Document Type
Article
Publication Date
8-1-2012
Abstract
A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n>2, there is an integer f(n) so that if {pipe}E(M){pipe}>f(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}>g(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.
Publication Source (Journal or Book title)
European Journal of Combinatorics
First Page
1100
Last Page
1112
Recommended Citation
Chun, D., Oxley, J., & Whittle, G. (2012). Capturing matroid elements in unavoidable 3-connected minors. European Journal of Combinatorics, 33 (6), 1100-1112. https://doi.org/10.1016/j.ejc.2012.01.012