Generalized Delta-Y exchange and k-regular matroids

This paper introduces a generalization of the matroid operation of Delta-Y exchange. This new operation, segment-cosegment exchange, replaces a coin-d ependent 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 th eorem of the first half of the paper is that, for every field, or indeed pa rtial field, F, the class of matroids representable over F is closed under segment-cosegment exchanges. It follows that, for all prime powers q, the s et of excluded miners for GF(q)-representability has at least 2(q-4) member s. In the second half of the paper, the operation of segment-cosegment exch ange is shown to Flay a fundamental role in an excluded-minor result for k- regular matroids, where such matroids generalize regular matroids and Whitt le's near-regular matroids. (C) 2000 Academic Press.