Some Local Extremal Connectivity Results for Matroids

Abstract

Tutte proved that if e is an element of a 3-connected matroid M such that neither M\e nor M/e is 3-connected, then e is in a 3-circuit or a 3-cocircuit. In this paper, we prove a broad generalization of this result. Among the consequences of this generalization are that if X is an (n − 1)-element subset of an n-connected matroid M such that neither M\X nor M/X is connected, then, provided |E(M)| ≥ 2(n − 1)≥ 4, X is in both an n-element circuit and an n-element cocircuit. When n = 3, we describe the structure of M more closely using Δ − Y exchanges. Several related results are proved and we also show that, for all fields F other than GF(2), the set of excluded minors for F-representability is closed under both Δ − Y and Y − Δ exchanges.