[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85903-en":3,"doc-seo-85903-105":30,"detail-sidebar-cat-0-en-105":87},{"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},85903,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Reconfiguring Subgraphs with Extra Resources","Subgraph reconfiguration asks whether one subgraph can be transformed into another through local changes while preserving a graph property. This work studies the edge-specified token model and proves hardness results tied to pathwidth and treewidth: connected graphs of pathwidth at most k are NP-hard for fixed k≥1, and the same holds for k≥2; minor-closed reductions imply NP-hardness for planar graphs. After negative results, the paper analyzes buffer resources, showing Ω(n) extra space is required for planar graphs and graphs with bounded pathwidth and treewidth, while O(1) buffer suffices for cacti in a restricted setting.","arXiv :2607 . 10399v1 [ cs .DM] 11 Jul 2026  \nReconfiguring Subgraphs with Extra Resources  \nJason Fong \\#   \nGeorgia Institute of Technology, United States Jeffrey Kam \\#   \nUniversity of Cambridge, United Kingdom Steven Wong \\#  \nUniversity of Waterloo, Canada  \n~~ Abstract ~~  \nThe subgraph reconfiguration problem asks whether one subgraph can be transformed into another via a sequence of local changes while maintaining a specified graph property. In this work, we focus on the setting where the subgraph is specified by its set of edges. Our contributions in this paper are twofold. First, motivated by the contrast that path reconfiguration is NP-hard while tree reconfiguration is solvable in linear time, we prove two generalizations: (1) for any fixed k at least one, reconfiguring connected graphs with pathwidth at most k is NP-hard, and (2) for any fixed k at least two, reconfiguring graphs with pathwidth at most k is also NP-hard. En route to proving (2), we show a general hardness result that applies to a range of minor-closed graph classes, which we use to show planar graph reconfiguration is also NP-hard. Second, given our negative results, we extend the problem to a resource-focused setting, asking how much additional buffer space is needed to turn a non-reconfigurable instance into a reconfigurable one. We show that Ω(n) extra buffer space is needed for planar graphs and graphs with bounded pathwidth and treewidth, while O(1) extra buffer space is sufficient for cactus graphs in a restricted setting.  \n2012 ACM Subject Classification Theory of computation → Graph algorithms analysis; Theory of computation → Problems, reductions and completeness  \nKeywords and phrases Reconfiguration, Subgraph Reconfiguration, Treewidth, Pathwidth, Reconfiguration with Extra Buffer  \nAcknowledgements Part of this work was presented in the Banff Reconfiguration Workshop (22w5090) and the authors would like to thank the participants for their suggestions. The authors would like to thank Naomi Nishimura and Benjamin Moore for the early discussions and feedback.  \n 1  Introduction  \nCombinatorial reconfiguration [18] is the study of transformations between configurations by making one local change at a time, providing a unifying framework for problems ranging from toy problems like the 15 puzzle to practical use-cases in power supply [12] and quantum computing [6, 1] . Our focus in this paper is on the Subgraph reconfiguration problem, first introduced by Hanaka et al. [11] . In this setting, we are given an input graph G and some graph property Π (e.g. a graph is a tree) . A configuration is a subgraph H ⊆ G such that H satisfies Π . Specifically, we only consider the edge variant setting, where each configuration (a subgraph) is specified by its set of edges. To align with existing literature, we imagine a placement of tokens on the edges of G and the placement of tokens is a valid configuration if the token-induced subgraph, formed by edges with a token, satisfies Π . The local change in our case, also known as a reconfiguration step, is the movement of a token from one edge to an unoccupied edge, also known as the Token Jumping model. We are most interested in the problem where given a source subgraph Es and target subgraph Et of the input graph G, both satisfying Π, whether there exists a sequence of local changes, also known as a reconfiguration sequence, that transforms Es to Et. We call the problem the Π reconfiguration problem.  \n2 Reconfiguring Subgraphs with Extra Resources  \n\n| Property | Complexity | Constraints | Extra Buffer |\n| --- | --- | --- | --- |\n\n\n| Path | NP-hard [11] | Input is connected | Ω(n) [Theorem 17] |\n| --- | --- | --- | --- |\n| Tree | P [11] | Input is connected | 0 [Theorem 17] |\n| Cycle | P [11] | Input is connected | ∞ [Theorem 17] |\n| Connected\u003Cbr>Pathwidth ≤ k | NP-hard [Theorem 6] | Input is connected | Ω(n) [Theorem 18] |\n| Treewidth ≤ k | NP-hard [Theorem 14] | None | Ω(n) [Theorem 18] |\n| Planar | ","cbCaifisUHMU0iDr","https://ap.wps.com/l/cbCaifisUHMU0iDr","pdf",793830,2,1,28,"English","en",105,"# Introduction\n## Reconfiguring Subgraphs with Extra Resources","[{\"question\":\"How does the paper relate complexity to structural parameters like pathwidth and treewidth?\",\"answer\":\"It proves NP-hardness for connected graphs of bounded pathwidth (for fixed k≥1 and also for k≥2) and shows NP-hardness for graphs with bounded treewidth.\"},{\"question\":\"What is the role of extra buffer space in making an instance reconfigurable?\",\"answer\":\"After proving negative complexity results, the paper asks how much additional buffer is needed to enable reconfiguration. It shows Ω(n) extra buffer for planar graphs and bounded pathwidth/treewidth graphs, but O(1) buffer suffices for cacti under a restricted condition.\"}]",1784207060,71,{"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":82,"head_meta":84,"extra_data":86,"updated_unix":28},"reconfiguring-subgraphs-with-extra-resources","",{"@graph":36,"@context":81},[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/reconfiguring-subgraphs-with-extra-resources/85903/",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","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77],{"name":72,"@type":73,"acceptedAnswer":74},"How does the paper relate complexity to structural parameters like pathwidth and treewidth?","Question",{"text":75,"@type":76},"It proves NP-hardness for connected graphs of bounded pathwidth (for fixed k≥1 and also for k≥2) and shows NP-hardness for graphs with bounded treewidth.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the role of extra buffer space in making an instance reconfigurable?",{"text":80,"@type":76},"After proving negative complexity results, the paper asks how much additional buffer is needed to enable reconfiguration. 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