Unavoidable minors of graphs of large type Original Research Article

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 Gi−1, and where Gn 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.