Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings
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??(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) 232248), proved that any S??(G) is a subset of a maximum stable set of G. In Levit and Mandrescu (Discrete Appl. Math., 124 (2002) 91101) we have shown that the family ?(T) of a forest T forms a greedoid on its vertex set. In this paper, we demonstrate that for a bipartite graph G, ?(G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted.
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