[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82653-en":3,"doc-seo-82653-105":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},82653,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Separating Geodesic Structure and Product Structure","Geodesic treewidth captures how a graph can be partitioned into geodesics so that contracting each part yields a quotient of small treewidth. Row treewidth captures an analogous product-style structure using disjoint geodesic unions aligned with a layering. The work separates these notions by proving bounded row treewidth does not imply bounded geodesic treewidth, and by giving a polynomial-time decision procedure for whether a treewidth-2 graph has geodesic treewidth 1, contrasting known NP-hardness for row treewidth. It further studies XP/FPT-type decidability, NP-hardness of geodesic treewidth, and improved lower bounds for planar graphs.","arXiv :2607 .02098v1 [math .CO] 2 Jul 2026  \nSeparating Geodesic Structure and Product Structure  \nLaura Merker \\#   \nKarlsruhe Institute of Technology, Germany Lena Scherzer \\#   \nUniversity of Hamburg, Germany Samuel Schneider \\#   \nKarlsruhe Institute of Technology, Germany  \n~~ Abstract ~~  \nThe geodesic treewidth of a graph G is the smallest k for which there is a partition P into geodesics such that G/P has treewidth k, where G/P is obtained from G by contracting each part of P. Based on this notion, row treewidth was developed and is defined for a graph G as the smallest k such that G ⊆ H ⊠ P for some graph H of treewidth k and a path P. Equivalently, the row treewidth of a graph G is the smallest k for which there is a partition P into disjoint unions of geodesics that are aligned with respect to some layering such that G/P has treewidth k.  \nWe separate the two notions by showing that bounded row treewidth does not imply bounded geodesic treewidth and by presenting a polynomial-time algorithm to decide whether a graph of treewidth 2 has geodesic treewidth 1, which is known to be NP-hard for row treewidth [Biedl, Eppstein, Ueckerdt, 2025] . More generally, we provide an algorithm to decide whether a given graph has geodesic treewidth at most d that is XP in the treewidth, whereas there is no such algorithm for row treewidth, unless P = NP [Biedl, Eppstein, Ueckerdt, 2025] . On the other hand, we show that computing the geodesic treewidth is NP-hard and that every graph with geodesic treewidth 1 has bounded row treewidth. Moreover, we improve the best known lower bound on the geodesic treewidth of planar graphs to 5 .  \n2012 ACM Subject Classification Mathematics of computing → Graph theory; Mathematics of computing → Graph algorithms  \nKeywords and phrases product structure, row treewidth, geodesic structure, geodesic treewidth  \nRelated Version An extended abstract of this paper appears in the proceedings of the 34th Annual European Symposium on Algorithms (ESA 2026) .  \n 1  Introduction  \nBounded treewidth allows using a wide range of algorithmic techniques to solve numerous important problems efficiently. However, many common graph classes, like planar graphs, do not have bounded treewidth. Hence, we are interested in generalizations of treewidth that apply to broader classes of graphs but still provide comparable structural benefits. In particular, we need to be able to handle large grids as they are planar graphs of high treewidth. A popular way to define such generalizations is to take a partition P of a graph G into (disjoint unions of) geodesics 1 and aim for small treewidth of the quotient G/P, i. e. , the graph that is obtained by contracting each (possibly disconnected) part of P into a single vertex. For a first example, note that a grid admits a partition into geodesics such that the quotient is a path and thus has small treewidth. The two generalizations of treewidth we consider in this paper are a concept introduced by Pilipczuk and Siebertz [35], which we  \n1 A geodesic is a shortest path between any two vertices in the graph.  \n2 Separating Geodesic Structure and Product Structure  \nFigure 1 Left: A graph G with a partition P into geodesics. Right: The graph G/P.  \nFigure 2 Left: A graph G with a layering L = (L1 , L2 , L3 , L4 ) and a partition P of layered width 1 . Right: The corresponding quotient G/P resulting from contracting each part of P.  \ncall geodesic treewidth, and row treewidth, introduced by Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [19] and Bose, Dujmović, Javarsineh, Morin, and Wood [11] .  \nThe geodesic treewidth of a graph G is the smallest k for which there is a partition Pinto geodesics such that G/P has treewidth k, see Figure 1 . This concept is first introduced by Pilipczuk and Siebertz [35] to prove structural and algorithmic results on proper minorclosed graph classes, in particular planar graphs, concerning p-centered colorings, bounded expansion, and subgraph-isomorp","cbCaiuZNeZ4uc7A6","https://ap.wps.com/l/cbCaiuZNeZ4uc7A6","pdf",1557908,4,1,36,"English","en",105,"# Introduction\n## Geodesic treewidth\n## Row treewidth and product structure\n## Relationship between the two notions","[{\"question\":\"What is geodesic treewidth of a graph?\",\"answer\":\"Geodesic treewidth is the smallest k such that the graph can be partitioned into geodesics and, after contracting each part, the quotient graph has treewidth k.\"},{\"question\":\"How does row treewidth differ from geodesic treewidth?\",\"answer\":\"Row treewidth uses a partition into disjoint unions of geodesics aligned with a layering (layered width 1). Contracting each aligned part gives a quotient whose treewidth is minimized over such partitions.\"},{\"question\":\"What key complexity and implication results separate the two notions?\",\"answer\":\"Bounded row treewidth does not imply bounded geodesic treewidth. The paper also provides a polynomial-time algorithm to decide whether a treewidth-2 graph has geodesic treewidth 1, while geodesic treewidth is NP-hard to compute and determining geodesic treewidth at most d is XP in treewidth.\"}]",1784182097,91,{"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},"separating-geodesic-structure-and-product-structure","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/separating-geodesic-structure-and-product-structure/82653/",{"url":52,"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 is geodesic treewidth of a graph?","Question",{"text":75,"@type":76},"Geodesic treewidth is the smallest k such that the graph can be partitioned into geodesics and, after contracting each part, the quotient graph has treewidth k.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does row treewidth differ from geodesic treewidth?",{"text":80,"@type":76},"Row treewidth uses a partition into disjoint unions of geodesics aligned with a layering (layered width 1). Contracting each aligned part gives a quotient whose treewidth is minimized over such partitions.",{"name":82,"@type":73,"acceptedAnswer":83},"What key complexity and implication results separate the two notions?",{"text":84,"@type":76},"Bounded row treewidth does not imply bounded geodesic treewidth. The paper also provides a polynomial-time algorithm to decide whether a treewidth-2 graph has geodesic treewidth 1, while geodesic treewidth is NP-hard to compute and determining geodesic treewidth at most d is XP in treewidth.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]