Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids

A matching M is uniquely restricted in a graph G if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself [M.C. Golumbic, T. Hirst, M. Lewenstein, Uniquely restricted matchings, Algorithmica 31 (2001) 139–154]. G is a König–Egerváry graph   provided α(G)+μ(G)=|V(G)|α(G)+μ(G)=|V(G)| [R.W. Deming, Independence numbers of graphs—an extension of the König–Egerváry theorem, Discrete Math. 27 (1979) 23–33; F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, J. Combin. Theory Ser. B 27 (1979) 228–229], where μ(G)μ(G) is the size of a maximum matching and α(G)α(G) is the cardinality of a maximum stable set. S is a local maximum stable set of G  , and we write S∈Ψ(G)S∈Ψ(G), if S   is a maximum stable set of the subgraph spanned by S∪N(S)S∪N(S), where N(S)N(S) is the neighborhood of S  . Nemhauser and Trotter [Vertex packings: structural properties and algorithms, Math. Programming 8 (1975) 232–248], proved that any S∈Ψ(G)S∈Ψ(G) is a subset of a maximum stable set of G  . In [V.E. Levit, E. Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math. 132 (2003) 163–174] we have proved that for a bipartite graph G,Ψ(G)G,Ψ(G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted. In this paper we demonstrate that if G   is a triangle-free graph, then Ψ(G)Ψ(G) is a greedoid if and only if all its maximum matchings are uniquely restricted and for any S∈Ψ(G)S∈Ψ(G), the subgraph spanned by S∪N(S)S∪N(S) is a König–Egerváry graph.