Triangles in 3-connected matroids

Abstract

AbstractA collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S8 or the generalized parallel connection across T of F7 and M(K4)