{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T11:07:14Z","timestamp":1780744034044,"version":"3.54.1"},"reference-count":19,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2010,3,1]],"date-time":"2010-03-01T00:00:00Z","timestamp":1267401600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2010,3]]},"abstract":"\n Let\n C<\/jats:italic>\n be a class of labeled connected graphs, and let C\n \n n<\/jats:italic>\n <\/jats:sub>\n be a graph drawn uniformly at random from graphs in\n C<\/jats:italic>\n that contain exactly\n n<\/jats:italic>\n vertices. Denote by\n b<\/jats:italic>\n (\u2113; C\n \n n<\/jats:italic>\n <\/jats:sub>\n ) the number of blocks (i.e., maximal biconnected subgraphs) of C\n \n n<\/jats:italic>\n <\/jats:sub>\n that contain exactly \u2113 vertices, and let\n lb<\/jats:italic>\n (C\n \n n<\/jats:italic>\n <\/jats:sub>\n ) be the number of vertices in a largest block of C\n \n n<\/jats:italic>\n <\/jats:sub>\n . We show that under certain general assumptions on\n C<\/jats:italic>\n , C\n \n n<\/jats:italic>\n <\/jats:sub>\n belongs with high probability to one of the following categories:\n <\/jats:p>\n \n (1)\n lb<\/jats:italic>\n (C\n \n n<\/jats:italic>\n <\/jats:sub>\n ) \u223c\n cn<\/jats:italic>\n , for some explicitly given\n c<\/jats:italic>\n =\n c<\/jats:italic>\n (\n C<\/jats:italic>\n ), and the second largest block is of order\n n<\/jats:italic>\n \u03b1<\/jats:sup>\n , where 1 > \u03b1 = \u03b1(\n C<\/jats:italic>\n ), or\n <\/jats:p>\n \n (2)\n lb<\/jats:italic>\n (C\n \n n<\/jats:italic>\n <\/jats:sub>\n ) =\n O<\/jats:italic>\n (log\n n<\/jats:italic>\n ), that is, all blocks contain at most logarithmically many vertices.\n <\/jats:p>\n \n Moreover, in both cases we show that the quantity\n b<\/jats:italic>\n (\u2113; C\n \n n<\/jats:italic>\n <\/jats:sub>\n ) is concentrated for all \u2113 and we determine its expected value. As a corollary we obtain that the class of planar graphs belongs to category (1). In contrast to that, outerplanar and series-parallel graphs belong to category (2).\n <\/jats:p>","DOI":"10.1145\/1721837.1721847","type":"journal-article","created":{"date-parts":[[2010,4,7]],"date-time":"2010-04-07T02:56:32Z","timestamp":1270608992000},"page":"1-21","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":15,"title":["Maximal biconnected subgraphs of random planar graphs"],"prefix":"10.1145","volume":"6","author":[{"given":"Konstantinos","family":"Panagiotou","sequence":"first","affiliation":[{"name":"Max-Planck-Institute for Informatics, Saarbr\u00fccken, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Angelika","family":"Steger","sequence":"additional","affiliation":[{"name":"ETH Zurich, Z\u00fcrich, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2010,4,6]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10021"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.37236\/1659"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240070402"},{"key":"e_1_2_1_4_1","volume-title":"Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'08)","author":"Bernasconi N."},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-85363-3_25"},{"key":"e_1_2_1_6_1","volume-title":"DMTCS Proceedings (EuroComb'05)","volume":"388","author":"Bodirsky M."},{"key":"e_1_2_1_7_1","first-page":"61","article-title":"The random planar graph","volume":"113","author":"Denise A.","year":"1996","journal-title":"Congressus Numerantium"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548304006315"},{"key":"e_1_2_1_9_1","first-page":"17","article-title":"On the evolution of random graphs. 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Discrete Mathematics and Theoritical Computer Science","volume":"156","author":"Gim\u00e9nez O."},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.endm.2007.07.080"},{"key":"e_1_2_1_16_1","doi-asserted-by":"crossref","unstructured":"Harary F. and Palmer E. 1973. Graphical Enumeration. Academic Press New York. Harary F. and Palmer E. 1973. Graphical Enumeration. 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