{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:18:41Z","timestamp":1759335521607},"reference-count":31,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2009,7,1]],"date-time":"2009-07-01T00:00:00Z","timestamp":1246406400000},"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":[[2009,7]]},"abstract":"<jats:p>A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a 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=\"S0963548309009717_char1\" \/><\/jats:private-char>of labelled graphs has smooth growth; and, for the random graph<jats:italic>R<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>sampled uniformly from the<jats:italic>n<\/jats:italic>-vertex graphs in<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548309009717_char1\" \/><\/jats:private-char>, the fragment not in the giant component asymptotically has a simple \u2018Boltzmann Poisson distribution\u2019. In particular, as<jats:italic>n<\/jats:italic>\u2192 \u221e the probability that<jats:italic>R<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>is connected tends to 1\/<jats:italic>A<\/jats:italic>(\u03c1), where<jats:italic>A<\/jats:italic>(<jats:italic>x<\/jats:italic>) is the exponential generating function for<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548309009717_char1\" \/><\/jats:private-char>and \u03c1 is its radius of convergence.<\/jats:p>","DOI":"10.1017\/s0963548309009717","type":"journal-article","created":{"date-parts":[[2009,2,19]],"date-time":"2009-02-19T11:18:01Z","timestamp":1235042281000},"page":"583-599","source":"Crossref","is-referenced-by-count":24,"title":["Random Graphs from a Minor-Closed Class"],"prefix":"10.1017","volume":"18","author":[{"given":"COLIN","family":"McDIARMID","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,7,1]]},"reference":[{"key":"S0963548309009717_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2007.04.011"},{"key":"S0963548309009717_ref25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2004.09.007"},{"key":"S0963548309009717_ref18","unstructured":"[18] Kurauskas V. and McDiarmid C. (2008) Random graphs without disjoint cycles. In preparation."},{"key":"S0963548309009717_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2007.09.003"},{"key":"S0963548309009717_ref7","doi-asserted-by":"crossref","unstructured":"[7] Bodirsky M. , Gim\u00e9nez O. , Kang M. and Noy M. (2005) On the number of series-parallel and outerplanar graphs. In Proc. European Conference on Combinatorics, Graph Theory, and Applications (EuroComb 2005), Discrete Math. Theor. Comput. Sci. Proc., Vol. AE, pp. 383\u2013388.","DOI":"10.46298\/dmtcs.3451"},{"key":"S0963548309009717_ref27","unstructured":"[27] Moon J. W. (1970) Counting Labelled Trees, Vol. 1 of Canadian Mathematical Monographs."},{"key":"S0963548309009717_ref30","doi-asserted-by":"publisher","DOI":"10.1007\/BF01202354"},{"key":"S0963548309009717_ref17","volume-title":"Random Graphs","author":"Kolchin","year":"1999"},{"key":"S0963548309009717_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(87)90119-1"},{"key":"S0963548309009717_ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2006.01.006"},{"key":"S0963548309009717_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718"},{"key":"S0963548309009717_ref21","unstructured":"[21] McDiarmid C. (2008) Minor-closed unlabelled graph classes. In preparation."},{"key":"S0963548309009717_ref4","doi-asserted-by":"crossref","first-page":"#21","DOI":"10.37236\/745","article-title":"Coefficients of functional compositions often grow smoothly","volume":"15","author":"Bender","year":"2008","journal-title":"Electron. J. Combin."},{"key":"S0963548309009717_ref1","unstructured":"[1] Addario-Berry L. , McDiarmid C. and Reed B. (2007) Connectivity of addable monotone graph classes. Manuscript."},{"key":"S0963548309009717_ref6","unstructured":"[6] Bernardi O. , Noy M. and Welsh D. (2007) On the growth rate of minor-closed classes of graphs. Manuscript: arXiv:0710.2995."},{"key":"S0963548309009717_ref13","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-08-00624-3"},{"key":"S0963548309009717_ref24","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548308009097"},{"key":"S0963548309009717_ref16","volume-title":"Random Mappings","author":"Kolchin","year":"1986"},{"key":"S0963548309009717_ref29","first-page":"73","article-title":"Some remarks on the theory of trees","volume":"4","author":"R\u00e9nyi","year":"1959","journal-title":"Publications of the Mathematical Institute of the Hungarian Academy of Sciences"},{"key":"S0963548309009717_ref3","doi-asserted-by":"crossref","first-page":"#33","DOI":"10.37236\/1511","article-title":"Asymptotics for the probability of connectedness and the distribution of number of components","volume":"7","author":"Bell","year":"2000","journal-title":"Electron. J. Combin."},{"key":"S0963548309009717_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-33700-8_15"},{"key":"S0963548309009717_ref11","volume-title":"Graph Theory","author":"Diestel","year":"2005"},{"key":"S0963548309009717_ref5","doi-asserted-by":"crossref","first-page":"#43","DOI":"10.37236\/1659","article-title":"The number of labeled 2-connected planar graphs","volume":"9","author":"Bender","year":"2002","journal-title":"Electron. J. Combin."},{"key":"S0963548309009717_ref14","doi-asserted-by":"crossref","unstructured":"[14] Gim\u00e9nez O. , Noy M. and Ru\u00e9 J. (2007) Graph classes with given 3-connected components: Asymptotic counting and critical phenomena. Manuscript.","DOI":"10.1016\/j.endm.2007.07.080"},{"key":"S0963548309009717_ref23","unstructured":"[23] McDiarmid C. (2008) A combinatorial proof of smoothness. In preparation."},{"key":"S0963548309009717_ref22","unstructured":"[22] McDiarmid C. (2008) Forests of paths and caterpillars. In preparation."},{"key":"S0963548309009717_ref9","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20149"},{"key":"S0963548309009717_ref12","doi-asserted-by":"crossref","first-page":"R114","DOI":"10.37236\/838","article-title":"The number of graphs not containing K3,3 as a minor","volume":"15","author":"Gerke","year":"2008","journal-title":"Electron. J. Combin."},{"key":"S0963548309009717_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2007.11.006"},{"key":"S0963548309009717_ref31","unstructured":"[31] Robertson N. and Seymour P. D. (1983\u20132004) Graph minors I\u2013XX. J. Combin. Theory Ser. B."},{"key":"S0963548309009717_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01350657"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548309009717","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T02:55:26Z","timestamp":1684896926000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548309009717\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,7]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,7]]}},"alternative-id":["S0963548309009717"],"URL":"https:\/\/doi.org\/10.1017\/s0963548309009717","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,7]]}}}