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Random graphs containing few disjoint excluded minors

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Abstract

The Erdős-Pósa theorem (1965) states that in each graph G which contains at most k disjoint cycles, there is a ‘blocking’ set B of at most f(k) vertices such that the graph GB is acyclic. Robertson and Seymour (1986) give an extension concerning any minor-closed class math formula of graphs, as long as math formula does not contain all planar graphs: in each graph G which contains at most k disjoint excluded minors for math formula, there is a set B of at most g(k) vertices such that GB is in math formula.

In an earlier paper (Kurauskas and McDiarmid, Combin, Probab Comput 20 (2011) 763–775), we showed that, amongst all graphs on vertex set math formula which contain at most k disjoint cycles, all but an exponentially small proportion contain a blocking set of just k vertices.

In the present paper we build on the previous work, and give an extension concerning any minor-closed graph class math formula with 2-connected excluded minors, as long as math formula does not contain all fans (here a ‘fan’ is a graph consisting of a path together with a vertex joined to each vertex on the path). We show that amongst all graphs G on math formula which contain at most k disjoint excluded minors for math formula, all but an exponentially small proportion contain a set B of k vertices such that GB is in math formula. (This is not the case when math formula contains all fans.) For a random graph Rn sampled uniformly from the graphs on math formula with at most k disjoint excluded minors for math formula, we consider also vertex degrees and the uniqueness of small blockers, the clique number and chromatic number, and the probability of being connected. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 240-268, 2014

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