[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86401-en":3,"doc-seo-86401-105":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},86401,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Adjacency Labelling for Proper Minor-Closed Graph Classes","This paper establishes tight results on adjacency labelling schemes and induced-universal graphs for proper minor-closed classes. Every proper minor-closed class admits an adjacency labelling scheme using (1+o(1)) log2 n bits per vertex, and the dependence on the class appears only in the lower-order term. Equivalently, for each such class and each n, there exists an induced-universal graph U with n1+o(1) vertices for all n-vertex graphs in the class, and both statements are optimal up to lower-order terms.","arXiv :2605 .06616v3 [ cs .DM] 13 Jul 2026  \n\n| ADJACENCY LABELLING FOR PROPER MINOR-CLOSED GRAPH CLASSES\u003Cbr>Vida Dujmović Cyril GavoilleÊ Gwenaël JoretÄ Piotr MicekÅ Pat MorinÆ David R. WoodÇ |  |\n| --- | --- |\n| Abstract. We show that every proper minor-closed class of graphs admits a (1+o(1)) log2 nbit adjacency labelling scheme. Equivalently, for every proper minor-closed class G and every positive integer n there exists an n 1+o(1)-vertex graph U such that every n-vertex graph in G is isomorphic to an induced subgraph of U. Both results are optimal up to the lower order term. They generalize the corresponding results for planar graphs and apex-minorfree classes (Dujmović et al., J. ACM 2021) to all proper minor-closed classes, answering the open question raised in that paper and anticipated earlier by Bonamy, Gavoille, and Pilipczuk (SODA 2020) . |  |\n| 1 Introduction\u003Cbr>Let G be a class of graphs and let f : N → N be a function. We say that G admits an f(n)-bit adjacency labelling scheme if there exists a function A : ({0, 1}∗ )2 → {0, 1} such that for all positive integers n, for every n-vertex graph G ∈ G, there exists a function ℓ : V ( G) → {0, 1}∗ such that |ℓ(v)| ⩽ f(n) for each vertex v in G, and such that for every two vertices u, v in G, |  |\n| 􀀸􀀾 0 A (ℓ(u ), ℓ (v)) = \u003Cbr>􀀾􀀺 1 | if uv \u003C E( G),\u003Cbr>if uv ∈ E( G). |\n| A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. A class of graphs G is minor-closed if every minor of every graph in G is also in G, and it is proper if it is not the class of all graphs. Some examples of proper proper minor-closed include planar graphs, graphs of treewidth at |  |\n\n School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada ([vida. dujmovic@uottawa. ca](vida. dujmovic@uottawa. ca)). Research supported by NSERC and a University of Ottawa Research Chair.  \nÊ LaBRI, University of Bordeaux, France ([gavoille@labri. fr](gavoille@labri. fr)) . This work is supported by the French ANR Projets ENEDISC (ANR-24-CE48-7768-01) and TEMPOGRAL (ANR-22-CE48-0001) .  \nÄ Département d’Informatique, Université libre de Bruxelles, Belgium ([gwenael. joret@ulb. be](gwenael. joret@ulb. be)) . G. Joretis supported by the Belgian National Fund for Scientific Research (FNRS) and by the Australian Research Council.  \nÅ Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland ([piotr. micek@uj. edu. pl](piotr. micek@uj. edu. pl)). Research supported by the National Science Center of Poland under grant UMO-2023/05/Y/ST6/00079 within the WEAVE-UNISONO program.  \nÆ School of Computer Science, Carleton University, Ottawa, Canada (morin@scs. carleton. ca) . Research supported by NSERC.  \nÇ School of Mathematics, Monash University, Melbourne, Australia ([david. wood@monash. edu](david. wood@monash. edu)) . Research supported by the Australian Research Council and by NSERC.  \nmost k, graphs embeddable in a fixed surface, linklessly embeddable graphs, knotlessly embeddable graphs and, for every fixed graph X, the class of graphs with no X-minor. Note that every proper minor-closed class is contained in the class of Kt-minor-free graphs for some fixed t.  \nIn this paper we prove the following result. (All logarithms are in base 2.)  \nTheorem 1. Every proper minor-closed class of graphs admits a (1 + o(1)) log n-bit adjacency labelling scheme.  \nNote that all the dependence on the fixed proper minor-closed class in Theorem 1 is in the o(log n) term. Also, Theorem 1 is optimal up to the o(log n) term, which is O 􀀐(log n)3/4􀀑 . The proof of Theorem 1 is constructive. For every fixed proper minor-closed class G, thereis a polynomial-time algorithm that takes a graph G ∈ G as input and constructs the labelling ℓ : V ( G) → {0, 1}∗ .  \nA consequence 1 of Theorem 1 is the existence of induced-universal graphs with a nearlinear number of vertices for ","cbCail88y2BYlwII","https://ap.wps.com/l/cbCail88y2BYlwII","pdf",395259,5,1,49,"English","en",105,"# Abstract\n# Introduction\n## State of the Art\n# Theorem and Consequences","[{\"question\":\"What main result does the paper prove about adjacency labelling schemes for proper minor-closed graph classes?\",\"answer\":\"It proves that every proper minor-closed class admits a (1+o(1)) log2 n-bit adjacency labelling scheme, with the class-specific dependence appearing only in the lower-order term.\"},{\"question\":\"How is the adjacency labelling result connected to induced-universal graphs?\",\"answer\":\"The paper states an equivalence: for every proper minor-closed class and every positive integer n, there exists an induced-universal graph U with n1+o(1) vertices such that every n-vertex graph in the class appears as an induced subgraph of U.\"},{\"question\":\"Is the stated bound optimal, and what does “constructive” mean in this context?\",\"answer\":\"The bounds are optimal up to the lower-order term. The proof is constructive: for any fixed proper minor-closed class, a polynomial-time algorithm constructs the required vertex labelling from an input graph in the class.\"}]",1784211521,123,{"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},"adjacency-labelling-for-proper-minor-closed-graph-classes","",{"@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/adjacency-labelling-for-proper-minor-closed-graph-classes/86401/",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","2026-07-27","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 main result does the paper prove about adjacency labelling schemes for proper minor-closed graph classes?","Question",{"text":76,"@type":77},"It proves that every proper minor-closed class admits a (1+o(1)) log2 n-bit adjacency labelling scheme, with the class-specific dependence appearing only in the lower-order term.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the adjacency labelling result connected to induced-universal graphs?",{"text":81,"@type":77},"The paper states an equivalence: for every proper minor-closed class and every positive integer n, there exists an induced-universal graph U with n1+o(1) vertices such that every n-vertex graph in the class appears as an induced subgraph of U.",{"name":83,"@type":74,"acceptedAnswer":84},"Is the stated bound optimal, and what does “constructive” mean in this context?",{"text":85,"@type":77},"The bounds are optimal up to the lower-order term. The proof is constructive: for any fixed proper minor-closed class, a polynomial-time algorithm constructs the required vertex labelling from an input graph in the class.","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},[94,98,102,106,110,115,120,123,128,131,135],{"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":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"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":20,"slug":138},19,"General","general"]