[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85266-en":3,"doc-seo-85266-105":29,"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":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},85266,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes","Monadic dependence serves as a structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A class is monadically dependent when its vertex-colored members cannot interpret all graphs using a single fixed first-order formula. Two consequences are proved: every monadically dependent class has almost linear neighborhood complexity, and every n-vertex graph admits radius-1 merge-width n^{o(1)}. The approach is algorithmic, giving an O(n^5) method producing a radius-1 merge-width O(n^{1-1/d} log n) construction sequence under neighborhood bounds.","arXiv :2607 . 10941v1 [ cs .DM] 12 Jul 2026  \nNeighborhood Complexity and Radius-1 Merge-Width  \nin Monadically Dependent Graph Classes ∗ Jan Dreier, Nikolas Mählmann, Rose McCarty, Michał Pilipczuk, Szymon Toruńczyk  \nAbstract  \nMonadic dependence is a proposed structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A graph class is monadically dependent if the class of all graphs cannot be interpreted in its vertex-colored members using a fixed first-order formula. We prove two structural consequences of monadic dependence. First, every monadically dependent class has almost linear neighborhood complexity: for every graph G in the class and every set A ⊆ V(G), the family {NG (v)∩A : v ∈ V(G)} has size |A| 1+o(1) . Second, every n-vertex graph in a monadically dependent class has radius-1 merge-width no(1) . Here, merge-width is the decomposition parameter of Dreier and Toruńczyk based on construction sequences; its radius-r version measures local reachability among parts through already resolved pairs. This settles the radius-1 case of the conjectured connection between monadic dependence and almost bounded merge-width and provides the first decomposition-based structural description of monadically dependent graph classes. Our proof is algorithmic: we give an O (n5 )-time algorithm that, given an n-vertex graph G such that |{NG (v) ∩ A : v ∈ V (G)}| ⩽ O(|A|d ) for every A ⊆ V (G), computes a construction sequence witnessing radius-1 merge-width O (n1−1/d log n) .  \n1 Introduction  \nIn the first-order model checking problem, the input is a graph (or other structure) and a sentence φ of first-order logic, and the task is to decide whether φ is true in G. This fundamental problem captures many concrete problems of interest, including k-Clique, k-Dominating Set, and k-Independent Set, and has been the focus of decades of research aimed at understanding which structural restrictions on the input graph render the problem tractable.  \nA landmark result of Grohe, Kreutzer, and Siebertz [22] established that first-order model checking is fixed-parameter tractable on every nowhere dense graph class; the running time of the algorithm is of the form f (|φ|) · n1+o(1), where f is a function depending on the class and the o(1) term also depends on the class. For monotone (subgraph-closed) graph classes, nowhere denseness is precisely the dividing line between (fixed-parameter) tractability and intractability [19, 22] . However, there are natural tractable classes—such as classes of bounded clique-width [11] or twin-width [9]—that are not monotone, contain dense graphs, and hence lie beyond the scope of this classification.  \n∗ NM received funding from the European Union through an ERA Fellowship with grant agreement No. 101334340 – LoCoMoDe. RM was supported by the National Science Foundation under Grant No. DMS-2452111 . MP was supported by the project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement No. 948057. SzT received funding from the European Research Council (ERC) with grant agreement No. 101126229 – BUKA.  \nThe search for the exact tractability boundary among all hereditary (induced-subgraph-closed) graph classes has converged on a notion from Shelah’s classification theory: monadic dependence [4] . A graph class C is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from C using a fixed first-order formula. Monadic dependence precisely captures the known tractability boundaries in all settings where a complete classification exists: it is equivalent to nowhere denseness for monotone classes [2], to monadic stability for orderless classes [27], and to bounded twin-width for classes of ordered graphs [8] . This has led to the following conjecture, which is now the central open problem in the area.  \nConjecture 1 (e.g., [1, 8 ","cbCaikhlwJROsq4X","https://ap.wps.com/l/cbCaikhlwJROsq4X","pdf",884217,1,24,"English","en",105,"# Abstract\n# Introduction\n## First-order model checking and tractability\n## Monadic dependence and the conjecture\n## Merge-width and construction sequences\n## Contributions","[{\"question\":\"What does monadic dependence mean for a graph class?\",\"answer\":\"A graph class is monadically dependent if not all graphs can be interpreted in its vertex-colored members using a single fixed first-order formula.\"},{\"question\":\"What structural consequences are proved for monadically dependent classes?\",\"answer\":\"The paper proves that monadically dependent classes have almost linear neighborhood complexity and that every n-vertex graph in such a class has radius-1 merge-width n^{o(1)}.\"},{\"question\":\"How does the paper make the result constructive?\",\"answer\":\"It gives an O(n^5)-time algorithm that, under uniform neighborhood-size assumptions, computes a construction sequence witnessing the claimed radius-1 merge-width bound.\"}]",1784202166,60,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"neighborhood-complexity-and-radius-1-merge-width-in-monadically-dependent-graph-classes","",{"@graph":35,"@context":85},[36,53,68],{"@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/neighborhood-complexity-and-radius-1-merge-width-in-monadically-dependent-graph-classes/85266/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 monadic dependence mean for a graph class?","Question",{"text":75,"@type":76},"A graph class is monadically dependent if not all graphs can be interpreted in its vertex-colored members using a single fixed first-order formula.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What structural consequences are proved for monadically dependent classes?",{"text":80,"@type":76},"The paper proves that monadically dependent classes have almost linear neighborhood complexity and that every n-vertex graph in such a class has radius-1 merge-width n^{o(1)}.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper make the result constructive?",{"text":84,"@type":76},"It gives an O(n^5)-time algorithm that, under uniform neighborhood-size assumptions, computes a construction sequence witnessing the claimed radius-1 merge-width 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