[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82314-en":3,"doc-seo-82314-105":29,"detail-sidebar-cat-0-en-105":95},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82314,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Spanning Paths and Cycles Structural Limitations of the Irrelevant Vertex Technique","The paper studies the Irrelevant Vertex Technique in algorithmic graph theory under spanning routing constraints. Spanning Disjoint Paths asks for internally disjoint paths linking given terminal pairs whose union spans all annotated red vertices. A new structural parameter of annotated graphs, depth2, is introduced to capture this phenomenon exactly. A complete combinatorial dichotomy holds for every red-minor-closed class: the technique applies iff the class has bounded depth2. The work provides a depth2-based algorithm with matching lower bounds.","arXiv :2607 .09342v 1 [ cs .DS] 10 Jul 2026  \nSpanning Paths and Cycles: Structural Limitations of the Irrelevant Vertex Technique  \nDimitrios M. Thilikos∗†  \nLIRMM, Université de Montpellier, CNRS, Montpellier, France  \nSebastian Wiederrecht‡§  \nSchool of Computing, KAIST, South Korea  \nThe Irrelevant Vertex Technique is one of the cornerstones of algorithmic graph theory, underlying Robertson and Seymour’s algorithm for Disjoint Paths and much of the algorithmic Graph Minors theory. We show that, in the setting of spanning routing, this technique exhibits an exact combinatorial limitation.  \nUnlike classical routing problems, spanning routing is not governed by the number of distinguished vertices but by the way they are distributed throughout the graph. Here, the input is a triple (G, R, T ) where (G, R) is an annotated graph and T is a set of terminal pairs. The goal is to determine if G contains a family of internally disjoint paths connecting the pairs in T such that the union of the paths spans the entire set R. We identify a new structural parameter of annotated graphs, called depth2 , that measures precisely this phenomenon. Our main result is a complete combinatorial dichotomy: for every red-minor-closed class of annotated graphs, the Irrelevant Vertex Technique applies to Spanning Disjoint Paths if and only if the class has bounded depth2 . Thus depth2 forms the exact structural boundary between classes where the Robertson-Seymour paradigm survives and those where it breaks down. Our proof combines a new local structure theorem for annotated graphs of bounded depth2 with a spanning analogue of the celebrated Vital Linkage Theorem. The resulting algorithm solves Spanning Disjoint Paths in time 22poly (k+d) · n2 where d denotes the depth2 of the input instance. We provide matching lower bounds which show that beyond bounded depth2 no irrelevant-vertex rule can exist, even on planar graphs. In particular, depth2 is the exact combinatorial barrier for the Irrelevant Vertex Technique under spanning constraints.  \n∗ Supported by French National Research Agency (ANR) under project GODASse ANR-24-CE48-4377 and under the France 2030 grant reference number ANR-24-RRII-0002 operated by the Inria Quadrant Program.  \n†[sedthilk@thilikos.info](sedthilk@thilikos.info)  \n‡Supported by the Institute for Basic Science (IBS-R029-C1) .  \n§[wiederrecht@kaist.ac.kr](wiederrecht@kaist.ac.kr)  \nContents  \n1 Introduction 1  \n1.1 Proof outline and organisation ............................ 9  \n2 Preliminaries 13  \n2. 1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13  \n2.2 Tools more specific to graph minor structure theory ................ 14  \n2.3 The structure of annotated graphs of small bidimensionality ............ 18  \n3 The local structure of annotated graphs with small depth2 19  \n3.1 Part 1: Placing candles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22  \n3.2 Part 2: Cleaning a vortex ............................... 34  \n4 Annotated graphs with vital solutions 40  \n4.1 What to do with a clique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  \n4.2 Desolate instances and their loci . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  \n4.3 Bringing desolation to a vital instance . . . . . . . . . . . . . . . . . . . . . . . . 45  \n4.4 Entering the wasteland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47  \n5 Irrelevant vertices for spanning linkages 61  \n5.1 Combing spanning linkages .............................. 62  \n5.2 Avoiding an insulated vertex .............................. 69  \n6 The Spanning Linkage Algorithm 74  \n6.1 Finding an irrelevant vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76  \n6.2 The case of bounded treewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78  \n7 Irrevelant-vertex tightness 79  \n7.1 Constructing a vital instance ............................. 80  \n7.2 Treewidth ","cbCaiuEHpd65gLVQ","https://ap.wps.com/l/cbCaiuEHpd65gLVQ","pdf",10729871,1,94,"English","en",105,"# Introduction\n## Proof outline and organisation\n# Preliminaries\n## Basics\n## Tools more specific to graph minor structure theory\n## The structure of annotated graphs of small bidimensionality\n# The local structure of annotated graphs with small depth2\n## Placing candles\n## Cleaning a vortex\n# Annotated graphs with vital solutions\n## What to do with a clique\n## Desolate instances and their loci\n## Bringing desolation to a vital instance\n## Entering the wasteland\n# Irrelevant vertices for spanning linkages\n## Combing spanning linkages\n## Avoiding an insulated vertex\n# The Spanning Linkage Algorithm\n## Finding an irrelevant vertex\n## The case of bounded treewidth\n# Irrevelant-vertex tightness\n## Constructing a vital instance\n## Treewidth of the instances\n## Vitality of the instances\n# Conclusion","[{\"question\":\"What is Spanning Disjoint Paths in the paper’s setting?\",\"answer\":\"Given an annotated graph (G,R) and terminal pairs T, the task is to find internally disjoint paths connecting each pair in T such that the union of the paths spans all red vertices R.\"},{\"question\":\"What structural parameter determines when the Irrelevant Vertex Technique works?\",\"answer\":\"The paper introduces depth2, a parameter measuring how spanning constraints interact with the annotated graph structure.\"},{\"question\":\"What is the main combinatorial dichotomy result?\",\"answer\":\"For any red-minor-closed class of annotated graphs, the Irrelevant Vertex Technique applies to Spanning Disjoint Paths if and only if the class has bounded depth2.\"},{\"question\":\"Does the paper provide performance and limitation bounds?\",\"answer\":\"Yes. It gives an algorithm running in time 2^{O(poly(k+d))}·n^2 (with depth2 denoted d) and matching lower bounds showing that beyond bounded depth2 no irrelevant-vertex rule can exist, even for planar graphs.\"}]",1784179557,237,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":90,"head_meta":92,"extra_data":94,"updated_unix":27},"spanning-paths-and-cycles-structural-limitations-of-the-irrelevant-vertex-technique","",{"@graph":35,"@context":89},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/spanning-paths-and-cycles-structural-limitations-of-the-irrelevant-vertex-technique/82314/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What is Spanning Disjoint Paths in the paper’s setting?","Question",{"text":75,"@type":76},"Given an annotated graph (G,R) and terminal pairs T, the task is to find internally disjoint paths connecting each pair in T such that the union of the paths spans all red vertices R.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What structural parameter determines when the Irrelevant Vertex Technique works?",{"text":80,"@type":76},"The paper introduces depth2, a parameter measuring how spanning constraints interact with the annotated graph structure.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main combinatorial dichotomy result?",{"text":84,"@type":76},"For any red-minor-closed class of annotated graphs, the Irrelevant Vertex Technique applies to Spanning Disjoint Paths if and only if the class has bounded depth2.",{"name":86,"@type":73,"acceptedAnswer":87},"Does the paper provide performance and limitation bounds?",{"text":88,"@type":76},"Yes. 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