[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83192-en":3,"doc-seo-83192-105":30,"detail-sidebar-cat-0-en-105":91},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83192,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","A Coarse Block-Cut Tree Theorem","We prove a coarse analogue of the block-cut tree fact that every graph decomposes along cut-vertices into 2-connected components. For any connected graph G and positive integer d, G has a tree decomposition whose adhesion sets have weak diameter at most 3d+2. Vertices within the same bag cannot be separated by any set of weak diameter at most d that lies farther than d from both vertices. Using the coarse Menger’s theorem for two paths, this yields a dual formulation via pairs of far paths connecting neighborhoods.","A coarse block-cut tree theorem  \nJlia Baligcs  \nUniversity of Oxford [jbaligacs@gmail.com](jbaligacs@gmail.com)  \nVclav Blaej  \nUniversity of Warsaw [v.blazej@uw.edu.pl](v.blazej@uw.edu.pl)  \nJadwiga Czyewska  \nUniversity of Warsaw  \n[j.czyzewska@mimuw.edu.pl](j.czyzewska@mimuw.edu.pl)  \nMichał Pilipczuk  \nUniversity of Warsaw  \n[michal.pilipczuk@mimuw.edu.pl](michal.pilipczuk@mimuw.edu.pl)  \nEvangelos Protopapas  \nUniversity of Warsaw  \n[eprotopapas@mimuw.edu.pl](eprotopapas@mimuw.edu.pl)  \n[math .CO] 8 Jul 2026  \nAbstract  \nWe prove a coarse analogue of the classic fact that every graph can be decomposed along its cutvertices into 2-connected components. Precisely, we prove that for every graph G and a positive integer d, G admits a tree decomposition whose adhesion sets have weak diameter at most 3d + 2 so that no two vertices u, v lying in the same bag can be separated by a set of weak diameter at most d whose distance from u and v is more than d. By the Coarse Menger’s Theorem for two paths, this condition admits also a dual formulation, phrased in terms of the existence of two paths that are far from eachother and connect the vicinity of u with the vicinity of v.  \narXiv :2607 .07111v1  \nFunding information:  \nJB (during employment in Warsaw), VB, MP: ERC project BOBR, funded from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme with grant agreement No. 948057.  \nJB (during employment in Oxford): ERC grant CCOO (grant no. 101165139)  \nJCz: Polish National Science Centre SONATA BIS-12, grant number 2022/46/E/ST6/00143 .  \nEP: ERC grant BUKA (No. 101126229) .  \n1 Introduction  \nCoarse graph theory is a young and vibrant direction in structural graph theory that aims to understand the structure of graphs viewed as metric spaces, by endowing them with the shortest-paths metric. As outlined in the foundational work of Georgakopoulous and Papasoglu [12], the hope is that many classic results of structural graph theory admit natural coarse analogues, obtained by replacing the conditions that objects intersect or are disjoint by requesting that they are close or far from each other. Recent investigations have revealed a mix of successes and failures. For instance, the Erds–Psa Theorem can be lifted to the coarse setting [1, 2, 10], and so can also Gallai’s Theorem [9] . The coarse variant of Menger’s Theorem fails in full generality [19, 22], but holds in various restricted settings [5, 7, 11, 12, 13, 15, 20, 21] . There are also natural coarse analogues of pathwidth and treewidth [14, 16, 17] . For these, the coarse analogue of the Excluded Tree Theorem holds [18], but the coarse analogue of the Excluded Grid Theorem fails [3] . The Alon-Seymour-Tarsi Separator Theorem can be lifted to the coarse setting, at least to a large degree [8] . The coarse analogue of the Kuratowski-Wagner Theorem remains tantalizingly open [12] .  \nIn this paper we add another positive example to this growing list. One of the most basic results of structural graph theory is the so-called block-cut tree: every graph can be decomposed along its cut-vertices in a tree-like fashion into its 2-connected components. We prove the following coarse analogue of this fact (see Section 2 for definitions):  \nTheorem 1. For every positive integer d and every connected graph G, there exists a tree-decomposition (T,β) of G such that  \n• each adhesion set of (T,β) has weak diameter at most 3d + 2; and  \n• for any two vertices u, v ∈ V (G) in a common bag of (T,β), there is no set S ⊆ V (G) with diamG (S) ⩽ d and distG (S,{u, v}) > d + 1 that separates u and v in G.  \nMoreover, given G and d, such a tree-decomposition can be computed in time O (n(n + m)), where n and m denote the vertex and the edge count of G, respectively.  \nSince the coarse analogue of Menger’s Theorem holds for two paths [5, 12], the second condition in Theorem 1 can be reformulated to roughly the following: for any two vertices u, v lyi","cbCaie6rOwVVhvqp","https://ap.wps.com/l/cbCaie6rOwVVhvqp","pdf",667687,2,1,10,"English","en",105,"# Introduction\n# Preliminaries","[{\"question\":\"What classical graph-theoretic result does the paper generalize in a coarse setting?\",\"answer\":\"It generalizes the block-cut tree principle that decomposes a graph along its cut-vertices into 2-connected components, replacing exact intersection/separation by coarse notions of being close or far.\"},{\"question\":\"What guarantee does the main theorem provide for a graph G and parameter d?\",\"answer\":\"It provides a tree decomposition where each adhesion set has weak diameter at most 3d+2, and any two vertices in the same bag cannot be separated by a set of weak diameter at most d that is more than d away from both vertices.\"},{\"question\":\"How is the separation condition related to paths in the paper’s dual formulation?\",\"answer\":\"Using the coarse Menger’s theorem for two paths, the condition is rephrased to require two paths connecting neighborhoods around the vertices, with the paths remaining far from each other at scale Ω(d).\"}]",1784185862,25,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-coarse-block-cut-tree-theorem","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-coarse-block-cut-tree-theorem/83192/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What classical graph-theoretic result does the paper generalize in a coarse setting?","Question",{"text":75,"@type":76},"It generalizes the block-cut tree principle that decomposes a graph along its cut-vertices into 2-connected components, replacing exact intersection/separation by coarse notions of being close or far.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What guarantee does the main theorem provide for a graph G and parameter d?",{"text":80,"@type":76},"It provides a tree decomposition where each adhesion set has weak diameter at most 3d+2, and any two vertices in the same bag cannot be separated by a set of weak diameter at most d that is more than d away from both vertices.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the separation condition related to paths in the paper’s dual formulation?",{"text":84,"@type":76},"Using the coarse Menger’s theorem for two paths, the condition is rephrased to require two paths connecting neighborhoods around the vertices, with the paths remaining far from each other at scale Ω(d).","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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