Capturing matroid elements in unavoidable 3-connected minors

A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n\u3e2, there is an integer f(n) so that if {pipe}E(M){pipe}\u3ef(n), then M has a minor isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K 3,n, or U 2,n or U n-2,n. In this paper, we build on this result to determine what can be said about a large structure using a specified element e of M. In particular, we prove that, for every integer n exceeding two, there is an integer g(n) so that if {pipe}E(M){pipe}\u3eg(n), then e is an element of a minor of M isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K 1,1,1,n, a specific single-element extension of M(K 3,n) or the dual of this extension, or U 2,n or U n-2,n. © 2012 Elsevier Ltd