A hypergraph version of the Gallai-Edmonds Theorem

A matching-cover of a hypergraph H is a matching F such that V(F)=V(H) and every edge of F is contained in an edge of H. This generalises the notion of a matching in a graph. We use this concept to extend the Gallai-Edmonds Theorem to hypergraphs, and deduce a number of interesting corollaries.