[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86504-en":3,"doc-seo-86504-105":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},86504,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Independent Set Reconfiguration on Threshold Signed Graphs","Token Jumping and Sliding Token are fundamental reconfiguration problems on the independent sets of an undirected graph, asking whether two independent sets of equal size can be transformed via a sequence of intermediate independent sets of the same size. While both problems are PSPACE-complete on general graphs, polynomial-time algorithms are known for several restricted graph classes. This work proves polynomial-time solvability on threshold signed graphs (Dilworth-2 graphs) using the inclusion-chain structure that characterizes the class.","arXiv :2607 . 10629v 1 [ cs .DS] 12 Jul 2026  \nIndependent Set Reconfiguration on Threshold Signed Graphs  \nZiad Ismaili Alaoui  \nSchool of Computer Science and Informatics, University of Liverpool, United Kingdom Department of Informatics, Philipps-Universität Marburg, Germany [ziad. ismaili-alaoui@liverpool. ac. uk](ziad. ismaili-alaoui@liverpool. ac. uk)  \nAbstract  \nThe Token Jumping and Sliding Token problems are fundamental reconfiguration problems defined on the independent sets of an undirected graph. Given two independent sets I and J, each of size k, these problems ask whether there exists a sequence of elementary operations transforming I into J such that every intermediate configuration is also an independent set of size k. In Sliding Token, an operation moves a token from a vertex u ∈ I to an adjacent vertex v  I; in Token Jumping, the token may instead move to any vertex v  I. While both problems are PSPACE-complete on general graphs, polynomial-time algorithms have been developed for several graph classes, including trees, block graphs, cacti, bipartite permutation graphs, cographs, P4-tidy graphs, and interval graphs.  \nIn this paper, we prove that both problems are solvable in polynomial time on threshold signed graphs, also known as Dilworth-2 graphs. A graph G = (V, E) is a threshold signed graph if there exist a mapping a : V → R and positive real constants S and T such that, for any distinct vertices u, v ∈ V , {u, v} ∈ E if and only if |a(u) + a(v)| ≥ S or |a(u) − a(v)| ≥ T. This graph class is a subclass of permutation graphs, for which the complexity of these problems remains open, and is incomparable with the class of bipartite permutation graphs studied by Fox-Epstein et al. (ISAAC, 2015) . The algorithm is based on the inclusion-chain structure that characterises threshold signed graphs, a structural property that may be of independent interest.  \n1 Introduction  \nReconfiguration problems typically ask whether one feasible solution can be transformed into another via a sequence of elementary steps. A central example is Independent Set Reconfiguration, whose seemingly simple formulation conceals rich combinatorial structure. Among its most widely studied variants are Token Jumping and Sliding Token. Imagine placing tokens on the vertices of a graph so that they form an independent set I, and then moving one token at a time (along an edge in Sliding Token, or to any vertex in Token Jumping) until reaching another independent set J of the same size. Throughout the process, the occupied vertices must remain an independent set. Can I be transformed into J in this way? More formally, the problem is defined as follows.  \n\n| Independent Set Reconfiguration (Token Jumping and Sliding Token) |\n| --- |\n| Instance: An undirected graph G and independent sets 1 I, J ⊆ V (G), with |I| = |J| = k.\u003Cbr>Question: Does there exist a finite sequence of independent sets I1 , ... , Im such that I1 = I , Im = J , |Ii| = k for all i ∈ [m], |Ii△Ii+1| = 2 for all i ∈ [m − 1], and, in the case of Sliding Token, Ii△Ii+1 = {u, v} ∈ E (G) for all i ∈ [m − 1]? |\n\n1We use the terms independent set and configuration interchangeably.  \nAlthough both variants are PSPACE-complete if we know nothing about the input graphs [HD05 , IDH+11 , KMM12], only a handful of graph classes possess sufficient structure to admit polynomialtime algorithms. Since the introduction of Sliding Token by Demaine et al. [HD05] and of Token Jumping independently by Ito et al. [IDH+11] and Kamiński et al. [KMM12], a series of results has gradually expanded this list to include trees, interval graphs, cographs, and several other graph classes (see Table 1 for an overview2 ) . Nevertheless, positive results remain surprisingly scarce. As Bartier, Bousquet, and Mouawad recently observed in [BBM23],“almost no positive result is known for Token Sliding [sic]3 even for incredibly simple cases like bounded-degree graphs.”  \n\n| Graph Class | Jumping | Sliding |\n| --- | ","cbCaibMHqDmdSeFg","https://ap.wps.com/l/cbCaibMHqDmdSeFg","pdf",654659,4,1,15,"English","en",105,"# Abstract\n# Introduction\n## Problem Definition and Variants\n## Related Work and Complexity Landscape\n## Contributions","[{\"question\":\"What are the Token Jumping and Sliding Token reconfiguration problems?\",\"answer\":\"They ask whether two independent sets of the same size can be transformed into each other through a sequence of intermediate independent sets of that same size, using allowed token moves. Sliding Token moves a token only along an edge, while Token Jumping allows moving to any vertex.\"},{\"question\":\"What is the complexity of these problems on general graphs?\",\"answer\":\"Both problems are PSPACE-complete on general graphs, according to the discussion of known results in the introduction.\"},{\"question\":\"What does the paper prove for threshold signed graphs?\",\"answer\":\"It proves that both Token Jumping and Sliding Token are solvable in polynomial time on threshold signed graphs (also called Dilworth-2 graphs), based on the inclusion-chain structural characterization of this graph class.\"}]",1784212241,38,{"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},"independent-set-reconfiguration-on-threshold-signed-graphs","",{"@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/independent-set-reconfiguration-on-threshold-signed-graphs/86504/",{"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","2026-07-28","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 are the Token Jumping and Sliding Token reconfiguration problems?","Question",{"text":75,"@type":76},"They ask whether two independent sets of the same size can be transformed into each other through a sequence of intermediate independent sets of that same size, using allowed token moves. Sliding Token moves a token only along an edge, while Token Jumping allows moving to any vertex.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the complexity of these problems on general graphs?",{"text":80,"@type":76},"Both problems are PSPACE-complete on general graphs, according to the discussion of known results in the introduction.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the paper prove for threshold signed graphs?",{"text":84,"@type":76},"It proves that both Token Jumping and Sliding Token are solvable in polynomial time on threshold signed graphs (also called Dilworth-2 graphs), based on the inclusion-chain structural characterization of this graph class.","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"]