Abstract
In this paper we prove the following theorem: Let M be a 3-connected matroid other than the cycle matroid of a wheel of rank greater than three. Let C be a circuit of M. If every deletion of a pair of elements of C disconnects M, then every pair of elements of C is in a triad. A result of Oxley and the author, which characterizes the duals of Sylvester matroids, is an immediate consequence of this theorem. It also follows that if such a matroid M is graphic, then C is a 3-edge cycle.
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