[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85230-en":3,"doc-seo-85230-105":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},85230,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Recognizability equals CMSO-definability for graphs of rank-width at most two","Proof establishes that for finite graphs with rank-width at most two, VR-recognizability and counting monadic second-order (CMSO) definability coincide. The argument advances the recognizability-versus-definability problem beyond rank-width-one by analyzing split-prime graphs and organizing exact cut-rank-two separations via maximal partial-tree theory. Canonical cores yield a CMSO-definable laminar family, whose local pieces admit uniformly bounded linear rank-width, enabling finite-state evaluation. A CMSO-transducible split decomposition extends the result to all graphs with rank-width ≤ 2.","arXiv :2607 . 10594v1 [math .CO] 12 Jul 2026  \nRecognizability equals CMSO-definability for graphs of rank-width at most two  \nAntonios Kalampakas  \nDepartment of Mathematics, American University of the Middle East  \nEgaila 54200, Kuwait  \n12 July 2026  \nAbstract  \nWe prove that, on finite graphs of rank-width at most two, VR-recognizability and counting monadic second-order definability coincide. This advances the recognizability-versus-definability problem from bounded linear clique-width to the first nontrivial bounded rank-width level beyond the rank-width-one split-decomposition case. The proof first treats split-prime graphs. The maximal partial-tree theory of Clark and Whittle organizes the non-sequential cut-rank-two separations, while a single strong separation orients all strong equivalence classes and yields a CMSO-definable laminar family of canonical cores. Although the auxiliary partial tree isnot itself transduced, it proves that every canonical local piece has a port-contiguous layout of uniformly bounded linear rank-width. The width argument uses partition atoms and the branch-width-three display theorem of Hall, Oxley, Semple, and Whittle and does not assume that graph torsos remain prime. Coherent ordered rank-two frames then permit a finite-state bottom-up evaluation whose local transitions are definable by the bounded-linear-clique-width theorem of Bojańczyk, Grohe, and Pilipczuk. Finally, the CMSO-transducible canonical split decomposition lifts the result from prime graphs to arbitrary graphs of rank-width at most two.  \nKeywords: rank-width; clique-width; VR-recognizability; counting monadic second-order logic; split decomposition; connectivity functions; graph transductions.  \n2020 Mathematics Subject Classification: 05C83; 03B70; 68Q45 .  \n1 Introduction and main result  \nCourcelle’s programme relates logical descriptions of graph properties to finite-state evaluation on algebraic graph decompositions. For sparse graphs, the resulting equivalence between recognizability and counting monadic second-order definability on every bounded-tree-width class was proved by Bojańczyk and Pilipczuk [3] . The corresponding dense setting is governed by clique-width and rank-width. Clique-width was developed by Courcelle and Olariu [8], while rank-width, introduced by Oum and Seymour [12, 14], measures the F2-rank of the cuts in a branch decomposition and is equivalent to clique-width up to an exponential change of parameter; see also the survey of Oum [13] .  \nFor vertex-replacement, or VR, recognizability, the forward logical implication is classical: every CMSO 1-definable graph property is VR-recognizable [7, 5] . The converse fails without a width restriction, and the central question is therefore whether it becomes true on boundedclique-width, equivalently bounded-rank-width, classes. A major positive result of Bojańczyk,  \nGrohe, and Pilipczuk proves the equivalence on every class of bounded linear clique-width [2] . That theorem does not settle bounded rank-width: even rank-width-one graphs have unbounded linear rank-width, as is already witnessed by trees [1] . Recent CMSO transductions for canonical split decompositions, combined with the standard finite-state evaluation of their bounded-width outputs, nevertheless provide the rank-width-one case [4] . The present paper establishes the next level, rank-width at most two.  \nWe fix the conventions used throughout. All graphs are finite, simple, and undirected, and CMSO means CMSO 1 : monadic second-order logic over the vertex-adjacency structure, extended by all fixed modular-cardinality predicates. For k ≥ 1 , a k-context is a finite k-labelled VR term with one distinguished input. Two k-labelled graphs are equivalent for an unlabelled property P if every k-context, after labels are forgotten, either places both graphs in P or places neither in P. Following Courcelle–Engelfriet [5, Definition 4.29], P is globally VR-recognizable when this contextual equival","cbCaitOXnZ4Yy0iu","https://ap.wps.com/l/cbCaitOXnZ4Yy0iu","pdf",393411,4,1,24,"English","en",105,"# Introduction and main result\n# Theorem 1.1 (Main theorem)\n# Proof strategy and structural framework","[{\"question\":\"What is the main result of the paper for graphs with rank-width at most two?\",\"answer\":\"On finite graphs with rank-width at most two, a language obtained from a globally VR-recognizable property matches exactly the class of graphs satisfying some CMSO sentence. Thus VR-recognizability and counting monadic second-order definability coincide on this class.\"},{\"question\":\"Why does the proof start with split-prime graphs?\",\"answer\":\"The proof first treats split-prime inputs to organize non-sequential exact cut-rank-two separations using the Clark–Whittle and maximal partial-tree theorems. This structure supports a CMSO-definable laminar family of canonical cores before the argument is lifted to general graphs.\"},{\"question\":\"How does the paper extend the result from split-prime graphs to all graphs of rank-width at most two?\",\"answer\":\"After deriving the canonical structure and bounded-layout properties for prime inputs, the result is lifted using CMSO-transducible canonical split decomposition. This transduction transfers the definability equivalence from prime graphs to arbitrary graphs with rank-width ≤ 2.\"}]",1784201879,60,{"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},"recognizability-equals-cmso-definability-for-graphs-of-rank-width-at-most-two","",{"@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/recognizability-equals-cmso-definability-for-graphs-of-rank-width-at-most-two/85230/",{"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","2026-07-24","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 is the main result of the paper for graphs with rank-width at most two?","Question",{"text":75,"@type":76},"On finite graphs with rank-width at most two, a language obtained from a globally VR-recognizable property matches exactly the class of graphs satisfying some CMSO sentence. Thus VR-recognizability and counting monadic second-order definability coincide on this class.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does the proof start with split-prime graphs?",{"text":80,"@type":76},"The proof first treats split-prime inputs to organize non-sequential exact cut-rank-two separations using the Clark–Whittle and maximal partial-tree theorems. This structure supports a CMSO-definable laminar family of canonical cores before the argument is lifted to general graphs.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper extend the result from split-prime graphs to all graphs of rank-width at most two?",{"text":84,"@type":76},"After deriving the canonical structure and bounded-layout properties for prime inputs, the result is lifted using CMSO-transducible canonical split decomposition. This transduction transfers the definability equivalence from prime graphs to arbitrary graphs with rank-width ≤ 2.","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},[93,97,101,105,109,114,119,122,127,130,134],{"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":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]