Bipartite graphs with uniquely restricted maximum matchings and their corresponding greedoids

A maximum stable set in a graph G is a stable set of maximum size. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph spanned by S∪N(S), where N(S) is the neighborhood of S. A matching M is uniquely restricted if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself. Nemhauser and Trotter Jr. [12], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [10] we have shown that the family Ψ(T ) of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that for a bipartite graph G,Ψ(G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted.