.1.1"><csymbol cd="ambiguous" id="S3.19.p7.14.m14.1.1.2.1.cmml" xref="S3.19.p7.14.m14.1.1">subscript</csymbol><ci id="S3.19.p7.14.m14.1.1.2.2.cmml" xref="S3.19.p7.14.m14.1.1.2.2">𝑇</ci><ci id="S3.19.p7.14.m14.1.1.2.3.cmml" xref="S3.19.p7.14.m14.1.1.2.3">𝑌</ci></apply><ci id="S3.19.p7.14.m14.1.1.3.cmml" xref="S3.19.p7.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.14.m14.1c">T_{Y}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.14.m14.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with the node <math alttext="y_{0}" class="ltx_Math" display="inline" id="S3.19.p7.15.m15.1"><semantics id="S3.19.p7.15.m15.1a"><msub id="S3.19.p7.15.m15.1.1" xref="S3.19.p7.15.m15.1.1.cmml"><mi id="S3.19.p7.15.m15.1.1.2" xref="S3.19.p7.15.m15.1.1.2.cmml">y</mi><mn id="S3.19.p7.15.m15.1.1.3" xref="S3.19.p7.15.m15.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.19.p7.15.m15.1b"><apply id="S3.19.p7.15.m15.1.1.cmml" xref="S3.19.p7.15.m15.1.1"><csymbol cd="ambiguous" id="S3.19.p7.15.m15.1.1.1.cmml" xref="S3.19.p7.15.m15.1.1">subscript</csymbol><ci id="S3.19.p7.15.m15.1.1.2.cmml" xref="S3.19.p7.15.m15.1.1.2">𝑦</ci><cn id="S3.19.p7.15.m15.1.1.3.cmml" type="integer" xref="S3.19.p7.15.m15.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.15.m15.1c">y_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.15.m15.1d">italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> from the tree <math alttext="T^{y}" class="ltx_Math" display="inline" id="S3.19.p7.16.m16.1"><semantics id="S3.19.p7.16.m16.1a"><msup id="S3.19.p7.16.m16.1.1" xref="S3.19.p7.16.m16.1.1.cmml"><mi id="S3.19.p7.16.m16.1.1.2" xref="S3.19.p7.16.m16.1.1.2.cmml">T</mi><mi id="S3.19.p7.16.m16.1.1.3" xref="S3.19.p7.16.m16.1.1.3.cmml">y</mi></msup><annotation-xml encoding="MathML-Content" id="S3.19.p7.16.m16.1b"><apply id="S3.19.p7.16.m16.1.1.cmml" xref="S3.19.p7.16.m16.1.1"><csymbol cd="ambiguous" id="S3.19.p7.16.m16.1.1.1.cmml" xref="S3.19.p7.16.m16.1.1">superscript</csymbol><ci id="S3.19.p7.16.m16.1.1.2.cmml" xref="S3.19.p7.16.m16.1.1.2">𝑇</ci><ci id="S3.19.p7.16.m16.1.1.3.cmml" xref="S3.19.p7.16.m16.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.16.m16.1c">T^{y}</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.16.m16.1d">italic_T start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT</annotation></semantics></math>. Then <math alttext="\mathcal{T}_{Y}:=(B_{y}:y\in V(T_{Y}))" class="ltx_math_unparsed" display="inline" id="S3.19.p7.17.m17.1"><semantics id="S3.19.p7.17.m17.1a"><mrow id="S3.19.p7.17.m17.1b"><msub id="S3.19.p7.17.m17.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.19.p7.17.m17.1.1.2">𝒯</mi><mi id="S3.19.p7.17.m17.1.1.3">Y</mi></msub><mo id="S3.19.p7.17.m17.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S3.19.p7.17.m17.1.3"><mo id="S3.19.p7.17.m17.1.3.1" stretchy="false">(</mo><msub id="S3.19.p7.17.m17.1.3.2"><mi id="S3.19.p7.17.m17.1.3.2.2">B</mi><mi id="S3.19.p7.17.m17.1.3.2.3">y</mi></msub><mo id="S3.19.p7.17.m17.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.19.p7.17.m17.1.3.4">y</mi><mo id="S3.19.p7.17.m17.1.3.5">∈</mo><mi id="S3.19.p7.17.m17.1.3.6">V</mi><mrow id="S3.19.p7.17.m17.1.3.7"><mo id="S3.19.p7.17.m17.1.3.7.1" stretchy="false">(</mo><msub id="S3.19.p7.17.m17.1.3.7.2"><mi id="S3.19.p7.17.m17.1.3.7.2.2">T</mi><mi id="S3.19.p7.17.m17.1.3.7.2.3">Y</mi></msub><mo id="S3.19.p7.17.m17.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S3.19.p7.17.m17.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.19.p7.17.m17.1c">\mathcal{T}_{Y}:=(B_{y}:y\in V(T_{Y}))</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.17.m17.1d">caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT := ( italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT : italic_y ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) )</annotation></semantics></math> is a tree decomposition of <math alttext="G[Y]" class="ltx_Math" display="inline" id="S3.19.p7.18.m18.1"><semantics id="S3.19.p7.18.m18.1a"><mrow id="S3.19.p7.18.m18.1.2" xref="S3.19.p7.18.m18.1.2.cmml"><mi id="S3.19.p7.18.m18.1.2.2" xref="S3.19.p7.18.m18.1.2.2.cmml">G</mi><mo id="S3.19.p7.18.m18.1.2.1" xref="S3.19.p7.18.m18.1.2.1.cmml">⁢</mo><mrow id="S3.19.p7.18.m18.1.2.3.2" xref="S3.19.p7.18.m18.1.2.3.1.cmml"><mo id="S3.19.p7.18.m18.1.2.3.2.1" stretchy="false" xref="S3.19.p7.18.m18.1.2.3.1.1.cmml">[</mo><mi id="S3.19.p7.18.m18.1.1" xref="S3.19.p7.18.m18.1.1.cmml">Y</mi><mo id="S3.19.p7.18.m18.1.2.3.2.2" stretchy="false" xref="S3.19.p7.18.m18.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.19.p7.18.m18.1b"><apply id="S3.19.p7.18.m18.1.2.cmml" xref="S3.19.p7.18.m18.1.2"><times id="S3.19.p7.18.m18.1.2.1.cmml" xref="S3.19.p7.18.m18.1.2.1"></times><ci id="S3.19.p7.18.m18.1.2.2.cmml" xref="S3.19.p7.18.m18.1.2.2">𝐺</ci><apply id="S3.19.p7.18.m18.1.2.3.1.cmml" xref="S3.19.p7.18.m18.1.2.3.2"><csymbol cd="latexml" id="S3.19.p7.18.m18.1.2.3.1.1.cmml" xref="S3.19.p7.18.m18.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.19.p7.18.m18.1.1.cmml" xref="S3.19.p7.18.m18.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.19.p7.18.m18.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="S3.19.p7.18.m18.1d">italic_G [ italic_Y ]</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_clm" id="Thmthm11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm11.1.1.1">Claim 11</span></span><span class="ltx_text ltx_font_bold" id="Thmthm11.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm11.p1"> <p class="ltx_p" id="Thmthm11.p1.2"><span class="ltx_text ltx_font_italic" id="Thmthm11.p1.2.2">The width of <math alttext="\mathcal{T}_{Y}" class="ltx_Math" display="inline" id="Thmthm11.p1.1.1.m1.1"><semantics id="Thmthm11.p1.1.1.m1.1a"><msub id="Thmthm11.p1.1.1.m1.1.1" xref="Thmthm11.p1.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmthm11.p1.1.1.m1.1.1.2" xref="Thmthm11.p1.1.1.m1.1.1.2.cmml">𝒯</mi><mi id="Thmthm11.p1.1.1.m1.1.1.3" xref="Thmthm11.p1.1.1.m1.1.1.3.cmml">Y</mi></msub><annotation-xml encoding="MathML-Content" id="Thmthm11.p1.1.1.m1.1b"><apply id="Thmthm11.p1.1.1.m1.1.1.cmml" xref="Thmthm11.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmthm11.p1.1.1.m1.1.1.1.cmml" xref="Thmthm11.p1.1.1.m1.1.1">subscript</csymbol><ci id="Thmthm11.p1.1.1.m1.1.1.2.cmml" xref="Thmthm11.p1.1.1.m1.1.1.2">𝒯</ci><ci id="Thmthm11.p1.1.1.m1.1.1.3.cmml" xref="Thmthm11.p1.1.1.m1.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm11.p1.1.1.m1.1c">\mathcal{T}_{Y}</annotation><annotation encoding="application/x-llamapun" id="Thmthm11.p1.1.1.m1.1d">caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math> is less than <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="Thmthm11.p1.2.2.m2.1"><semantics id="Thmthm11.p1.2.2.m2.1a"><mrow id="Thmthm11.p1.2.2.m2.1.1" xref="Thmthm11.p1.2.2.m2.1.1.cmml"><mrow id="Thmthm11.p1.2.2.m2.1.1.1.1" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.cmml"><mo id="Thmthm11.p1.2.2.m2.1.1.1.1.2" stretchy="false" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="Thmthm11.p1.2.2.m2.1.1.1.1.1" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.cmml"><mrow id="Thmthm11.p1.2.2.m2.1.1.1.1.1.2" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.cmml"><mn id="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.2" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.2.cmml">2</mn><mo id="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.1" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.3" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="Thmthm11.p1.2.2.m2.1.1.1.1.1.1" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.1.cmml">+</mo><mn id="Thmthm11.p1.2.2.m2.1.1.1.1.1.3" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="Thmthm11.p1.2.2.m2.1.1.1.1.3" stretchy="false" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmthm11.p1.2.2.m2.1.1.2" xref="Thmthm11.p1.2.2.m2.1.1.2.cmml">⁢</mo><mi id="Thmthm11.p1.2.2.m2.1.1.3" xref="Thmthm11.p1.2.2.m2.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmthm11.p1.2.2.m2.1b"><apply id="Thmthm11.p1.2.2.m2.1.1.cmml" xref="Thmthm11.p1.2.2.m2.1.1"><times id="Thmthm11.p1.2.2.m2.1.1.2.cmml" xref="Thmthm11.p1.2.2.m2.1.1.2"></times><apply id="Thmthm11.p1.2.2.m2.1.1.1.1.1.cmml" xref="Thmthm11.p1.2.2.m2.1.1.1.1"><plus id="Thmthm11.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.1"></plus><apply id="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.cmml" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.2"><times id="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.1.cmml" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.1"></times><cn id="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.2.cmml" type="integer" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.2">2</cn><ci id="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.3.cmml" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="Thmthm11.p1.2.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="Thmthm11.p1.2.2.m2.1.1.1.1.1.3">1</cn></apply><ci id="Thmthm11.p1.2.2.m2.1.1.3.cmml" xref="Thmthm11.p1.2.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm11.p1.2.2.m2.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="Thmthm11.p1.2.2.m2.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.20.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Claim:.</h6> <div class="ltx_para" id="S3.20.4.p1"> <p class="ltx_p" id="S3.20.4.p1.6">The inductive hypothesis ensures that all bags of the tree decomposition have size at most <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="S3.20.4.p1.1.m1.1"><semantics id="S3.20.4.p1.1.m1.1a"><mrow id="S3.20.4.p1.1.m1.1.1" xref="S3.20.4.p1.1.m1.1.1.cmml"><mrow id="S3.20.4.p1.1.m1.1.1.1.1" xref="S3.20.4.p1.1.m1.1.1.1.1.1.cmml"><mo id="S3.20.4.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S3.20.4.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.20.4.p1.1.m1.1.1.1.1.1" xref="S3.20.4.p1.1.m1.1.1.1.1.1.cmml"><mrow id="S3.20.4.p1.1.m1.1.1.1.1.1.2" xref="S3.20.4.p1.1.m1.1.1.1.1.1.2.cmml"><mn id="S3.20.4.p1.1.m1.1.1.1.1.1.2.2" xref="S3.20.4.p1.1.m1.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.20.4.p1.1.m1.1.1.1.1.1.2.1" xref="S3.20.4.p1.1.m1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.20.4.p1.1.m1.1.1.1.1.1.2.3" xref="S3.20.4.p1.1.m1.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.20.4.p1.1.m1.1.1.1.1.1.1" xref="S3.20.4.p1.1.m1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.20.4.p1.1.m1.1.1.1.1.1.3" xref="S3.20.4.p1.1.m1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.20.4.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S3.20.4.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.20.4.p1.1.m1.1.1.2" xref="S3.20.4.p1.1.m1.1.1.2.cmml">⁢</mo><mi id="S3.20.4.p1.1.m1.1.1.3" xref="S3.20.4.p1.1.m1.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.1.m1.1b"><apply id="S3.20.4.p1.1.m1.1.1.cmml" xref="S3.20.4.p1.1.m1.1.1"><times id="S3.20.4.p1.1.m1.1.1.2.cmml" xref="S3.20.4.p1.1.m1.1.1.2"></times><apply id="S3.20.4.p1.1.m1.1.1.1.1.1.cmml" xref="S3.20.4.p1.1.m1.1.1.1.1"><plus id="S3.20.4.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.20.4.p1.1.m1.1.1.1.1.1.1"></plus><apply id="S3.20.4.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.20.4.p1.1.m1.1.1.1.1.1.2"><times id="S3.20.4.p1.1.m1.1.1.1.1.1.2.1.cmml" xref="S3.20.4.p1.1.m1.1.1.1.1.1.2.1"></times><cn id="S3.20.4.p1.1.m1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.20.4.p1.1.m1.1.1.1.1.1.2.2">2</cn><ci id="S3.20.4.p1.1.m1.1.1.1.1.1.2.3.cmml" xref="S3.20.4.p1.1.m1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.20.4.p1.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S3.20.4.p1.1.m1.1.1.1.1.1.3">1</cn></apply><ci id="S3.20.4.p1.1.m1.1.1.3.cmml" xref="S3.20.4.p1.1.m1.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.1.m1.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.1.m1.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math> except for those associated with non-leaf nodes of <math alttext="T^{\prime}_{Y}" class="ltx_Math" display="inline" id="S3.20.4.p1.2.m2.1"><semantics id="S3.20.4.p1.2.m2.1a"><msubsup id="S3.20.4.p1.2.m2.1.1" xref="S3.20.4.p1.2.m2.1.1.cmml"><mi id="S3.20.4.p1.2.m2.1.1.2.2" xref="S3.20.4.p1.2.m2.1.1.2.2.cmml">T</mi><mi id="S3.20.4.p1.2.m2.1.1.3" xref="S3.20.4.p1.2.m2.1.1.3.cmml">Y</mi><mo id="S3.20.4.p1.2.m2.1.1.2.3" xref="S3.20.4.p1.2.m2.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.2.m2.1b"><apply id="S3.20.4.p1.2.m2.1.1.cmml" xref="S3.20.4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.20.4.p1.2.m2.1.1.1.cmml" xref="S3.20.4.p1.2.m2.1.1">subscript</csymbol><apply id="S3.20.4.p1.2.m2.1.1.2.cmml" xref="S3.20.4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.20.4.p1.2.m2.1.1.2.1.cmml" xref="S3.20.4.p1.2.m2.1.1">superscript</csymbol><ci id="S3.20.4.p1.2.m2.1.1.2.2.cmml" xref="S3.20.4.p1.2.m2.1.1.2.2">𝑇</ci><ci id="S3.20.4.p1.2.m2.1.1.2.3.cmml" xref="S3.20.4.p1.2.m2.1.1.2.3">′</ci></apply><ci id="S3.20.4.p1.2.m2.1.1.3.cmml" xref="S3.20.4.p1.2.m2.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.2.m2.1c">T^{\prime}_{Y}</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.2.m2.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="y" class="ltx_Math" display="inline" id="S3.20.4.p1.3.m3.1"><semantics id="S3.20.4.p1.3.m3.1a"><mi id="S3.20.4.p1.3.m3.1.1" xref="S3.20.4.p1.3.m3.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.3.m3.1b"><ci id="S3.20.4.p1.3.m3.1.1.cmml" xref="S3.20.4.p1.3.m3.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.3.m3.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.3.m3.1d">italic_y</annotation></semantics></math> be a non-leaf node in <math alttext="T_{Y}^{\prime}" class="ltx_Math" display="inline" id="S3.20.4.p1.4.m4.1"><semantics id="S3.20.4.p1.4.m4.1a"><msubsup id="S3.20.4.p1.4.m4.1.1" xref="S3.20.4.p1.4.m4.1.1.cmml"><mi id="S3.20.4.p1.4.m4.1.1.2.2" xref="S3.20.4.p1.4.m4.1.1.2.2.cmml">T</mi><mi id="S3.20.4.p1.4.m4.1.1.2.3" xref="S3.20.4.p1.4.m4.1.1.2.3.cmml">Y</mi><mo id="S3.20.4.p1.4.m4.1.1.3" xref="S3.20.4.p1.4.m4.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.4.m4.1b"><apply id="S3.20.4.p1.4.m4.1.1.cmml" xref="S3.20.4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.20.4.p1.4.m4.1.1.1.cmml" xref="S3.20.4.p1.4.m4.1.1">superscript</csymbol><apply id="S3.20.4.p1.4.m4.1.1.2.cmml" xref="S3.20.4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.20.4.p1.4.m4.1.1.2.1.cmml" xref="S3.20.4.p1.4.m4.1.1">subscript</csymbol><ci id="S3.20.4.p1.4.m4.1.1.2.2.cmml" xref="S3.20.4.p1.4.m4.1.1.2.2">𝑇</ci><ci id="S3.20.4.p1.4.m4.1.1.2.3.cmml" xref="S3.20.4.p1.4.m4.1.1.2.3">𝑌</ci></apply><ci id="S3.20.4.p1.4.m4.1.1.3.cmml" xref="S3.20.4.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.4.m4.1c">T_{Y}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> whose depth is <math alttext="d<h" class="ltx_Math" display="inline" id="S3.20.4.p1.5.m5.1"><semantics id="S3.20.4.p1.5.m5.1a"><mrow id="S3.20.4.p1.5.m5.1.1" xref="S3.20.4.p1.5.m5.1.1.cmml"><mi id="S3.20.4.p1.5.m5.1.1.2" xref="S3.20.4.p1.5.m5.1.1.2.cmml">d</mi><mo id="S3.20.4.p1.5.m5.1.1.1" xref="S3.20.4.p1.5.m5.1.1.1.cmml"><</mo><mi id="S3.20.4.p1.5.m5.1.1.3" xref="S3.20.4.p1.5.m5.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.5.m5.1b"><apply id="S3.20.4.p1.5.m5.1.1.cmml" xref="S3.20.4.p1.5.m5.1.1"><lt id="S3.20.4.p1.5.m5.1.1.1.cmml" xref="S3.20.4.p1.5.m5.1.1.1"></lt><ci id="S3.20.4.p1.5.m5.1.1.2.cmml" xref="S3.20.4.p1.5.m5.1.1.2">𝑑</ci><ci id="S3.20.4.p1.5.m5.1.1.3.cmml" xref="S3.20.4.p1.5.m5.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.5.m5.1c">d<h</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.5.m5.1d">italic_d < italic_h</annotation></semantics></math>. If <math alttext="d=0" class="ltx_Math" display="inline" id="S3.20.4.p1.6.m6.1"><semantics id="S3.20.4.p1.6.m6.1a"><mrow id="S3.20.4.p1.6.m6.1.1" xref="S3.20.4.p1.6.m6.1.1.cmml"><mi id="S3.20.4.p1.6.m6.1.1.2" xref="S3.20.4.p1.6.m6.1.1.2.cmml">d</mi><mo id="S3.20.4.p1.6.m6.1.1.1" xref="S3.20.4.p1.6.m6.1.1.1.cmml">=</mo><mn id="S3.20.4.p1.6.m6.1.1.3" xref="S3.20.4.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.6.m6.1b"><apply id="S3.20.4.p1.6.m6.1.1.cmml" xref="S3.20.4.p1.6.m6.1.1"><eq id="S3.20.4.p1.6.m6.1.1.1.cmml" xref="S3.20.4.p1.6.m6.1.1.1"></eq><ci id="S3.20.4.p1.6.m6.1.1.2.cmml" xref="S3.20.4.p1.6.m6.1.1.2">𝑑</ci><cn id="S3.20.4.p1.6.m6.1.1.3.cmml" type="integer" xref="S3.20.4.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.6.m6.1c">d=0</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.6.m6.1d">italic_d = 0</annotation></semantics></math>, then</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|B_{y}|\leq|W\cup Z|+|B^{\prime}_{y}|\leq(2ta-1)+a<(2t+1)a\enspace." class="ltx_Math" display="block" id="S3.Ex23.m1.1"><semantics id="S3.Ex23.m1.1a"><mrow id="S3.Ex23.m1.1.1.1" xref="S3.Ex23.m1.1.1.1.1.cmml"><mrow id="S3.Ex23.m1.1.1.1.1" xref="S3.Ex23.m1.1.1.1.1.cmml"><mrow id="S3.Ex23.m1.1.1.1.1.1.1" xref="S3.Ex23.m1.1.1.1.1.1.2.cmml"><mo id="S3.Ex23.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.1.2.1.cmml">|</mo><msub id="S3.Ex23.m1.1.1.1.1.1.1.1" xref="S3.Ex23.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex23.m1.1.1.1.1.1.1.1.2" xref="S3.Ex23.m1.1.1.1.1.1.1.1.2.cmml">B</mi><mi id="S3.Ex23.m1.1.1.1.1.1.1.1.3" xref="S3.Ex23.m1.1.1.1.1.1.1.1.3.cmml">y</mi></msub><mo id="S3.Ex23.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex23.m1.1.1.1.1.7" xref="S3.Ex23.m1.1.1.1.1.7.cmml">≤</mo><mrow id="S3.Ex23.m1.1.1.1.1.3" xref="S3.Ex23.m1.1.1.1.1.3.cmml"><mrow id="S3.Ex23.m1.1.1.1.1.2.1.1" xref="S3.Ex23.m1.1.1.1.1.2.1.2.cmml"><mo id="S3.Ex23.m1.1.1.1.1.2.1.1.2" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.2.1.2.1.cmml">|</mo><mrow id="S3.Ex23.m1.1.1.1.1.2.1.1.1" xref="S3.Ex23.m1.1.1.1.1.2.1.1.1.cmml"><mi id="S3.Ex23.m1.1.1.1.1.2.1.1.1.2" xref="S3.Ex23.m1.1.1.1.1.2.1.1.1.2.cmml">W</mi><mo id="S3.Ex23.m1.1.1.1.1.2.1.1.1.1" xref="S3.Ex23.m1.1.1.1.1.2.1.1.1.1.cmml">∪</mo><mi id="S3.Ex23.m1.1.1.1.1.2.1.1.1.3" xref="S3.Ex23.m1.1.1.1.1.2.1.1.1.3.cmml">Z</mi></mrow><mo id="S3.Ex23.m1.1.1.1.1.2.1.1.3" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.2.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex23.m1.1.1.1.1.3.3" xref="S3.Ex23.m1.1.1.1.1.3.3.cmml">+</mo><mrow id="S3.Ex23.m1.1.1.1.1.3.2.1" xref="S3.Ex23.m1.1.1.1.1.3.2.2.cmml"><mo id="S3.Ex23.m1.1.1.1.1.3.2.1.2" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.3.2.2.1.cmml">|</mo><msubsup id="S3.Ex23.m1.1.1.1.1.3.2.1.1" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1.cmml"><mi id="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.2" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.2.cmml">B</mi><mi id="S3.Ex23.m1.1.1.1.1.3.2.1.1.3" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1.3.cmml">y</mi><mo id="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.3" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.Ex23.m1.1.1.1.1.3.2.1.3" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.3.2.2.1.cmml">|</mo></mrow></mrow><mo id="S3.Ex23.m1.1.1.1.1.8" xref="S3.Ex23.m1.1.1.1.1.8.cmml">≤</mo><mrow id="S3.Ex23.m1.1.1.1.1.4" xref="S3.Ex23.m1.1.1.1.1.4.cmml"><mrow id="S3.Ex23.m1.1.1.1.1.4.1.1" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.cmml"><mo id="S3.Ex23.m1.1.1.1.1.4.1.1.2" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.cmml">(</mo><mrow id="S3.Ex23.m1.1.1.1.1.4.1.1.1" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.cmml"><mrow id="S3.Ex23.m1.1.1.1.1.4.1.1.1.2" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.cmml"><mn id="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.2" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.2.cmml">2</mn><mo id="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.1" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.1.cmml">⁢</mo><mi id="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.3" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.3.cmml">t</mi><mo id="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.1a" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.1.cmml">⁢</mo><mi id="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.4" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.2.4.cmml">a</mi></mrow><mo id="S3.Ex23.m1.1.1.1.1.4.1.1.1.1" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.1.cmml">−</mo><mn id="S3.Ex23.m1.1.1.1.1.4.1.1.1.3" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.3.cmml">1</mn></mrow><mo id="S3.Ex23.m1.1.1.1.1.4.1.1.3" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.4.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex23.m1.1.1.1.1.4.2" xref="S3.Ex23.m1.1.1.1.1.4.2.cmml">+</mo><mi id="S3.Ex23.m1.1.1.1.1.4.3" xref="S3.Ex23.m1.1.1.1.1.4.3.cmml">a</mi></mrow><mo id="S3.Ex23.m1.1.1.1.1.9" xref="S3.Ex23.m1.1.1.1.1.9.cmml"><</mo><mrow id="S3.Ex23.m1.1.1.1.1.5" xref="S3.Ex23.m1.1.1.1.1.5.cmml"><mrow id="S3.Ex23.m1.1.1.1.1.5.1.1" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.cmml"><mo id="S3.Ex23.m1.1.1.1.1.5.1.1.2" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.cmml">(</mo><mrow id="S3.Ex23.m1.1.1.1.1.5.1.1.1" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.cmml"><mrow id="S3.Ex23.m1.1.1.1.1.5.1.1.1.2" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.cmml"><mn id="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.2" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.2.cmml">2</mn><mo id="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.1" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.1.cmml">⁢</mo><mi id="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.3" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.Ex23.m1.1.1.1.1.5.1.1.1.1" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.1.cmml">+</mo><mn id="S3.Ex23.m1.1.1.1.1.5.1.1.1.3" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.3.cmml">1</mn></mrow><mo id="S3.Ex23.m1.1.1.1.1.5.1.1.3" stretchy="false" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex23.m1.1.1.1.1.5.2" xref="S3.Ex23.m1.1.1.1.1.5.2.cmml">⁢</mo><mi id="S3.Ex23.m1.1.1.1.1.5.3" xref="S3.Ex23.m1.1.1.1.1.5.3.cmml">a</mi></mrow></mrow><mo id="S3.Ex23.m1.1.1.1.2" lspace="0.500em" xref="S3.Ex23.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex23.m1.1b"><apply id="S3.Ex23.m1.1.1.1.1.cmml" xref="S3.Ex23.m1.1.1.1"><and id="S3.Ex23.m1.1.1.1.1a.cmml" xref="S3.Ex23.m1.1.1.1"></and><apply id="S3.Ex23.m1.1.1.1.1b.cmml" xref="S3.Ex23.m1.1.1.1"><leq id="S3.Ex23.m1.1.1.1.1.7.cmml" xref="S3.Ex23.m1.1.1.1.1.7"></leq><apply id="S3.Ex23.m1.1.1.1.1.1.2.cmml" xref="S3.Ex23.m1.1.1.1.1.1.1"><abs id="S3.Ex23.m1.1.1.1.1.1.2.1.cmml" xref="S3.Ex23.m1.1.1.1.1.1.1.2"></abs><apply id="S3.Ex23.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex23.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex23.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex23.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex23.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex23.m1.1.1.1.1.1.1.1.2">𝐵</ci><ci id="S3.Ex23.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex23.m1.1.1.1.1.1.1.1.3">𝑦</ci></apply></apply><apply id="S3.Ex23.m1.1.1.1.1.3.cmml" xref="S3.Ex23.m1.1.1.1.1.3"><plus id="S3.Ex23.m1.1.1.1.1.3.3.cmml" xref="S3.Ex23.m1.1.1.1.1.3.3"></plus><apply id="S3.Ex23.m1.1.1.1.1.2.1.2.cmml" xref="S3.Ex23.m1.1.1.1.1.2.1.1"><abs id="S3.Ex23.m1.1.1.1.1.2.1.2.1.cmml" xref="S3.Ex23.m1.1.1.1.1.2.1.1.2"></abs><apply id="S3.Ex23.m1.1.1.1.1.2.1.1.1.cmml" xref="S3.Ex23.m1.1.1.1.1.2.1.1.1"><union id="S3.Ex23.m1.1.1.1.1.2.1.1.1.1.cmml" xref="S3.Ex23.m1.1.1.1.1.2.1.1.1.1"></union><ci id="S3.Ex23.m1.1.1.1.1.2.1.1.1.2.cmml" xref="S3.Ex23.m1.1.1.1.1.2.1.1.1.2">𝑊</ci><ci id="S3.Ex23.m1.1.1.1.1.2.1.1.1.3.cmml" xref="S3.Ex23.m1.1.1.1.1.2.1.1.1.3">𝑍</ci></apply></apply><apply id="S3.Ex23.m1.1.1.1.1.3.2.2.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1"><abs id="S3.Ex23.m1.1.1.1.1.3.2.2.1.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1.2"></abs><apply id="S3.Ex23.m1.1.1.1.1.3.2.1.1.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1"><csymbol cd="ambiguous" id="S3.Ex23.m1.1.1.1.1.3.2.1.1.1.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1">subscript</csymbol><apply id="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1"><csymbol cd="ambiguous" id="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.1.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1">superscript</csymbol><ci id="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.2.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.2">𝐵</ci><ci id="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.3.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1.2.3">′</ci></apply><ci id="S3.Ex23.m1.1.1.1.1.3.2.1.1.3.cmml" xref="S3.Ex23.m1.1.1.1.1.3.2.1.1.3">𝑦</ci></apply></apply></apply></apply><apply id="S3.Ex23.m1.1.1.1.1c.cmml" xref="S3.Ex23.m1.1.1.1"><leq id="S3.Ex23.m1.1.1.1.1.8.cmml" xref="S3.Ex23.m1.1.1.1.1.8"></leq><share href="https://arxiv.org/html/2503.17112v1#S3.Ex23.m1.1.1.1.1.3.cmml" id="S3.Ex23.m1.1.1.1.1d.cmml" xref="S3.Ex23.m1.1.1.1"></share><apply id="S3.Ex23.m1.1.1.1.1.4.cmml" xref="S3.Ex23.m1.1.1.1.1.4"><plus id="S3.Ex23.m1.1.1.1.1.4.2.cmml" 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xref="S3.Ex23.m1.1.1.1.1.9"></lt><share href="https://arxiv.org/html/2503.17112v1#S3.Ex23.m1.1.1.1.1.4.cmml" id="S3.Ex23.m1.1.1.1.1f.cmml" xref="S3.Ex23.m1.1.1.1"></share><apply id="S3.Ex23.m1.1.1.1.1.5.cmml" xref="S3.Ex23.m1.1.1.1.1.5"><times id="S3.Ex23.m1.1.1.1.1.5.2.cmml" xref="S3.Ex23.m1.1.1.1.1.5.2"></times><apply id="S3.Ex23.m1.1.1.1.1.5.1.1.1.cmml" xref="S3.Ex23.m1.1.1.1.1.5.1.1"><plus id="S3.Ex23.m1.1.1.1.1.5.1.1.1.1.cmml" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.1"></plus><apply id="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.cmml" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.2"><times id="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.1.cmml" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.1"></times><cn id="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.2.cmml" type="integer" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.2">2</cn><ci id="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.3.cmml" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.2.3">𝑡</ci></apply><cn id="S3.Ex23.m1.1.1.1.1.5.1.1.1.3.cmml" type="integer" xref="S3.Ex23.m1.1.1.1.1.5.1.1.1.3">1</cn></apply><ci id="S3.Ex23.m1.1.1.1.1.5.3.cmml" xref="S3.Ex23.m1.1.1.1.1.5.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex23.m1.1c">|B_{y}|\leq|W\cup Z|+|B^{\prime}_{y}|\leq(2ta-1)+a<(2t+1)a\enspace.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex23.m1.1d">| italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | ≤ | italic_W ∪ italic_Z | + | italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | ≤ ( 2 italic_t italic_a - 1 ) + italic_a < ( 2 italic_t + 1 ) italic_a .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.20.4.p1.9">If <math alttext="d\geq 1" class="ltx_Math" display="inline" id="S3.20.4.p1.7.m1.1"><semantics id="S3.20.4.p1.7.m1.1a"><mrow id="S3.20.4.p1.7.m1.1.1" xref="S3.20.4.p1.7.m1.1.1.cmml"><mi id="S3.20.4.p1.7.m1.1.1.2" xref="S3.20.4.p1.7.m1.1.1.2.cmml">d</mi><mo id="S3.20.4.p1.7.m1.1.1.1" xref="S3.20.4.p1.7.m1.1.1.1.cmml">≥</mo><mn id="S3.20.4.p1.7.m1.1.1.3" xref="S3.20.4.p1.7.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.7.m1.1b"><apply id="S3.20.4.p1.7.m1.1.1.cmml" xref="S3.20.4.p1.7.m1.1.1"><geq id="S3.20.4.p1.7.m1.1.1.1.cmml" xref="S3.20.4.p1.7.m1.1.1.1"></geq><ci id="S3.20.4.p1.7.m1.1.1.2.cmml" xref="S3.20.4.p1.7.m1.1.1.2">𝑑</ci><cn id="S3.20.4.p1.7.m1.1.1.3.cmml" type="integer" xref="S3.20.4.p1.7.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.7.m1.1c">d\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.7.m1.1d">italic_d ≥ 1</annotation></semantics></math> then, by <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm5" title="Lemma 5. ‣ 2 Preliminaries ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5</span></a>, <math alttext="|B^{\prime}_{y}|\leq(d+1)a" class="ltx_Math" display="inline" id="S3.20.4.p1.8.m2.2"><semantics id="S3.20.4.p1.8.m2.2a"><mrow id="S3.20.4.p1.8.m2.2.2" xref="S3.20.4.p1.8.m2.2.2.cmml"><mrow id="S3.20.4.p1.8.m2.1.1.1.1" xref="S3.20.4.p1.8.m2.1.1.1.2.cmml"><mo id="S3.20.4.p1.8.m2.1.1.1.1.2" stretchy="false" xref="S3.20.4.p1.8.m2.1.1.1.2.1.cmml">|</mo><msubsup id="S3.20.4.p1.8.m2.1.1.1.1.1" xref="S3.20.4.p1.8.m2.1.1.1.1.1.cmml"><mi id="S3.20.4.p1.8.m2.1.1.1.1.1.2.2" xref="S3.20.4.p1.8.m2.1.1.1.1.1.2.2.cmml">B</mi><mi id="S3.20.4.p1.8.m2.1.1.1.1.1.3" xref="S3.20.4.p1.8.m2.1.1.1.1.1.3.cmml">y</mi><mo id="S3.20.4.p1.8.m2.1.1.1.1.1.2.3" xref="S3.20.4.p1.8.m2.1.1.1.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.20.4.p1.8.m2.1.1.1.1.3" stretchy="false" xref="S3.20.4.p1.8.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.20.4.p1.8.m2.2.2.3" xref="S3.20.4.p1.8.m2.2.2.3.cmml">≤</mo><mrow id="S3.20.4.p1.8.m2.2.2.2" xref="S3.20.4.p1.8.m2.2.2.2.cmml"><mrow id="S3.20.4.p1.8.m2.2.2.2.1.1" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.cmml"><mo id="S3.20.4.p1.8.m2.2.2.2.1.1.2" stretchy="false" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.cmml">(</mo><mrow id="S3.20.4.p1.8.m2.2.2.2.1.1.1" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.cmml"><mi id="S3.20.4.p1.8.m2.2.2.2.1.1.1.2" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.2.cmml">d</mi><mo id="S3.20.4.p1.8.m2.2.2.2.1.1.1.1" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.1.cmml">+</mo><mn id="S3.20.4.p1.8.m2.2.2.2.1.1.1.3" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S3.20.4.p1.8.m2.2.2.2.1.1.3" stretchy="false" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S3.20.4.p1.8.m2.2.2.2.2" xref="S3.20.4.p1.8.m2.2.2.2.2.cmml">⁢</mo><mi id="S3.20.4.p1.8.m2.2.2.2.3" xref="S3.20.4.p1.8.m2.2.2.2.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.8.m2.2b"><apply id="S3.20.4.p1.8.m2.2.2.cmml" xref="S3.20.4.p1.8.m2.2.2"><leq id="S3.20.4.p1.8.m2.2.2.3.cmml" xref="S3.20.4.p1.8.m2.2.2.3"></leq><apply id="S3.20.4.p1.8.m2.1.1.1.2.cmml" 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xref="S3.20.4.p1.8.m2.2.2.2.1.1"><plus id="S3.20.4.p1.8.m2.2.2.2.1.1.1.1.cmml" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.1"></plus><ci id="S3.20.4.p1.8.m2.2.2.2.1.1.1.2.cmml" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.2">𝑑</ci><cn id="S3.20.4.p1.8.m2.2.2.2.1.1.1.3.cmml" type="integer" xref="S3.20.4.p1.8.m2.2.2.2.1.1.1.3">1</cn></apply><ci id="S3.20.4.p1.8.m2.2.2.2.3.cmml" xref="S3.20.4.p1.8.m2.2.2.2.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.8.m2.2c">|B^{\prime}_{y}|\leq(d+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.8.m2.2d">| italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | ≤ ( italic_d + 1 ) italic_a</annotation></semantics></math> and <math alttext="|\operatorname{int}_{\mathcal{T}_{Y}^{\prime}}(y)|\leq|W_{\ell}|\cdot(\tfrac{2% }{3})^{d}" class="ltx_Math" display="inline" id="S3.20.4.p1.9.m3.4"><semantics id="S3.20.4.p1.9.m3.4a"><mrow id="S3.20.4.p1.9.m3.4.4" 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id="S3.20.4.p1.9.m3.4c">|\operatorname{int}_{\mathcal{T}_{Y}^{\prime}}(y)|\leq|W_{\ell}|\cdot(\tfrac{2% }{3})^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.9.m3.4d">| roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_y ) | ≤ | italic_W start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT | ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm9" title="Claim 9. ‣ Proof of Theorem 1. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Claim</span> <span class="ltx_text ltx_ref_tag">9</span></a>,</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="|B_{y}|\leq|\operatorname{int}_{\mathcal{T}_{Y}^{\prime}}(y)\cap(W\cup Z)|+|B^% {\prime}_{y}|\leq(2+\tfrac{1}{6})ta\cdot(\tfrac{2}{3})^{d}+3da+(d+1)a\leq(2+% \tfrac{1}{6})ta\cdot(\tfrac{2}{3})^{d}+(4d+1)a\enspace." class="ltx_Math" display="block" id="S3.Ex24.m1.4"><semantics id="S3.Ex24.m1.4a"><mrow id="S3.Ex24.m1.4.4.1" xref="S3.Ex24.m1.4.4.1.1.cmml"><mrow id="S3.Ex24.m1.4.4.1.1" xref="S3.Ex24.m1.4.4.1.1.cmml"><mrow id="S3.Ex24.m1.4.4.1.1.1.1" 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{\prime}_{y}|\leq(2+\tfrac{1}{6})ta\cdot(\tfrac{2}{3})^{d}+3da+(d+1)a\leq(2+% \tfrac{1}{6})ta\cdot(\tfrac{2}{3})^{d}+(4d+1)a\enspace.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex24.m1.4d">| italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | ≤ | roman_int start_POSTSUBSCRIPT caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_y ) ∩ ( italic_W ∪ italic_Z ) | + | italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | ≤ ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) italic_t italic_a ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT + 3 italic_d italic_a + ( italic_d + 1 ) italic_a ≤ ( 2 + divide start_ARG 1 end_ARG start_ARG 6 end_ARG ) italic_t italic_a ⋅ ( divide start_ARG 2 end_ARG start_ARG 3 end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT + ( 4 italic_d + 1 ) italic_a .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.20.4.p1.13">With the choices of <math alttext="t" class="ltx_Math" display="inline" id="S3.20.4.p1.10.m1.1"><semantics id="S3.20.4.p1.10.m1.1a"><mi id="S3.20.4.p1.10.m1.1.1" xref="S3.20.4.p1.10.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.10.m1.1b"><ci id="S3.20.4.p1.10.m1.1.1.cmml" xref="S3.20.4.p1.10.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.10.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.10.m1.1d">italic_t</annotation></semantics></math> and <math alttext="h" class="ltx_Math" display="inline" id="S3.20.4.p1.11.m2.1"><semantics id="S3.20.4.p1.11.m2.1a"><mi id="S3.20.4.p1.11.m2.1.1" xref="S3.20.4.p1.11.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.11.m2.1b"><ci id="S3.20.4.p1.11.m2.1.1.cmml" xref="S3.20.4.p1.11.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.11.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.11.m2.1d">italic_h</annotation></semantics></math> above, the right hand side of this equation is less than <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="S3.20.4.p1.12.m3.1"><semantics id="S3.20.4.p1.12.m3.1a"><mrow id="S3.20.4.p1.12.m3.1.1" xref="S3.20.4.p1.12.m3.1.1.cmml"><mrow id="S3.20.4.p1.12.m3.1.1.1.1" xref="S3.20.4.p1.12.m3.1.1.1.1.1.cmml"><mo id="S3.20.4.p1.12.m3.1.1.1.1.2" stretchy="false" xref="S3.20.4.p1.12.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.20.4.p1.12.m3.1.1.1.1.1" xref="S3.20.4.p1.12.m3.1.1.1.1.1.cmml"><mrow id="S3.20.4.p1.12.m3.1.1.1.1.1.2" xref="S3.20.4.p1.12.m3.1.1.1.1.1.2.cmml"><mn id="S3.20.4.p1.12.m3.1.1.1.1.1.2.2" xref="S3.20.4.p1.12.m3.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.20.4.p1.12.m3.1.1.1.1.1.2.1" xref="S3.20.4.p1.12.m3.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.20.4.p1.12.m3.1.1.1.1.1.2.3" xref="S3.20.4.p1.12.m3.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.20.4.p1.12.m3.1.1.1.1.1.1" xref="S3.20.4.p1.12.m3.1.1.1.1.1.1.cmml">+</mo><mn id="S3.20.4.p1.12.m3.1.1.1.1.1.3" xref="S3.20.4.p1.12.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.20.4.p1.12.m3.1.1.1.1.3" stretchy="false" xref="S3.20.4.p1.12.m3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.20.4.p1.12.m3.1.1.2" xref="S3.20.4.p1.12.m3.1.1.2.cmml">⁢</mo><mi id="S3.20.4.p1.12.m3.1.1.3" xref="S3.20.4.p1.12.m3.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.12.m3.1b"><apply id="S3.20.4.p1.12.m3.1.1.cmml" xref="S3.20.4.p1.12.m3.1.1"><times id="S3.20.4.p1.12.m3.1.1.2.cmml" xref="S3.20.4.p1.12.m3.1.1.2"></times><apply id="S3.20.4.p1.12.m3.1.1.1.1.1.cmml" xref="S3.20.4.p1.12.m3.1.1.1.1"><plus id="S3.20.4.p1.12.m3.1.1.1.1.1.1.cmml" xref="S3.20.4.p1.12.m3.1.1.1.1.1.1"></plus><apply id="S3.20.4.p1.12.m3.1.1.1.1.1.2.cmml" xref="S3.20.4.p1.12.m3.1.1.1.1.1.2"><times id="S3.20.4.p1.12.m3.1.1.1.1.1.2.1.cmml" xref="S3.20.4.p1.12.m3.1.1.1.1.1.2.1"></times><cn id="S3.20.4.p1.12.m3.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.20.4.p1.12.m3.1.1.1.1.1.2.2">2</cn><ci id="S3.20.4.p1.12.m3.1.1.1.1.1.2.3.cmml" xref="S3.20.4.p1.12.m3.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.20.4.p1.12.m3.1.1.1.1.1.3.cmml" type="integer" xref="S3.20.4.p1.12.m3.1.1.1.1.1.3">1</cn></apply><ci id="S3.20.4.p1.12.m3.1.1.3.cmml" xref="S3.20.4.p1.12.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.12.m3.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.12.m3.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math> for all <math alttext="d\in\{1,\ldots,h-1\}" class="ltx_Math" display="inline" id="S3.20.4.p1.13.m4.3"><semantics id="S3.20.4.p1.13.m4.3a"><mrow id="S3.20.4.p1.13.m4.3.3" xref="S3.20.4.p1.13.m4.3.3.cmml"><mi id="S3.20.4.p1.13.m4.3.3.3" xref="S3.20.4.p1.13.m4.3.3.3.cmml">d</mi><mo id="S3.20.4.p1.13.m4.3.3.2" xref="S3.20.4.p1.13.m4.3.3.2.cmml">∈</mo><mrow id="S3.20.4.p1.13.m4.3.3.1.1" xref="S3.20.4.p1.13.m4.3.3.1.2.cmml"><mo id="S3.20.4.p1.13.m4.3.3.1.1.2" stretchy="false" xref="S3.20.4.p1.13.m4.3.3.1.2.cmml">{</mo><mn id="S3.20.4.p1.13.m4.1.1" xref="S3.20.4.p1.13.m4.1.1.cmml">1</mn><mo id="S3.20.4.p1.13.m4.3.3.1.1.3" xref="S3.20.4.p1.13.m4.3.3.1.2.cmml">,</mo><mi id="S3.20.4.p1.13.m4.2.2" mathvariant="normal" xref="S3.20.4.p1.13.m4.2.2.cmml">…</mi><mo id="S3.20.4.p1.13.m4.3.3.1.1.4" xref="S3.20.4.p1.13.m4.3.3.1.2.cmml">,</mo><mrow id="S3.20.4.p1.13.m4.3.3.1.1.1" xref="S3.20.4.p1.13.m4.3.3.1.1.1.cmml"><mi id="S3.20.4.p1.13.m4.3.3.1.1.1.2" xref="S3.20.4.p1.13.m4.3.3.1.1.1.2.cmml">h</mi><mo id="S3.20.4.p1.13.m4.3.3.1.1.1.1" xref="S3.20.4.p1.13.m4.3.3.1.1.1.1.cmml">−</mo><mn id="S3.20.4.p1.13.m4.3.3.1.1.1.3" xref="S3.20.4.p1.13.m4.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S3.20.4.p1.13.m4.3.3.1.1.5" stretchy="false" xref="S3.20.4.p1.13.m4.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.20.4.p1.13.m4.3b"><apply id="S3.20.4.p1.13.m4.3.3.cmml" xref="S3.20.4.p1.13.m4.3.3"><in id="S3.20.4.p1.13.m4.3.3.2.cmml" xref="S3.20.4.p1.13.m4.3.3.2"></in><ci id="S3.20.4.p1.13.m4.3.3.3.cmml" xref="S3.20.4.p1.13.m4.3.3.3">𝑑</ci><set id="S3.20.4.p1.13.m4.3.3.1.2.cmml" xref="S3.20.4.p1.13.m4.3.3.1.1"><cn id="S3.20.4.p1.13.m4.1.1.cmml" type="integer" xref="S3.20.4.p1.13.m4.1.1">1</cn><ci id="S3.20.4.p1.13.m4.2.2.cmml" xref="S3.20.4.p1.13.m4.2.2">…</ci><apply id="S3.20.4.p1.13.m4.3.3.1.1.1.cmml" xref="S3.20.4.p1.13.m4.3.3.1.1.1"><minus id="S3.20.4.p1.13.m4.3.3.1.1.1.1.cmml" xref="S3.20.4.p1.13.m4.3.3.1.1.1.1"></minus><ci id="S3.20.4.p1.13.m4.3.3.1.1.1.2.cmml" xref="S3.20.4.p1.13.m4.3.3.1.1.1.2">ℎ</ci><cn id="S3.20.4.p1.13.m4.3.3.1.1.1.3.cmml" type="integer" xref="S3.20.4.p1.13.m4.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.20.4.p1.13.m4.3c">d\in\{1,\ldots,h-1\}</annotation><annotation encoding="application/x-llamapun" id="S3.20.4.p1.13.m4.3d">italic_d ∈ { 1 , … , italic_h - 1 }</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S3.21.p8"> <p class="ltx_p" id="S3.21.p8.8">Let <math alttext="y_{r}" class="ltx_Math" display="inline" id="S3.21.p8.1.m1.1"><semantics id="S3.21.p8.1.m1.1a"><msub id="S3.21.p8.1.m1.1.1" xref="S3.21.p8.1.m1.1.1.cmml"><mi id="S3.21.p8.1.m1.1.1.2" xref="S3.21.p8.1.m1.1.1.2.cmml">y</mi><mi id="S3.21.p8.1.m1.1.1.3" xref="S3.21.p8.1.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S3.21.p8.1.m1.1b"><apply id="S3.21.p8.1.m1.1.1.cmml" xref="S3.21.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S3.21.p8.1.m1.1.1.1.cmml" xref="S3.21.p8.1.m1.1.1">subscript</csymbol><ci id="S3.21.p8.1.m1.1.1.2.cmml" xref="S3.21.p8.1.m1.1.1.2">𝑦</ci><ci id="S3.21.p8.1.m1.1.1.3.cmml" xref="S3.21.p8.1.m1.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.21.p8.1.m1.1c">y_{r}</annotation><annotation encoding="application/x-llamapun" id="S3.21.p8.1.m1.1d">italic_y start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> be the root of <math alttext="T_{Y}" class="ltx_Math" display="inline" id="S3.21.p8.2.m2.1"><semantics id="S3.21.p8.2.m2.1a"><msub id="S3.21.p8.2.m2.1.1" xref="S3.21.p8.2.m2.1.1.cmml"><mi id="S3.21.p8.2.m2.1.1.2" xref="S3.21.p8.2.m2.1.1.2.cmml">T</mi><mi id="S3.21.p8.2.m2.1.1.3" xref="S3.21.p8.2.m2.1.1.3.cmml">Y</mi></msub><annotation-xml encoding="MathML-Content" id="S3.21.p8.2.m2.1b"><apply id="S3.21.p8.2.m2.1.1.cmml" xref="S3.21.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S3.21.p8.2.m2.1.1.1.cmml" xref="S3.21.p8.2.m2.1.1">subscript</csymbol><ci id="S3.21.p8.2.m2.1.1.2.cmml" xref="S3.21.p8.2.m2.1.1.2">𝑇</ci><ci id="S3.21.p8.2.m2.1.1.3.cmml" xref="S3.21.p8.2.m2.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.21.p8.2.m2.1c">T_{Y}</annotation><annotation encoding="application/x-llamapun" id="S3.21.p8.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT</annotation></semantics></math>. Then, <math alttext="B_{y_{r}}:=B^{\prime}_{y_{r}}\cap Y\supseteq W\cup Z" class="ltx_Math" display="inline" id="S3.21.p8.3.m3.1"><semantics id="S3.21.p8.3.m3.1a"><mrow id="S3.21.p8.3.m3.1.1" xref="S3.21.p8.3.m3.1.1.cmml"><msub id="S3.21.p8.3.m3.1.1.2" xref="S3.21.p8.3.m3.1.1.2.cmml"><mi id="S3.21.p8.3.m3.1.1.2.2" xref="S3.21.p8.3.m3.1.1.2.2.cmml">B</mi><msub id="S3.21.p8.3.m3.1.1.2.3" xref="S3.21.p8.3.m3.1.1.2.3.cmml"><mi id="S3.21.p8.3.m3.1.1.2.3.2" xref="S3.21.p8.3.m3.1.1.2.3.2.cmml">y</mi><mi id="S3.21.p8.3.m3.1.1.2.3.3" xref="S3.21.p8.3.m3.1.1.2.3.3.cmml">r</mi></msub></msub><mo id="S3.21.p8.3.m3.1.1.3" lspace="0.278em" rspace="0.278em" xref="S3.21.p8.3.m3.1.1.3.cmml">:=</mo><mrow id="S3.21.p8.3.m3.1.1.4" xref="S3.21.p8.3.m3.1.1.4.cmml"><msubsup id="S3.21.p8.3.m3.1.1.4.2" xref="S3.21.p8.3.m3.1.1.4.2.cmml"><mi id="S3.21.p8.3.m3.1.1.4.2.2.2" xref="S3.21.p8.3.m3.1.1.4.2.2.2.cmml">B</mi><msub id="S3.21.p8.3.m3.1.1.4.2.3" xref="S3.21.p8.3.m3.1.1.4.2.3.cmml"><mi id="S3.21.p8.3.m3.1.1.4.2.3.2" xref="S3.21.p8.3.m3.1.1.4.2.3.2.cmml">y</mi><mi