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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. © 2002 Elsevier Science B.V.

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Discrete Mathematics

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