Splicing Matroids

TL;DR

This paper introduces the concept of free splice in matroid theory, establishing its existence, properties, and interactions with other matroid operations, and characterizing when it is associative.

Contribution

It defines and analyzes the free splice operation in matroids, including existence, characterization, and its algebraic properties, expanding understanding of matroid amalgams.

Findings

Existence of free splices for certain matroid pairs

Characterization of when a matroid is a free splice

Weakened associativity property of free splice

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\U 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.