Capturing matroid elements in unavoidable 3-connected minors

  • Authors:

  • Deborah Chun;James Oxley;Geoff Whittle

  • Affiliations:

  • Mathematics Department, Louisiana State University, Baton Rouge, LA, United States;Mathematics Department, Louisiana State University, Baton Rouge, LA, United States;School of Mathematics, Statistics, and Operations Research, Victoria University of Wellington, Wellington, New Zealand

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2012

Quantified Score

Hi-index 0.00

Visualization

Abstract

A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n2, there is an integer f(n) so that if |E(M)|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 |E(M)|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.