On the Structure of 3-connected Matroids and Graphs

An element e of a 3 -connected matroid M is essential if neither the deletionM \ e nor the contraction M / e is 3 -connected. Tutte’s Wheels and Whirls Theorem proves that the only 3 -connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3 -connected matroids that have some non-essential elements, showing that every such matroid M must have at least two such elements. We prove that the 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. We also prove that if an essential element e of M is in more than one fan, then that fan has three or five elements; in the latter case, e is in exactly three fans. Moreover, we show that if M has a fan with 2 k or 2 k + 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. The results proved here will be used elsewhere to completely determine all 3 -connected matroids with exactly two non-essential elements.