Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings

Abstract

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. (Math. Programming 8(1975) 232–248), proved that any S∈Ψ( G) is a subset of a maximum stable set of G. In Levit and Mandrescu (Discrete Appl. Math., 124 (2002) 91–101) 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.