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In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is defined to be the minimum number n such that there is a sequence of graphs G = G0, G1,...,Gn, where Gi is obtained by contracting one edge in or deleting one edge from each block of G/_i, and where G is edgeless. We show that a 3-connected graph has large type if and only if it has a minor isomorphic to a large fan. Furthermore, we show that if a graph has large type, then it has a minor isomorphic to a large fan or to a large member of one of two specified families of graphs. © 2002 Elsevier Science B.V. All rights reserved.

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Discrete Mathematics

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