Abstract

An edge e in a 3-connected graph G is contractible if the contraction G/ e is still 3-connected. The problem of bounding the number of contractible edges in a 3-connected graph has been studied by numerous authors. In this paper, the corresponding problem for matroids is considered and new graph results are obtained. An element e in a 3-connected matroid M is contractible or vertically contractible if its contraction M/ e is, respectively, 3-connected or vertically 3-connected. Cunningham and Seymour independently proved that every 3-connected matroid has a vertically contractible element. In this paper, we study the contractible and vertically contractible elements in 3-connected matroids and get best-possible lower bounds for the number of vertically contractible elements in 3-connected and minimally 3-connected matroids. We also prove generalizations of Tutte's Wheels and Whirls Theorem for matroids and Tutte's Wheels Theorem for graphs.

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