{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:20:05Z","timestamp":1759335605701},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2011,1,24]],"date-time":"2011-01-24T00:00:00Z","timestamp":1295827200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2011,5]]},"abstract":"<jats:p>The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.<\/jats:p><jats:p>In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we call<jats:italic>cores<\/jats:italic>. In fact, we prove a general theorem regarding random biconnected graphs from various classes. If<jats:monospace>B<\/jats:monospace><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>is a graph drawn uniformly at random from a suitable class<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548310000532_char1\" \/><\/jats:private-char>of labelled biconnected graphs, then we show that with probability 1 \u2212<jats:italic>o<\/jats:italic>(1) as<jats:italic>n<\/jats:italic>\u2192 \u221e,<jats:monospace>B<\/jats:monospace><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>belongs to exactly one of the following categories:<jats:list list-type=\"number\"><jats:list-item><jats:label>(i)<\/jats:label><jats:p>either there is a unique<jats:italic>giant<\/jats:italic>core in<jats:monospace>B<\/jats:monospace><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, that is, there is a 0 &lt;<jats:italic>c<\/jats:italic>=<jats:italic>c<\/jats:italic>(<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548310000532_char1\" \/><\/jats:private-char>) &lt; 1 such that the largest core contains ~<jats:italic>cn<\/jats:italic>vertices, and every other core contains at most<jats:italic>n<\/jats:italic><jats:sup>\u03b1<\/jats:sup>vertices, where 0 &lt; \u03b1 = \u03b1(<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548310000532_char1\" \/><\/jats:private-char>) &lt; 1;<\/jats:p><\/jats:list-item><jats:list-item><jats:label>(ii)<\/jats:label><jats:p>or all cores of<jats:monospace>B<\/jats:monospace><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>contain<jats:italic>O<\/jats:italic>(log<jats:italic>n<\/jats:italic>) vertices.<\/jats:p><\/jats:list-item><\/jats:list>Moreover, we find the critical condition that determines the category to which<jats:monospace>B<\/jats:monospace><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>belongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 and<jats:italic>n<\/jats:italic>. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particular<jats:italic>c<\/jats:italic>= 0.765.\u00a0.\u00a0. and \u03b1 = 2\/3.<\/jats:p>","DOI":"10.1017\/s0963548310000532","type":"journal-article","created":{"date-parts":[[2011,1,24]],"date-time":"2011-01-24T13:03:31Z","timestamp":1295874211000},"page":"381-412","source":"Crossref","is-referenced-by-count":7,"title":["3-Connected Cores In Random Planar Graphs"],"prefix":"10.1017","volume":"20","author":[{"given":"NIKOLAOS","family":"FOUNTOULAKIS","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"KONSTANTINOS","family":"PANAGIOTOU","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2011,1,24]]},"reference":[{"key":"S0963548310000532_ref13","doi-asserted-by":"crossref","unstructured":"[13] Panagiotou K. and Steger A. (2009) Maximal biconnected subgraphs of random planar graphs. In Proc. 20th Annual ACM\u2013SIAM Symposium on Discrete Algorithms (SODA '09), pp. 432\u2013440.","DOI":"10.1137\/1.9781611973068.48"},{"key":"S0963548310000532_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548304006315"},{"key":"S0963548310000532_ref3","doi-asserted-by":"crossref","unstructured":"[3] Bodirsky M. , Gim\u00e9nez O. , Kang M. and Noy M. (2005) On the number of series parallel and outerplanar graphs. In 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Vol. AE of DMTCS Proceedings, pp. 383\u2013388.","DOI":"10.46298\/dmtcs.3451"},{"key":"S0963548310000532_ref2","unstructured":"[2] Bernasconi N. , Panagiotou K. and Steger A. (2008) On properties of random dissections and triangulations. In Proc. 19th Annual ACM\u2013SIAM Symposium on Discrete Algorithms (SODA '08), pp. 132\u2013141."},{"key":"S0963548310000532_ref10","unstructured":"[10] Gim\u00e9nez O. , Noy M. and Ru\u00e9 J. (2009) Graph classes with given 3-connected components: asymptotic enumeration and random graphs. Manuscript, available at: http:\/\/arxiv.org\/abs\/0907.0376."},{"key":"S0963548310000532_ref1","doi-asserted-by":"crossref","DOI":"10.37236\/1659","article-title":"The number of 2-connected planar graphs","volume":"9","author":"Bender","year":"2002","journal-title":"Electron. J. Combin."},{"key":"S0963548310000532_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-211-75357-6"},{"key":"S0963548310000532_ref15","doi-asserted-by":"crossref","DOI":"10.3138\/9781487584863","volume-title":"Connectivity in Graphs","author":"Tutte","year":"1966"},{"key":"S0963548310000532_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2004.09.007"},{"key":"S0963548310000532_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(82)90072-7"},{"key":"S0963548310000532_ref14","first-page":"226","article-title":"Towards a theory of non-repeating contact schemes.","volume":"51","author":"Trakhtenbrot","year":"1958","journal-title":"Trudi Mat. Inst. Akad. Nauk SSSR"},{"key":"S0963548310000532_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801655"},{"key":"S0963548310000532_ref4","first-page":"61","article-title":"The random planar graph.","volume":"113","author":"Denise","year":"1996","journal-title":"Congress. 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