[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85154-en":3,"doc-seo-85154-105":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},85154,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","A Note on Diameter Certification in Trees","Local certification verifies global network properties using only bounded-radius communication and per-vertex certificates. For general graphs, certifying that the diameter is at most d requires large certificates, while trees allow more efficient verification. This note provides a 1-local certification scheme for trees: it certifies whether a given tree’s diameter is at most d using certificates of at most 3⌈log2(d+1)⌉ bits. The work frames the result within proof-labeling schemes and MSO-logic meta-theorems.","arXiv :2607 .09929v1 [ cs .DC] 10 Jul 2026  \nA Note on Diameter Certification in Trees  \nJosef Erik Sedláček \\# 􀀚  \nFaculty of Information Technology, CTU in Prague, Prague, Czech Republic  \n~~ Abstract ~~  \nIn the local certification model, certifying the diameter of general graphs requires large certificates, but trees admit more efficient solutions. In this note, we provide a 1-local certification scheme that certifies whether the diameter of a given tree is at most d using certificates of at most 3⌈log2 (d + 1)⌉ bits.  \n2012 ACM Subject Classification Theory of computation → Distributed algorithms; Theory of computation → Graph algorithms analysis  \nKeywords and phrases local certification, locally checkable proofs, proof-labeling schemes, graph diameter, trees  \nFunding This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS23/205/OHK3/3T/18 .  \n 1  Introduction  \nLocal certification and network structure  \nThis note focuses on local certification, a framework in distributed computing used to verify global properties of a network. In this context, the network topology is naturally modeled asa graph G = (V, E), where vertices represent the computing nodes and edges represent the communication links. We are interested in checking global graph properties such as acyclicity, planarity and bounded diameter.  \nSince vertices typically only have a local view of their immediate neighborhood, most global properties cannot be verified without external assistance. For example, a vertex cannot determine if the entire graph is bipartite just by communicating with its direct neighbors. To overcome this limitation, the local certification model [4] has been introduced. In this model, a centralized oracle (the prover) assigns a label, called a certificate, to each vertex. The vertices then communicate with their neighbors up to a constant radius and decide whether to accept or reject the configuration. Returning to the bipartiteness example, theprover could simply assign a color {1, 2} to each vertex as its certificate. A vertex accepts if all of its neighbors have a certificate of a different color from its own.  \nA certification scheme is correct if there exists a certificate assignment that makes all vertices accept when the property holds, and for any assignment, at least one vertex rejects when the property does not hold. The primary measure of efficiency in this setting is the certificate size. Specifically, we consider the maximum size of a certificate over all vertices in the graph, measured in bits, and the objective is to minimize this maximum size. The threshold of Θ(log n) bits, where n is the number of vertices, has emerged as the standard baseline for compact local certification [7] . This size is significant because it enables the certification of structures like spanning trees.  \nThis general verification mechanism can be formalized through various models, most notably proof-labeling schemes [8] and locally checkable proofs [7] . While these settings typically assume that each vertex is equipped with a unique piece of information called an identifier of size O (log n) bits, in this note we operate in the more restrictive anonymous model, meaning that vertices do not possess any identifiers. Note that since our goal is to establish an upper bound, presenting a scheme for a model without identifiers yields a  \n2 A Note on Diameter Certification in Trees  \nstronger result, as the correctness and size bounds immediately carry over to settings where identifiers are available. For a comprehensive overview of these different certification models and their variants, we refer the reader to [4] .  \nIn this note, our main objective is to certify that the diameter of a given tree is at most a constant d. The classical approach to local certification often considered general graphs. It has been proven th˜at certifying that the diameter of a general graph is at most d requires  \ncertificat","cbCairCzcKQTgtsm","https://ap.wps.com/l/cbCairCzcKQTgtsm","pdf",351647,4,1,5,"English","en",105,"# Introduction\n## Local certification and network structure\n## Diameter certification on trees\n## Meta-theorems and MSO logic\n## Motivation for the result","[{\"question\":\"What problem does the note address?\",\"answer\":\"It addresses how to locally certify that the diameter of a given tree is at most a fixed value d, using small per-vertex certificates.\"},{\"question\":\"How does the proposed scheme work and what is its locality?\",\"answer\":\"It presents a 1-local certification scheme where each vertex decides based on information from its immediate neighborhood and the certificates assigned to involved vertices.\"},{\"question\":\"What certificate size bound is achieved for trees?\",\"answer\":\"The scheme uses certificates of at most 3⌈log2(d+1)⌉ bits per vertex to verify whether the tree’s diameter is at most d.\"}]",1784201437,13,{"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},"a-note-on-diameter-certification-in-trees","",{"@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/a-note-on-diameter-certification-in-trees/85154/",{"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","2026-07-23","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 problem does the note address?","Question",{"text":75,"@type":76},"It addresses how to locally certify that the diameter of a given tree is at most a fixed value d, using small per-vertex certificates.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed scheme work and what is its locality?",{"text":80,"@type":76},"It presents a 1-local certification scheme where each vertex decides based on information from its immediate neighborhood and the certificates assigned to involved vertices.",{"name":82,"@type":73,"acceptedAnswer":83},"What certificate size bound is achieved for trees?",{"text":84,"@type":76},"The scheme uses certificates of at most 3⌈log2(d+1)⌉ bits per vertex to verify whether the tree’s diameter is at most d.","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":22,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Comic",60,"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":22,"slug":137},19,"General","general"]