Generalized Δ–Y Exchange and k-Regular Matroids

This paper introduces a generalization of the matroid operation of Δ–Y exchange. This new operation, segment–cosegment exchange, replaces a coindependent set of k collinear points in a matroid by an independent set of k points that are collinear in the dual of the resulting matroid. The main theorem of the first half of the paper is that, for every field, or indeed partial field, F, the class of matroids representable over F is closed under segment–cosegment exchanges. It follows that, for all prime powers q, the set of excluded minors for GF(q)-representability has at least 2q−4 members. In the second half of the paper, the operation of segment–cosegment exchange is shown to play a fundamental role in an excluded-minor result for k-regular matroids, where such matroids generalize regular matroids and Whittleʹs near-regular matroids.