On Nonbinary 3-connected Matroids
نویسنده
چکیده
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 U2,4minor. 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-minor. This paper extends Seymour's theorem by proving that if {x,y,z} 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.
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