[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84592-en":3,"doc-seo-84592-105":29,"detail-sidebar-cat-0-en-105":83},{"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},84592,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Characterization and linear-time recognition of balanced distance-hereditary graphs","Balanced graphs are defined via clique-matrices: they contain no odd-order square submatrix whose every row and column has exactly two 1s. Although balancedness is known to be equivalent to forbidding induced extended odd suns, minimal forbidden induced subgraph characterizations remained open. This work proves that, for distance-hereditary graphs, balancedness is exactly hereditary clique-Helly graphs, equivalently characterized by the single forbidden induced subgraph 3K2. It further yields an explicit linear-time recognition algorithm that outputs an induced 3K2 witness when the graph is not balanced.","arXiv :2607 .00730v1 [math .CO] 1 Jul 2026  \nCharacterization and linear-time recognition of balanced distance-hereditary graphs  \nLuc´ıa Busolini∗ Guillermo Dur´an† Mart´ın D. Safe‡  \nAbstract  \nA graph is balanced if its clique-matrix contains no square submatrix of odd order with exactly two 1’s in each row and in each column. Although it is known that a graph is balanced if and only if it contains no induced extended odd sun, a characterization of balanced graphs by minimal forbidden induced subgraphs is still unknown. In this work, we prove that, within the class of distance-hereditary graphs, balanced graphs are exactly the hereditary cliqueHelly graphs. Equivalently, they are characterized by a single forbidden induced subgraph, namely 3K2 . From this result, we derive an explicit linear-time algorithm that decides balancedness within the class of distance-hereditary graphs and returns an induced 3K2 when the input graph is not balanced.  \n1 Introduction  \nLet G be a graph. Let Q 1 ,..., Qk be the maximal cliques and let v 1 , . . . , vn be the vertices of G. A clique-matrix of G is the matrix A = (aij) whose rows are indexed by Q 1 ,..., Qk, whose columns are indexed by v 1 , . . . , vn, and such that aij = 1 if vj ∈ Qi and aij = 0 otherwise. A graph G is balanced [6] if its clique-matrix contains no square submatrix of odd order with exactly two 1’s in each row and in each column. The name ‘balanced graphs’ was introduced by Berge and Chv´atal [6] . Berge and Las Vergnas [7] proved that balanced graphs are perfect and clique-perfect, and Berge [4] proved that they are also hereditary clique-Helly (see Section 2) .  \nSince balanced graphs are perfect, they have no odd holes and no odd antiholes, where an odd hole in a graph G is a chordless cycle of G whose length is odd and at least 5 and an odd antihole in G is an odd hole in the complement G. Moreover, Lehel and Tuza [26] showed that balanced graphs contain no induced odd suns. Finally, Bonomo et al. [8] characterized balanced graphs by means of extended odd suns, a family of graphs that generalizes odd holes and odd suns.  \nTheorem 1.1 ([8]) . A graph is balanced if and only if it contains no induced extended odd sun.  \nHowever, this characterization is not by minimal forbidden induced subgraphs. Indeed, some extended odd suns contain other extended odd suns as proper induced subgraphs (see Figure 1) . In fact, the characterization of balanced graphs by minimal forbidden induced subgraphs remains unknown, but some partial results are known [12] .  \n∗ Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina and Instituto de C´alculo, CONICET-Universidad de Buenos Aires, Buenos Aires, Argentina. E-mail: [lbusolini@dm.uba.ar](lbusolini@dm.uba.ar)  \n†Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina and Instituto de C´alculo, CONICET-Universidad de Buenos Aires, Buenos Aires, Argentina. E-mail: [gduran@dm.uba.ar](gduran@dm.uba.ar)  \n‡Departamento de Matem´atica, Universidad Nacional del Sur (UNS), Bah´ıa Blanca, Argentina and INMABB, Universidad Nacional del Sur (UNS)-CONICET, Bah´ıa Blanca, Argentina. E-mail: [msafe@uns.edu.ar](msafe@uns.edu.ar)  \nFigure 1: An extended odd sun that is not a minimal induced subgraph for the class of balanced graphs. Bold lines are the edges of a proper induced extended odd sun.  \nFigure 2: The graph 3K2 .  \nBalanced graphs were characterized by minimal forbidden induced subgraphs when restricted to diamond-free graphs [1], some subclasses of circular-arc graphs [10], paw-free graphs [11], P4-tidy graphs [11], claw-free graphs [13], complements of bipartite graphs, line graphs of multigraphs, and complements of line graphs of multigraphs [9] .  \nBalanced graphs can be recognized in polynomial time by applying Zambelli’s recognition algorithm for balanced matrices [30] . More precisely, as observed ","cbCaivXqFFUfMXyz","https://ap.wps.com/l/cbCaivXqFFUfMXyz","pdf",500109,1,25,"English","en",105,"# Introduction\n## Balanced graphs and clique-matrices\n## Forbidden induced subgraphs\n## Related recognition algorithms and known subclasses\n## Distance-hereditary graphs and main goal","[{\"question\":\"What does the proposed linear-time algorithm do?\",\"answer\":\"It decides balancedness for graphs in the distance-hereditary class in linear time and returns an induced 3K2 as a witness when the input graph is not balanced.\"}]",1784196980,63,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":27},"characterization-and-linear-time-recognition-of-balanced-distance-hereditary-graphs","",{"@graph":35,"@context":77},[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/characterization-and-linear-time-recognition-of-balanced-distance-hereditary-graphs/84592/",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],{"name":72,"@type":73,"acceptedAnswer":74},"What does the proposed linear-time algorithm do?","Question",{"text":75,"@type":76},"It decides balancedness for graphs in the distance-hereditary class in linear time and returns an induced 3K2 as a witness when the input graph is not balanced.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]