Splicing Matroids
Abstract
We introduce and study a natural variant of matroid amalgams. For matroids M(A) and N(B) such that M/(A-B)=N(B-A), we define a splice of M and N to be a matroid L on the union of A and B with L(B-A)=M and L/(A-B)=N. We show that splices exist for each such pair of matroids M and N; furthermore, there is a freest splice of M and N, which we call the free splice. We characterize when a matroid L(E) is the free splice of L and L/V for subsets U and V of E. We study minors of free splices and the interaction between free splice and several other matroid operations. Although free splice is not an associative operation, we prove a weakened counterpart of associativity that holds in general and we characterize the triples for which associativity holds. We also study free splice as it relates to various classes of matroids.