On nonbinary 3-connected matroids

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It is well known that a matroid is binary if and only if it has no minor isomorphic to U2,4, the 4-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a U2,4- minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary 3-connected matroid is in a U2,4-niinor. This paper extends Seymour’s theorem by proving that if {x, y, 2} is contained in a nonbinary 3-connected matroid M, then either M has a U2,4−minor using {x, y, z}, or M has a minor isomorphic to the rank-3 whirl that uses {x, y, z} as its rim or its spokes. © 1987 American Mathematical Society.

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Transactions of the American Mathematical Society

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