2014
DOI: 10.1017/s0963548314000303
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Asymptotic Properties of Some Minor-Closed Classes of Graphs

Abstract: Abstract. Let A be a minor-closed class of labelled graphs, and let Gn be a random graph sampled uniformly from the set of n-vertex graphs of A. When n is large, what is the probability that Gn is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected.Using exact enumeration, we study a collection of classes A excluding non-2-connected minors, and show that th… Show more

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Cited by 4 publications
(8 citation statements)
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“…This is clearly a minor-closed class that is decomposable, but not addable. It is in fact not smooth, but a similar property follows from Proposition 20 in [7].…”
Section: Again a Nearly Verbatim Repeat Of The Proof Of Corollary 21mentioning
confidence: 82%
See 3 more Smart Citations
“…This is clearly a minor-closed class that is decomposable, but not addable. It is in fact not smooth, but a similar property follows from Proposition 20 in [7].…”
Section: Again a Nearly Verbatim Repeat Of The Proof Of Corollary 21mentioning
confidence: 82%
“…caterpillars). As a direct corollary of their Propositions 23 and 26 we have: Bousquet-Mélou and Weller [7] also analysed the class of all graphs whose components have order at most t (fixed). This is clearly a minor-closed class that is decomposable, but not addable.…”
Section: Again a Nearly Verbatim Repeat Of The Proof Of Corollary 21mentioning
confidence: 90%
See 2 more Smart Citations
“…As far as we know, while these notions are part and parcel of the work of graph theorists, they are usually not exploited by analytic combinatorists. For forbidden minors, there is the penetrating article of Bousquet-Mélou and Weller [4]. For forbidden subgraphs or forbidden induced subgraphs, we know of few papers, except because of the simple nature of graphs [31], or because some other, alternate property is used instead [5], or only asymptotics are determined [32].…”
Section: Introductionmentioning
confidence: 99%