id="S3.21.p8.3.m3.1.1.4.2.3.3" xref="S3.21.p8.3.m3.1.1.4.2.3.3.cmml">r</mi></msub><mo id="S3.21.p8.3.m3.1.1.4.2.2.3" xref="S3.21.p8.3.m3.1.1.4.2.2.3.cmml">′</mo></msubsup><mo id="S3.21.p8.3.m3.1.1.4.1" xref="S3.21.p8.3.m3.1.1.4.1.cmml">∩</mo><mi id="S3.21.p8.3.m3.1.1.4.3" xref="S3.21.p8.3.m3.1.1.4.3.cmml">Y</mi></mrow><mo id="S3.21.p8.3.m3.1.1.5" xref="S3.21.p8.3.m3.1.1.5.cmml">⊇</mo><mrow id="S3.21.p8.3.m3.1.1.6" xref="S3.21.p8.3.m3.1.1.6.cmml"><mi id="S3.21.p8.3.m3.1.1.6.2" xref="S3.21.p8.3.m3.1.1.6.2.cmml">W</mi><mo id="S3.21.p8.3.m3.1.1.6.1" xref="S3.21.p8.3.m3.1.1.6.1.cmml">∪</mo><mi id="S3.21.p8.3.m3.1.1.6.3" xref="S3.21.p8.3.m3.1.1.6.3.cmml">Z</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.21.p8.3.m3.1b"><apply id="S3.21.p8.3.m3.1.1.cmml" xref="S3.21.p8.3.m3.1.1"><and id="S3.21.p8.3.m3.1.1a.cmml" xref="S3.21.p8.3.m3.1.1"></and><apply id="S3.21.p8.3.m3.1.1b.cmml" xref="S3.21.p8.3.m3.1.1"><csymbol cd="latexml" id="S3.21.p8.3.m3.1.1.3.cmml" xref="S3.21.p8.3.m3.1.1.3">assign</csymbol><apply id="S3.21.p8.3.m3.1.1.2.cmml" xref="S3.21.p8.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.21.p8.3.m3.1.1.2.1.cmml" xref="S3.21.p8.3.m3.1.1.2">subscript</csymbol><ci id="S3.21.p8.3.m3.1.1.2.2.cmml" xref="S3.21.p8.3.m3.1.1.2.2">𝐵</ci><apply id="S3.21.p8.3.m3.1.1.2.3.cmml" xref="S3.21.p8.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S3.21.p8.3.m3.1.1.2.3.1.cmml" xref="S3.21.p8.3.m3.1.1.2.3">subscript</csymbol><ci id="S3.21.p8.3.m3.1.1.2.3.2.cmml" xref="S3.21.p8.3.m3.1.1.2.3.2">𝑦</ci><ci id="S3.21.p8.3.m3.1.1.2.3.3.cmml" xref="S3.21.p8.3.m3.1.1.2.3.3">𝑟</ci></apply></apply><apply id="S3.21.p8.3.m3.1.1.4.cmml" xref="S3.21.p8.3.m3.1.1.4"><intersect id="S3.21.p8.3.m3.1.1.4.1.cmml" xref="S3.21.p8.3.m3.1.1.4.1"></intersect><apply id="S3.21.p8.3.m3.1.1.4.2.cmml" xref="S3.21.p8.3.m3.1.1.4.2"><csymbol cd="ambiguous" id="S3.21.p8.3.m3.1.1.4.2.1.cmml" xref="S3.21.p8.3.m3.1.1.4.2">subscript</csymbol><apply id="S3.21.p8.3.m3.1.1.4.2.2.cmml" xref="S3.21.p8.3.m3.1.1.4.2"><csymbol cd="ambiguous" id="S3.21.p8.3.m3.1.1.4.2.2.1.cmml" xref="S3.21.p8.3.m3.1.1.4.2">superscript</csymbol><ci id="S3.21.p8.3.m3.1.1.4.2.2.2.cmml" xref="S3.21.p8.3.m3.1.1.4.2.2.2">𝐵</ci><ci id="S3.21.p8.3.m3.1.1.4.2.2.3.cmml" xref="S3.21.p8.3.m3.1.1.4.2.2.3">′</ci></apply><apply id="S3.21.p8.3.m3.1.1.4.2.3.cmml" xref="S3.21.p8.3.m3.1.1.4.2.3"><csymbol cd="ambiguous" id="S3.21.p8.3.m3.1.1.4.2.3.1.cmml" xref="S3.21.p8.3.m3.1.1.4.2.3">subscript</csymbol><ci id="S3.21.p8.3.m3.1.1.4.2.3.2.cmml" xref="S3.21.p8.3.m3.1.1.4.2.3.2">𝑦</ci><ci id="S3.21.p8.3.m3.1.1.4.2.3.3.cmml" xref="S3.21.p8.3.m3.1.1.4.2.3.3">𝑟</ci></apply></apply><ci id="S3.21.p8.3.m3.1.1.4.3.cmml" xref="S3.21.p8.3.m3.1.1.4.3">𝑌</ci></apply></apply><apply id="S3.21.p8.3.m3.1.1c.cmml" xref="S3.21.p8.3.m3.1.1"><csymbol cd="latexml" id="S3.21.p8.3.m3.1.1.5.cmml" xref="S3.21.p8.3.m3.1.1.5">superset-of-or-equals</csymbol><share href="https://arxiv.org/html/2503.17112v1#S3.21.p8.3.m3.1.1.4.cmml" id="S3.21.p8.3.m3.1.1d.cmml" xref="S3.21.p8.3.m3.1.1"></share><apply id="S3.21.p8.3.m3.1.1.6.cmml" xref="S3.21.p8.3.m3.1.1.6"><union id="S3.21.p8.3.m3.1.1.6.1.cmml" xref="S3.21.p8.3.m3.1.1.6.1"></union><ci id="S3.21.p8.3.m3.1.1.6.2.cmml" xref="S3.21.p8.3.m3.1.1.6.2">𝑊</ci><ci id="S3.21.p8.3.m3.1.1.6.3.cmml" xref="S3.21.p8.3.m3.1.1.6.3">𝑍</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.21.p8.3.m3.1c">B_{y_{r}}:=B^{\prime}_{y_{r}}\cap Y\supseteq W\cup Z</annotation><annotation encoding="application/x-llamapun" id="S3.21.p8.3.m3.1d">italic_B start_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT := italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ italic_Y ⊇ italic_W ∪ italic_Z</annotation></semantics></math>. Therefore, <math alttext="\mathcal{T}_{Y}=(B_{y}:y\in V(T_{Y}))" class="ltx_math_unparsed" display="inline" id="S3.21.p8.4.m4.1"><semantics id="S3.21.p8.4.m4.1a"><mrow id="S3.21.p8.4.m4.1b"><msub id="S3.21.p8.4.m4.1.1"><mi class="ltx_font_mathcaligraphic" id="S3.21.p8.4.m4.1.1.2">𝒯</mi><mi id="S3.21.p8.4.m4.1.1.3">Y</mi></msub><mo id="S3.21.p8.4.m4.1.2">=</mo><mrow id="S3.21.p8.4.m4.1.3"><mo id="S3.21.p8.4.m4.1.3.1" stretchy="false">(</mo><msub id="S3.21.p8.4.m4.1.3.2"><mi id="S3.21.p8.4.m4.1.3.2.2">B</mi><mi id="S3.21.p8.4.m4.1.3.2.3">y</mi></msub><mo id="S3.21.p8.4.m4.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.21.p8.4.m4.1.3.4">y</mi><mo id="S3.21.p8.4.m4.1.3.5">∈</mo><mi id="S3.21.p8.4.m4.1.3.6">V</mi><mrow id="S3.21.p8.4.m4.1.3.7"><mo id="S3.21.p8.4.m4.1.3.7.1" stretchy="false">(</mo><msub id="S3.21.p8.4.m4.1.3.7.2"><mi id="S3.21.p8.4.m4.1.3.7.2.2">T</mi><mi id="S3.21.p8.4.m4.1.3.7.2.3">Y</mi></msub><mo id="S3.21.p8.4.m4.1.3.7.3" stretchy="false">)</mo></mrow><mo id="S3.21.p8.4.m4.1.3.8" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.21.p8.4.m4.1c">\mathcal{T}_{Y}=(B_{y}:y\in V(T_{Y}))</annotation><annotation encoding="application/x-llamapun" id="S3.21.p8.4.m4.1d">caligraphic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT = ( italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT : italic_y ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) )</annotation></semantics></math> is a tree decomposition of <math alttext="G[Y]" class="ltx_Math" display="inline" id="S3.21.p8.5.m5.1"><semantics id="S3.21.p8.5.m5.1a"><mrow id="S3.21.p8.5.m5.1.2" xref="S3.21.p8.5.m5.1.2.cmml"><mi id="S3.21.p8.5.m5.1.2.2" xref="S3.21.p8.5.m5.1.2.2.cmml">G</mi><mo id="S3.21.p8.5.m5.1.2.1" xref="S3.21.p8.5.m5.1.2.1.cmml">⁢</mo><mrow id="S3.21.p8.5.m5.1.2.3.2" xref="S3.21.p8.5.m5.1.2.3.1.cmml"><mo id="S3.21.p8.5.m5.1.2.3.2.1" stretchy="false" xref="S3.21.p8.5.m5.1.2.3.1.1.cmml">[</mo><mi id="S3.21.p8.5.m5.1.1" xref="S3.21.p8.5.m5.1.1.cmml">Y</mi><mo id="S3.21.p8.5.m5.1.2.3.2.2" stretchy="false" xref="S3.21.p8.5.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.21.p8.5.m5.1b"><apply id="S3.21.p8.5.m5.1.2.cmml" xref="S3.21.p8.5.m5.1.2"><times id="S3.21.p8.5.m5.1.2.1.cmml" xref="S3.21.p8.5.m5.1.2.1"></times><ci id="S3.21.p8.5.m5.1.2.2.cmml" xref="S3.21.p8.5.m5.1.2.2">𝐺</ci><apply id="S3.21.p8.5.m5.1.2.3.1.cmml" xref="S3.21.p8.5.m5.1.2.3.2"><csymbol cd="latexml" id="S3.21.p8.5.m5.1.2.3.1.1.cmml" xref="S3.21.p8.5.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.21.p8.5.m5.1.1.cmml" xref="S3.21.p8.5.m5.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.21.p8.5.m5.1c">G[Y]</annotation><annotation encoding="application/x-llamapun" id="S3.21.p8.5.m5.1d">italic_G [ italic_Y ]</annotation></semantics></math> that (by <a class="ltx_ref" href="https://arxiv.org/html/2503.17112v1#Thmthm11" title="Claim 11. ‣ Proof of Theorem 1. ‣ 3 The Proof ‣ SEPARATION NUMBER AND TREEWIDTH, REVISITEDThis research was partly funded by NSERC."