On the structure of triangle-free 3-connected matroids

Abstract

Let N be an arbitrary class of matroids closed under isomorphism. For a positive integer k, we say that M∈N is k-minor-irreducible if M has no minor N∈N such that 1≤|E(M)|−|E(N)|≤k. Tutte's Wheels and Whirls Theorem establish that, up to isomorphism, there are only two families of 1-minor-irreducible matroids in the class of 3-connected matroids. More recently, Lemos classified the 3-minor-irreducible matroids with at least 14 elements in the class of triangle-free 3-connected matroids. Here we prove a local characterization for the 2-minor-irreducible matroids with at least 11 elements in the class of triangle-free 3-connected matroids. This local characterization is used to establish two new families of 2-minor-irreducible matroids in this class.