Capturing matroid elements in unavoidable 3-connected minors

A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid MM has unavoidable structure. For every n>2n>2, there is an integer f(n)f(n) so that if |E(M)|>f(n)|E(M)|>f(n), then MM has a minor isomorphic to the rank-nn wheel or whirl, a rank-nn spike, the cycle or bond matroid of K3,nK3,n, or U2,nU2,n or Un−2,nUn−2,n. In this paper, we build on this result to determine what can be said about a large structure using a specified element ee of MM. In particular, we prove that, for every integer nn exceeding two, there is an integer g(n)g(n) so that if |E(M)|>g(n)|E(M)|>g(n), then ee is an element of a minor of MM isomorphic to the rank-nn wheel or whirl, a rank-nn spike, the cycle or bond matroid of K1,1,1,nK1,1,1,n, a specific single-element extension of M(K3,n)M(K3,n) or the dual of this extension, or U2,nU2,n or Un−2,nUn−2,n.

► Ding et al. (1997) showed that every sufficiently large 3-connected matroid has one of seven highly structured minors. ► Every element of such a matroid can be captured in one of nine highly structured minors. ► These nine minors are modifications of the original seven.