><span class="ltx_text ltx_ref_tag">Claim</span> <span class="ltx_text ltx_ref_tag">11</span></a>) has width less than <math alttext="(2t+1)a" class="ltx_Math" display="inline" id="S3.21.p8.6.m6.1"><semantics id="S3.21.p8.6.m6.1a"><mrow id="S3.21.p8.6.m6.1.1" xref="S3.21.p8.6.m6.1.1.cmml"><mrow id="S3.21.p8.6.m6.1.1.1.1" xref="S3.21.p8.6.m6.1.1.1.1.1.cmml"><mo id="S3.21.p8.6.m6.1.1.1.1.2" stretchy="false" xref="S3.21.p8.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S3.21.p8.6.m6.1.1.1.1.1" xref="S3.21.p8.6.m6.1.1.1.1.1.cmml"><mrow id="S3.21.p8.6.m6.1.1.1.1.1.2" xref="S3.21.p8.6.m6.1.1.1.1.1.2.cmml"><mn id="S3.21.p8.6.m6.1.1.1.1.1.2.2" xref="S3.21.p8.6.m6.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.21.p8.6.m6.1.1.1.1.1.2.1" xref="S3.21.p8.6.m6.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.21.p8.6.m6.1.1.1.1.1.2.3" xref="S3.21.p8.6.m6.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.21.p8.6.m6.1.1.1.1.1.1" xref="S3.21.p8.6.m6.1.1.1.1.1.1.cmml">+</mo><mn id="S3.21.p8.6.m6.1.1.1.1.1.3" xref="S3.21.p8.6.m6.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.21.p8.6.m6.1.1.1.1.3" stretchy="false" xref="S3.21.p8.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.21.p8.6.m6.1.1.2" xref="S3.21.p8.6.m6.1.1.2.cmml">⁢</mo><mi id="S3.21.p8.6.m6.1.1.3" xref="S3.21.p8.6.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.21.p8.6.m6.1b"><apply id="S3.21.p8.6.m6.1.1.cmml" xref="S3.21.p8.6.m6.1.1"><times id="S3.21.p8.6.m6.1.1.2.cmml" xref="S3.21.p8.6.m6.1.1.2"></times><apply id="S3.21.p8.6.m6.1.1.1.1.1.cmml" xref="S3.21.p8.6.m6.1.1.1.1"><plus id="S3.21.p8.6.m6.1.1.1.1.1.1.cmml" xref="S3.21.p8.6.m6.1.1.1.1.1.1"></plus><apply id="S3.21.p8.6.m6.1.1.1.1.1.2.cmml" xref="S3.21.p8.6.m6.1.1.1.1.1.2"><times id="S3.21.p8.6.m6.1.1.1.1.1.2.1.cmml" xref="S3.21.p8.6.m6.1.1.1.1.1.2.1"></times><cn id="S3.21.p8.6.m6.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.21.p8.6.m6.1.1.1.1.1.2.2">2</cn><ci id="S3.21.p8.6.m6.1.1.1.1.1.2.3.cmml" xref="S3.21.p8.6.m6.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.21.p8.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S3.21.p8.6.m6.1.1.1.1.1.3">1</cn></apply><ci id="S3.21.p8.6.m6.1.1.3.cmml" xref="S3.21.p8.6.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.21.p8.6.m6.1c">(2t+1)a</annotation><annotation encoding="application/x-llamapun" id="S3.21.p8.6.m6.1d">( 2 italic_t + 1 ) italic_a</annotation></semantics></math> and there exists <math alttext="y\in V(T_{Y})" class="ltx_Math" display="inline" id="S3.21.p8.7.m7.1"><semantics id="S3.21.p8.7.m7.1a"><mrow id="S3.21.p8.7.m7.1.1" xref="S3.21.p8.7.m7.1.1.cmml"><mi id="S3.21.p8.7.m7.1.1.3" xref="S3.21.p8.7.m7.1.1.3.cmml">y</mi><mo id="S3.21.p8.7.m7.1.1.2" xref="S3.21.p8.7.m7.1.1.2.cmml">∈</mo><mrow id="S3.21.p8.7.m7.1.1.1" xref="S3.21.p8.7.m7.1.1.1.cmml"><mi id="S3.21.p8.7.m7.1.1.1.3" xref="S3.21.p8.7.m7.1.1.1.3.cmml">V</mi><mo id="S3.21.p8.7.m7.1.1.1.2" xref="S3.21.p8.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S3.21.p8.7.m7.1.1.1.1.1" xref="S3.21.p8.7.m7.1.1.1.1.1.1.cmml"><mo id="S3.21.p8.7.m7.1.1.1.1.1.2" stretchy="false" xref="S3.21.p8.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="S3.21.p8.7.m7.1.1.1.1.1.1" xref="S3.21.p8.7.m7.1.1.1.1.1.1.cmml"><mi id="S3.21.p8.7.m7.1.1.1.1.1.1.2" xref="S3.21.p8.7.m7.1.1.1.1.1.1.2.cmml">T</mi><mi id="S3.21.p8.7.m7.1.1.1.1.1.1.3" xref="S3.21.p8.7.m7.1.1.1.1.1.1.3.cmml">Y</mi></msub><mo id="S3.21.p8.7.m7.1.1.1.1.1.3" stretchy="false" xref="S3.21.p8.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.21.p8.7.m7.1b"><apply id="S3.21.p8.7.m7.1.1.cmml" xref="S3.21.p8.7.m7.1.1"><in id="S3.21.p8.7.m7.1.1.2.cmml" xref="S3.21.p8.7.m7.1.1.2"></in><ci id="S3.21.p8.7.m7.1.1.3.cmml" xref="S3.21.p8.7.m7.1.1.3">𝑦</ci><apply id="S3.21.p8.7.m7.1.1.1.cmml" xref="S3.21.p8.7.m7.1.1.1"><times id="S3.21.p8.7.m7.1.1.1.2.cmml" xref="S3.21.p8.7.m7.1.1.1.2"></times><ci id="S3.21.p8.7.m7.1.1.1.3.cmml" xref="S3.21.p8.7.m7.1.1.1.3">𝑉</ci><apply id="S3.21.p8.7.m7.1.1.1.1.1.1.cmml" xref="S3.21.p8.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.21.p8.7.m7.1.1.1.1.1.1.1.cmml" xref="S3.21.p8.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S3.21.p8.7.m7.1.1.1.1.1.1.2.cmml" xref="S3.21.p8.7.m7.1.1.1.1.1.1.2">𝑇</ci><ci id="S3.21.p8.7.m7.1.1.1.1.1.1.3.cmml" xref="S3.21.p8.7.m7.1.1.1.1.1.1.3">𝑌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.21.p8.7.m7.1c">y\in V(T_{Y})</annotation><annotation encoding="application/x-llamapun" id="S3.21.p8.7.m7.1d">italic_y ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT )</annotation></semantics></math> such that <math alttext="W\cup Z\subseteq B_{y}" class="ltx_Math" display="inline" id="S3.21.p8.8.m8.1"><semantics id="S3.21.p8.8.m8.1a"><mrow id="S3.21.p8.8.m8.1.1" xref="S3.21.p8.8.m8.1.1.cmml"><mrow id="S3.21.p8.8.m8.1.1.2" xref="S3.21.p8.8.m8.1.1.2.cmml"><mi id="S3.21.p8.8.m8.1.1.2.2" xref="S3.21.p8.8.m8.1.1.2.2.cmml">W</mi><mo id="S3.21.p8.8.m8.1.1.2.1" xref="S3.21.p8.8.m8.1.1.2.1.cmml">∪</mo><mi id="S3.21.p8.8.m8.1.1.2.3" xref="S3.21.p8.8.m8.1.1.2.3.cmml">Z</mi></mrow><mo id="S3.21.p8.8.m8.1.1.1" xref="S3.21.p8.8.m8.1.1.1.cmml">⊆</mo><msub id="S3.21.p8.8.m8.1.1.3" xref="S3.21.p8.8.m8.1.1.3.cmml"><mi id="S3.21.p8.8.m8.1.1.3.2" xref="S3.21.p8.8.m8.1.1.3.2.cmml">B</mi><mi id="S3.21.p8.8.m8.1.1.3.3" xref="S3.21.p8.8.m8.1.1.3.3.cmml">y</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.21.p8.8.m8.1b"><apply id="S3.21.p8.8.m8.1.1.cmml" xref="S3.21.p8.8.m8.1.1"><subset id="S3.21.p8.8.m8.1.1.1.cmml" xref="S3.21.p8.8.m8.1.1.1"></subset><apply id="S3.21.p8.8.m8.1.1.2.cmml" xref="S3.21.p8.8.m8.1.1.2"><union id="S3.21.p8.8.m8.1.1.2.1.cmml" xref="S3.21.p8.8.m8.1.1.2.1"></union><ci id="S3.21.p8.8.m8.1.1.2.2.cmml" xref="S3.21.p8.8.m8.1.1.2.2">𝑊</ci><ci id="S3.21.p8.8.m8.1.1.2.3.cmml" xref="S3.21.p8.8.m8.1.1.2.3">𝑍</ci></apply><apply id="S3.21.p8.8.m8.1.1.3.cmml" xref="S3.21.p8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.21.p8.8.m8.1.1.3.1.cmml" xref="S3.21.p8.8.m8.1.1.3">subscript</csymbol><ci id="S3.21.p8.8.m8.1.1.3.2.cmml" xref="S3.21.p8.8.m8.1.1.3.2">𝐵</ci><ci id="S3.21.p8.8.m8.1.1.3.3.cmml" xref="S3.21.p8.8.m8.1.1.3.3">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.21.p8.8.m8.1c">W\cup Z\subseteq B_{y}</annotation><annotation encoding="application/x-llamapun" id="S3.21.p8.8.m8.1d">italic_W ∪ italic_Z ⊆ italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. This completes the proof. ∎</p> </div> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Chen et al. [2007]</span> <span class="ltx_bibblock"> Jiangzhuo Chen, Robert D. Kleinberg, László Lovász, Rajmohan Rajaraman, Ravi Sundaram, and Adrian Vetta. </span> <span class="ltx_bibblock">(almost) tight bounds and existence theorems for single-commodity confluent flows. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">J. ACM</em>, 54(4):16, 2007. </span> <span class="ltx_bibblock"><a class="ltx_ref" href="https:/doi.org/10.1145/1255443.1255444" title="">10.1145/1255443.1255444</a>. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Diestel [2017]</span> <span class="ltx_bibblock"> R. 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