Dominating Circuits In Regular Matroids
Advances in Applied Mathematics(2014)
摘要
In 1971, Nash-Williams proved that if G is a simple 2-connected graph on n vertices having minimum degree at least 1/3 (n + 2), then any longest cycle C in G is also edge-dominating; that is, each edge of G has at least one end-vertex incident with a vertex of C. We say that a circuit C in a matroid M is dominating if each component of M/C has rank at most one. In this paper, we show that an analogous theorem holds for regular matroids. More specifically, suppose M is a simple, connected, regular matroid and let C be a circuit in M. We show that if vertical bar C*vertical bar > r(M)/3 + 1, for all cocircuits C* in M which are, disjoint from C, then either C is a dominating circuit, or there is a circuit D such that the symmetric difference of C with D is a circuit which is strictly larger than C. (C) 2013 Elsevier Inc. All rights reserved.
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关键词
Matroid,Binary matroid,Minor
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