Unavoidable minors of graphs of large type

In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is deened to be the minimum number n such that there is a sequence of graphs G = G 0 , G 1 , : : : , G n , where G i is obtained by contracting one edge in or deleting one edge from each block of G i?1 , and where G n 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 speciied families of graphs.