{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9066","name":"Local Convergence and Stability of Tight Bridge-Addable Graph Classes","abstract":"A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if G is bridge-addable and G_n is a uniform n-vertex graph from G, then G_n is connected with probability at least (1+o(1))e^{-1\/2}. The constant e^{-1\/2} is best possible since it is reached for the class of forests.\r\n\r\nIn this paper we prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph G_n is connected with probability close to e^{-1\/2}, then G_n is asymptotically close to a uniform forest in some \"local\" sense. For example, if the probability converges to e^{-1\/2}, then G_n converges for the Benjamini-Schramm topology, to the uniform infinite random forest F_infinity. This result is reminiscent of so-called \"stability results\" in extremal graph theory, with the difference that here the \"stable\" extremum is not a graph but a graph class.","keywords":["bridge-addable classes","random graphs","stability","local convergence","random forests"],"author":[{"@type":"Person","name":"Chapuy, Guillaume","givenName":"Guillaume","familyName":"Chapuy"},{"@type":"Person","name":"Perarnau, Guillem","givenName":"Guillem","familyName":"Perarnau"}],"position":26,"pageStart":"26:1","pageEnd":"26:11","dateCreated":"2016-09-06","datePublished":"2016-09-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Chapuy, Guillaume","givenName":"Guillaume","familyName":"Chapuy"},{"@type":"Person","name":"Perarnau, Guillem","givenName":"Guillem","familyName":"Perarnau"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2016.26","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6263","volumeNumber":60,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2016)","dateCreated":"2016-09-06","datePublished":"2016-09-06","editor":[{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Mathieu, Claire","givenName":"Claire","familyName":"Mathieu"},{"@type":"Person","name":"Rolim, Jos\u00e9 D. P.","givenName":"Jos\u00e9 D. P.","familyName":"Rolim"},{"@type":"Person","name":"Umans, Chris","givenName":"Chris","familyName":"Umans"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9066","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6263"}}}