[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83730-en":3,"doc-seo-83730-105":30,"detail-sidebar-cat-0-en-105":92},{"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},83730,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","The Disjoint Separators Problem in Graphs","The disjoint separators problem in graphs studies whether, for a graph G and four pairwise disjoint vertex sets Sr, Tr, Sb, and Tb, there exist two disjoint separators: an (Sr, Tr)-separator and an (Sb, Tb)-separator. The task is equivalent to red/blue vertex coloring where Sr∪Tr is red and Sb∪Tb is blue, forbidding any red (Sr,Tr)-path and any blue (Sb,Tb)-path. The problem is proven NP-complete, with multiple restricted NP-complete cases, and for planar graphs a structural characterization plus a polynomial-time algorithm is provided for the singleton-terminal setting.","arXiv :2607 .03603v 1 [ cs .DM] 3 Jul 2026  \nThe disjoint separators problem in graphs  \nThomas Delépine1 , Florian Galliot2 , Yannick Mogge3 , Leandro Montero4 , and Nicolas  \nSchivre5  \n1LIRMM, Université de Montpellier, CNRS, Montpellier, France  \n2Aix-Marseille Université, CNRS, I2M, UMR 7373, Marseille, France  \n3LIRIS, Université Claude Bernard Lyon 1, UMR5205, Villeurbanne, France  \n4IMT Atlantique, LS2N-CNRS, La Chantrerie, Nantes, France  \n5LIMOS, Université Clermont Auvergne, CNRS, Clermont-Ferrand, France  \nAbstract  \nWe study the disjoint separators problem in graphs, an analogue of the famous disjoint paths problem. Given a graph G and four pairwise disjoint subsets of vertices Sr , Tr , Sb , Tb, we ask whether there exist an (Sr, Tr)-separator and an (Sb, Tb)-separator which are disjoint. This is equivalent to coloring the vertices in red or blue, with Sr ∪ Tr in red and Sb ∪ Tb in blue, such that there is no red (Sr, Tr)-path and no blue (Sb, Tb)-path. On the one hand, we show that the disjoint separators problem is NP-complete. We actually exhibit several NP-complete restrictions of this problem, including planar graphs of bounded maximum degree, and graphs of bounded maximum degree when |Sr| = |Tr| = |Sb| = |Tb| = 1 .  \nOn the other hand, these hardness results turn out to be quite tight, as we provide a structural characterization and a polynomial-time algorithm for planar graphs when |Sr| = |Tr| = |Sb| = |Tb| = 1 . This has an interesting consequence about the popular board game Hex: for the generalized game that may be played on any board, our result characterizes the planar boards on which draws are impossible, thus extending the well-known result about impossibility of draws on the standard commercialized board.  \n1 Introduction  \n1.1 Hex and its generalization  \nHex is a classic two-player board game invented in 1942 [9] . The board, shown in Figure 1(a), is a rhombus made of 11 × 11 hexagonal cells, with red borders on two opposite sides and blue borders on the other two sides. In turns, the players place a stone inside an unoccupied cell of their choice. The first player uses red stones, and she wins by connecting the two red borders with red stones, while the second player uses blue stones, and he wins by connecting the two blue borders with blue stones (where two cells are considered adjacent if they share a side) . Equivalently, Hex can be seen as played on the graph shown in Figure 1(b) . The vertices sr and tr are precolored in red, the vertices sb and tb are precolored in blue, and the players take turns coloring the other vertices. The first player colors vertices in red, and wins by getting a red sr tr-path, while the second player colors vertices in blue, and wins by getting a blue  \nsb tb-path. The game Hex can be generalized to any graph with four designated terminals, two of them precolored in red and the other two in blue: the first player wants to build a path between the two red terminals, while the second player wants to build a path between the two blue terminals (if all vertices are colored without either thing happening, then the game is declared a draw) . This game is part of a family of games which generalize strong positional games [8], also called Maker-Maker games, to instances where the winning combinations are not necessarily the same for both players. This family has recently been introduced in [6]: for example, if there is symmetry between the two colors (as is the case for Hex), then a strategy-stealing argument ensures that the second player cannot have a winning strategy.  \n(a)  \nFigure 1: (a) The Hex board. (b) An alternative formulation of Hex as a coloring game connecting terminals in a graph. The four outside edges have only been added to fit the statement of the upcoming Theorem 4.4 .  \nThe original Hex game is sometimes presented with Maker-Breaker rules: the first player wins if she connects the two red borders with her red stones, while the second player wins if h","cbCaieKZIC0M3a45","https://ap.wps.com/l/cbCaieKZIC0M3a45","pdf",602109,6,1,25,"English","en",105,"# Abstract\n# Introduction\n## Hex and its generalization","[{\"question\":\"What does the disjoint separators problem ask for in a graph?\",\"answer\":\"Given a graph G and four pairwise disjoint vertex subsets Sr, Tr, Sb, Tb, it asks whether there exist a (Sr,Tr)-separator and a (Sb,Tb)-separator that are disjoint.\"},{\"question\":\"How is the problem reformulated as a red/blue coloring question?\",\"answer\":\"Color vertices red with Sr∪Tr and blue with Sb∪Tb, so that there is no red (Sr,Tr)-path and no blue (Sb,Tb)-path.\"},{\"question\":\"What are the main computational results and the connection to Hex?\",\"answer\":\"The problem is NP-complete, including several restricted NP-complete cases. For planar graphs with |Sr|=|Tr|=|Sb|=|Tb|=1, a tight structural characterization and a polynomial-time algorithm are provided, yielding a characterization of planar boards where draws are impossible in a generalized Hex game.\"}]",1784190036,63,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"the-disjoint-separators-problem-in-graphs","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/the-disjoint-separators-problem-in-graphs/83730/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does the disjoint separators problem ask for in a graph?","Question",{"text":76,"@type":77},"Given a graph G and four pairwise disjoint vertex subsets Sr, Tr, Sb, Tb, it asks whether there exist a (Sr,Tr)-separator and a (Sb,Tb)-separator that are disjoint.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the problem reformulated as a red/blue coloring question?",{"text":81,"@type":77},"Color vertices red with Sr∪Tr and blue with Sb∪Tb, so that there is no red (Sr,Tr)-path and no blue (Sb,Tb)-path.",{"name":83,"@type":74,"acceptedAnswer":84},"What are the main computational results and the connection to Hex?",{"text":85,"@type":77},"The problem is NP-complete, including several restricted NP-complete cases. For planar graphs with |Sr|=|Tr|=|Sb|=|Tb|=1, a tight structural characterization and a polynomial-time algorithm are provided, yielding a characterization of planar boards where draws are impossible in a generalized Hex game.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},"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":107,"slug":138},19,"General","general"]