[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81865-en":3,"doc-seo-81865-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},81865,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Edge Geography Is XNLP-hard for Pathwidth and in XP for Tree-Partition Width","Directed Edge Geography and Undirected Edge Geography are classical PSPACE-complete two-player graph games where players move along edges and delete each used edge; the first player without a legal move loses. The paper proves both problems are XNLP-hard when parameterized by pathwidth, answering a question posed by Bodlaender over 30 years ago. It also shows Directed Edge Geography is fixed-parameter tractable for treewidth plus maximum degree, and that both games lie in XP on simple graphs under tree-partition width.","arXiv :2607 .03 189v 1 [ cs .CC] 3 Jul 2026  \nEdge Geography is XNLP-hard for Pathwidth and in XP for Tree-Partition Width  \nThobias Kvalvik Høivik \\# 􀀚  \nDepartment of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, Førde, Norway  \nErlend Raa Vågset \\# 􀀚  \nDepartment of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, Førde, Norway  \n~~ Abstract ~~  \nDirected Edge Geography and Undirected Edge Geography are classical PSPACE-complete two-player graph games in which players alternately make moves along edges, deleting each one after use; the first player unable to move loses. We prove that both problems are XNLP-hard when parameterized by pathwidth, addressing a question raised by Bodlaender over 30 years ago. On the positive side, we observe that Directed Edge Geography is fixed-parameter tractable when parameterized by treewidth and maximum degree. We also prove that both problems are in XP on simple graphs when parameterized by tree-partition width. These results develop modern lower-bound and decomposition-based algorithmic methods for width-based questions in PSPACE-complete graph games.  \n2012 ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms  \nKeywords and phrases Geography games, graph games, pathwidth, parameterized complexity, XNLP-hardness, PSPACE-complete games, tree-partition width  \nAcknowledgements We thank the anonymous reviewers for their careful reading and for their thoughtful and constructive feedback, which helped us improve the exposition and clarify several aspects of the paper.  \nConference version.  \nA shortened version of this paper has been accepted for presentation at ESA 2026 .  \n 1  Introduction  \nMany natural computational problems lie beyond NP, yet their parameterized complexity is still relatively poorly understood [12, 2] . Among them, PSPACE-hard problems are especially challenging, since their complexity is often driven by long sequences of interacting choices and evolving states. Such phenomena arise in puzzles such as Sokoban and Rush Hour [11, 16], in generalized games such as Chess and Go [17, 21], in motion planning [20, 25], in task and classical planning [26, 9], and in temporal reasoning [24] . In this paper, we study Directed Edge Geography and Undirected Edge Geography, classical PSPACE-complete graph games [19, 18] that offer a natural setting in which to ask what can and cannot be captured by structural width parameters.  \nWe are not the first to study PSPACE-hard problems from a parameterized perspective. Bäckström and Jonsson [2] develop finer complexity analyses for planning; Mouawad et al. [22] study natural process parameters for reconfiguration; and Bonnet et al. [8] analyze short winning-strategy problems in games. A particularly relevant line of study for us is quantified reasoning. Chen [10] gives an early tractability result for the quantified constraint satisfaction problem (QCSP) under bounded treewidth, and Pan and Vardi [23] show that  \n2 Edge Geography Parameterized by Width  \nfor quantified Boolean formulas (QBF), width remains useful when paired with additional control over alternation. Atserias and Oliva [1], however, show that ordinary graph width alone can still be too weak, proving PSPACE-completeness even at constant pathwidth. This in turn motivates the prefix-aware viewpoint of Eiben, Ganian, and Ordyniak [13], who introduce decomposition parameters that reflect the dependency structure induced by the quantifier prefix.  \nIn contrast to quantified reasoning, where graph structure and dependency structure need not align, Edge Geography is played directly on a graph. The issue is therefore not representational mismatch, but whether graph width alone can control an evolving game state. Bodlaender showed that stronger restrictions can make the problem easier. In particular, Edge Gene","cbCaiu1W1Fz1w7hp","https://ap.wps.com/l/cbCaiu1W1Fz1w7hp","pdf",1932726,1,65,"English","en",105,"# Introduction\n## Parameterized complexity background\n## Main results\n# Preliminaries\n## Edge Geography","[{\"question\":\"What are Directed Edge Geography and Undirected Edge Geography?\",\"answer\":\"They are two-player graph games on finite simple graphs where players alternately move along edges and delete the used edge; the first player who cannot move loses.\"},{\"question\":\"What complexity lower bound does the paper prove for pathwidth?\",\"answer\":\"Directed Edge Geography is XNLP-hard when parameterized by pathwidth, and via a parameter-preserving reduction the same holds for Undirected Edge Geography.\"},{\"question\":\"What tractability results are obtained using tree-partition width and related parameters?\",\"answer\":\"Both games on simple graphs, together with a rooted tree partition of bounded width, are solvable in XP; additionally, Directed Edge Geography is fixed-parameter tractable for treewidth together with maximum degree.\"}]",1784176734,164,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"edge-geography-is-xnlp-hard-for-pathwidth-and-in-xp-for-tree-partition-width","",{"@graph":35,"@context":84},[36,53,67],{"@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/edge-geography-is-xnlp-hard-for-pathwidth-and-in-xp-for-tree-partition-width/81865/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What are Directed Edge Geography and Undirected Edge Geography?","Question",{"text":74,"@type":75},"They are two-player graph games on finite simple graphs where players alternately move along edges and delete the used edge; the first player who cannot move loses.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What complexity lower bound does the paper prove for pathwidth?",{"text":79,"@type":75},"Directed Edge Geography is XNLP-hard when parameterized by pathwidth, and via a parameter-preserving reduction the same holds for Undirected Edge Geography.",{"name":81,"@type":72,"acceptedAnswer":82},"What tractability results are obtained using tree-partition width and related parameters?",{"text":83,"@type":75},"Both games on simple graphs, together with a rooted tree partition of bounded width, are solvable in XP; additionally, Directed Edge Geography is fixed-parameter tractable for treewidth together with maximum degree.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]