[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81929-en":3,"doc-seo-81929-105":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},81929,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Rerouting Curves on Surfaces","We study reconfiguring a crossing-free graph embedding on a surface by rerouting edges one at a time, keeping fixed vertex positions and maintaining crossing-free intermediate embeddings. Starting from a given embedding, a move replaces one edge curve with a new simple curve that does not intersect other edge curves. The work builds on prior results for matchings in the plane and torus, proving reconfiguration is always possible on the torus for matchings, trees, and forests, and extends sufficient conditions to other orientable and non-orientable surfaces. For general graphs, reconfiguration is not always possible.","Rerouting Curves on Surfaces  \narXiv :2607 .05362v 1 [ cs .CG] 6 Jul 2026  \nTimo Brand \\#   \nTechnische Universität München, DE Henry Förster \\#   \nJohn Cabot University, IT Technische Universität München, DE  \nUniversität Tübingen, DEAnna Lubiw \\#  University of Waterloo, CAJános Pach \\#   \nRényi Institute of Mathematics, HU EPFL, CH  \nGéza Tóth \\#   \nRényi Institute of Mathematics, HU Pavel Valtr \\#   \nCharles University Prague, CZ  \nStefan Felsner \\#   \nTechnische Universität Berlin, DE  \nStephen Kobourov \\#  University of Arizona, US Technische Universität München, DE  \nYoshio Okamoto \\#  The University of Electro-Communications, JPCsaba D. Tóth \\#   \nCal State Northridge, Los Angeles, CA, US Tufts University, Medford, MA, US Torsten Ueckerdt \\#   \nKarlsruhe Institute of Technology, DE  \n~~ Abstract ~~  \nWe study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In this process, we reroute one edge at a time while maintaining crossing-free intermediate embeddings. This problem was introduced by Ito et al. [TALG 2025], who showed that even if the graph is a matching of two edges, reconfiguration is not always possible in the plane, but is always possible on the torus. For matchings of two or more edges, they gave a necessary and sufficient condition for reconfigurable embeddings in the plane, but not on the torus. Our main result is that for matchings, trees and forests, reconfiguration is always possible on the torus, and consequently, on any orientable surface of genus at least one. In addition, we provide sufficient conditions for reconfiguration on orientable surfaces of genus at least one and in the projective plane. For more general graphs, we show that reconfiguration is not always possible.  \n2012 ACM Subject Classification Mathematics of computing → Geometric topology; Mathematics of computing → Graph algorithms; Mathematics of computing → Trees; Mathematics of computing → Combinatorial algorithms  \nKeywords and phrases Combinatorial reconfiguration, orientable surface, non-orientable surface, rerouting  \nFunding Yoshio Okamoto: Partially supported by JSPS KAKENHI Grant Numbers JP23K10982, JP26K23806, and JST ERATO Grant Number JPMJER2301 .  \nCsaba D. Tóth: Research supported in part by the NSF award DMS-2154347 .  \nTorsten Ueckerdt: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– 520708409  \nAcknowledgements This work started at Dagstuhl Seminar 24062 “Beyond-Planar Graphs: Models, Structures and Geometric Representations”.  \n 1  Introduction  \nWe study the problem of reconfiguring graph embeddings on a surface, where the vertices are fixed and each reconfiguration step redraws one edge (represented as a curve) . Consider a set P of points on a surface Σ, and two embeddings B and R of the same simple graph  \n2 Rerouting Curves on Surfaces  \n(c)  \n Figure 1 (a) Two embeddings (red & blue) that cannot be reconfigured on the plane. (b) Reconfiguration on the fundamental square. (c) Corresponding illustration on the torus.  \nG that share the same mapping of vertices to P. Here, an embedding means that each edge is drawn as a simple curve, which we call an edge curve, on the surface, and no two edge curves intersect except at a common endpoint. The edge curves of B may cross the edge curves of R and the correspondence between edge curves of B and R is implied by the identical mapping of vertices. A reconfiguration step or move replaces one edge curve γ of an embedded graph G by a new curve γ′ to obtain a new embedding of G; i.e. , γ′ may not cross any of the other edge curves in the embedding, though we allow γ and γ′ to intersect. We address the question whether B can be reconfigured to R via a sequence of moves.  \nIto et al. [22] showed that reconfiguration in the plane is not always possible, even if the graph","cbCaiaBOtzCZ5hla","https://ap.wps.com/l/cbCaiaBOtzCZ5hla","pdf",1572476,1,32,"English","en",105,"# Introduction\n## Problem definition and reconfiguration moves\n## Related work and motivation","[{\"question\":\"What does a reconfiguration step (move) mean in this problem?\",\"answer\":\"A move replaces one edge curve in the current crossing-free embedding by a new simple edge curve, keeping all other edge curves unchanged. The new curve must not cross any other edge curves (though the replaced curve and its replacement may intersect).\"},{\"question\":\"Why is reconfiguration sometimes impossible in the plane?\",\"answer\":\"Prior work by Ito et al. shows that even when the graph is a matching of two independent edges, some embedding pairs cannot be reconfigured on the plane while staying crossing-free throughout the process.\"},{\"question\":\"Which graph classes are guaranteed to be reconfigurable on the torus?\",\"answer\":\"The paper’s main result states that for matchings, trees, and forests, reconfiguration is always possible on the torus. This implies reconfiguration on any orientable surface of genus at least one for these classes.\"}]",1784177099,81,{"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},"rerouting-curves-on-surfaces","",{"@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/rerouting-curves-on-surfaces/81929/",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 does a reconfiguration step (move) mean in this problem?","Question",{"text":74,"@type":75},"A move replaces one edge curve in the current crossing-free embedding by a new simple edge curve, keeping all other edge curves unchanged. The new curve must not cross any other edge curves (though the replaced curve and its replacement may intersect).","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why is reconfiguration sometimes impossible in the plane?",{"text":79,"@type":75},"Prior work by Ito et al. shows that even when the graph is a matching of two independent edges, some embedding pairs cannot be reconfigured on the plane while staying crossing-free throughout the process.",{"name":81,"@type":72,"acceptedAnswer":82},"Which graph classes are guaranteed to be reconfigurable on the torus?",{"text":83,"@type":75},"The paper’s main result states that for matchings, trees, and forests, reconfiguration is always possible on the torus. This implies reconfiguration on any orientable surface of genus at least one for these classes.","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"]