A Characterisation of Strict Matching Matroids

It is well known that a matroid is a transversal matroid if and only if it is a matching matroid (in the sense that it is the restriction of the matching structure of some graph to a subset of its vertices). A simple proof of that result is now known and in this paper it is used to answer the long-standing question of which transversal matroids are “strict” matching matroids; i.e. actually equal to the matching structure of a graph. We develop a straightforward test of “coloop-surfeit” that can be applied to any transversal matroid, and our main theorem shows that a transversal matroid is a strict matching matroid if and only if it has even rank and coloop-surfeit. Furthermore, the proofs are algorithmic and enable the construction of an appropriate graph from any presentation of a strict matching matroid.