[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86083-en":3,"doc-seo-86083-105":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},86083,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Hereditary 2-WQO Graph Classes Have Bounded Clique-Width","A graph class is k-WQO when every k-labeled graph family from the class is well-quasi-ordered by label-preserving induced subgraph embeddings. This work proves that any hereditary graph class that is 2-WQO must have bounded clique-width. Together with Dumas and Lopez’s result, it settles Pouzet’s long-standing conjectures by showing equivalence between 2-WQO, k-WQO for all k≥2, and ∀-WQO. The proof uses a structure/non-structure dichotomy for monadic dependence and excludes large well-linked sets as the canonical clique-width obstruction.","Hereditary 2-WQO Graph Classes Have Bounded Clique-Width∗  \nJulien Duron University of Warsaw [j.duron@uw.edu.pl](j.duron@uw.edu.pl)  \nNikolas Mhlmann  \nUniversity of Warsaw [maehlmann@mimuw.edu.pl](maehlmann@mimuw.edu.pl)  \nSzymon Toruczyk University of Warsaw [szymtor@mimuw.edu.pl](szymtor@mimuw.edu.pl)  \narXiv :2607 . 10939v1 [math .CO] 12 Jul 2026  \nJuly 14, 2026  \nAbstract  \nA graph class is k-WQO ifits k-labeled graphs are well-quasi-ordered under label-preserving induced subgraph embeddings. We show that every hereditary graph class that is 2-WQO has bounded cliquewidth. Combined with the recent result of Dumas and Lopez, this confirms a long-standing conjecture of Pouzet: A hereditary graph class is 2-WQO if and only if it is k-WQO for all k ⩾ 2, if and only if it is ∀-WQO, that is, its labeled graphs are well-quasi-ordered for every possible well-quasi-ordered label set.  \nOur proof builds on a recent structure/non-structure dichotomy for the model theoretic notion of monadic dependence by Dreier, Mhlmann, and Toruczyk. Through the non-structure characterization by forbidden induced subgraphs, we show that every hereditary 2-WQO graph class is monadically dependent. Leveraging the Ramsey-theoretic structural properties provided by monadic dependence, we then establish bounded clique-width by ruling out the existence of large well-linked sets, which are the canonical obstructions for clique-width.  \nAcknowledgements. We thank Pierre Simon for suggesting the connection of Pouzet’s conjecture to monadic dependence.  \n1 Introduction  \nA graph class C is well-quasi-ordered (WQO) if it contains no infinite antichain under the induced subgraph (embedding) relation. Further, if k is a positive integer, we say that C is k-WQO if the class of all vertexlabelled graphs from C with labels in [k] is well-quasi-ordered under label-preserving embeddings.  \nFor example, the class of cycles is not WQO, while the class of paths is WQO, but it is not 2-WQO, since labelling the two endpoints with a distinguishing label creates an infinite antichain. On the other hand, the class of linear orders (viewed as directed graphs) is k-WQO for all k ⩾ 1 by Higman’s lemma, and so is the class of cographs [4](see also [2, Thm. 1]) .  \nThe notion of k-WQO was studied by Pouzet around 1972, who observed that already 2-WQO is a very strong assumption and asked whether for classes of relational structures which are hereditary (that is, closed under taking induced substructures) 2-WQO implies k-WQO for all k ⩾ 1. This is a rephrasing of a question posed in [11, §2.4], which has been repeated by Fra in [7, Chap. 12, §3.1] . In the setting of graphs, this has been repeated by Daligault, Rao, and Thomass [3] in the modern formulation, as follows:  \n∗JD and SzT received funding from the European Research Council (ERC) with grant agreement No. 101126229 – BUKA. NM received funding from the European Union through an ERA Fellowship with grant agreement No. 101334340 – LoCoMoDe.  \nConjecture 1.1 ([11], [3, Conj. 1]). For hereditary graph classes, 2-WQO implies k-WQOfor all k ⩾ 1.  \nIn fact, Pouzet [12, Problems 9 (1)] conjectured that for hereditary classes, 2-WQO even implies ∀-WQO1 , which means that the class remains WQO under embeddings which preserve or increase the labels, when the vertex labels are drawn from an arbitrary fixed well-quasi-order (another variant of this conjecture appears in [13, Problem 3]) .  \nDaligault, Rao, and Thomass [3, Conj. 5] proposed a route towards Pouzet’s conjecture, by posing the following structural conjecture:  \nConjecture 1.2 ([3]). Every hereditary 2-WQO graph class has bounded clique-width.  \nClique-width is one of the central width parameters for hereditary graph classes. It can be seen asan extension of tree-width to dense graphs. Roughly, graphs of bounded clique-width can be encoded in trees in a simple fashion (namely, by a fixed MSO formula) . Thus, the conjecture of Daligault, Rao, and Thomass gives an unexpecte","cbCaiagjE9eTv9rF","https://ap.wps.com/l/cbCaiagjE9eTv9rF","pdf",1094646,4,1,22,"English","en",105,"# Introduction\n## Background on WQO and k-WQO\n## Pouzet’s conjectures and related formulations\n## Clique-width and structural conjecture\n## Main results and consequences","[{\"question\":\"What does it mean for a graph class to be k-WQO?\",\"answer\":\"A graph class is k-WQO if all vertex-labeled graphs from the class with labels in [k] are well-quasi-ordered under label-preserving embeddings (using induced subgraph embeddings).\"},{\"question\":\"What is the main theorem proved in the document?\",\"answer\":\"Every hereditary 2-WQO graph class has bounded clique-width.\"},{\"question\":\"How does the result relate to Pouzet’s conjectures?\",\"answer\":\"Using Dumas and Lopez’s theorem, the paper shows that for hereditary graph classes, 2-WQO is equivalent to k-WQO for all k≥1 and to ∀-WQO, confirming conjectures of Pouzet.\"}]",1784208404,55,{"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},"hereditary-2-wqo-graph-classes-have-bounded-clique-width","",{"@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/hereditary-2-wqo-graph-classes-have-bounded-clique-width/86083/",{"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","2026-07-27","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 does it mean for a graph class to be k-WQO?","Question",{"text":75,"@type":76},"A graph class is k-WQO if all vertex-labeled graphs from the class with labels in [k] are well-quasi-ordered under label-preserving embeddings (using induced subgraph embeddings).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main theorem proved in the document?",{"text":80,"@type":76},"Every hereditary 2-WQO graph class has bounded clique-width.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the result relate to Pouzet’s conjectures?",{"text":84,"@type":76},"Using Dumas and Lopez’s theorem, the paper shows that for hereditary graph classes, 2-WQO is equivalent to k-WQO for all k≥1 and to ∀-WQO, confirming conjectures of Pouzet.","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"]