The structure of a 3-connected matroid with a 3-separating set of essential elements

An element e of a 3–connected matroid M is essential if neither the deletion nor the contraction of e from M is 3–connected. Tutte’s 1966 Wheels and Whirls Theorem proves that the only 3–connected matroids in which every element is essential are the wheels and whirls. It was proved by Oxley and Wu that if a 3–connected matroid M has a non-essential element, then it has at least two such elements. Moreover, the set of essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both. In addition, if M has a fan with 2k or 2k + 1 elements for some k ≥ 2, then M can be obtained by sticking together a (k + 1)–spoked wheel and a certain 3–connected minor of M . In this paper, it is shown how a slight modification of these ideas can be used to describe the structure of a 3–connected matroid M having a 3– separation (A, B) such that every element of A is essential. The motivation for this study derives from a desire to determine when one can remove an element from M so as to both maintain 3–connectedness and preserve one side of the 3–separation.