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) | [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) is the size of a maximum matching and 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), if S is a maximum stable set of the subgraph spanned by S ∪ N (S), where 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) 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) 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) is a greedoid if and only if all its maximum matchings are uniquely restricted and for any S ∈ Ψ (G), the subgraph spanned by S ∪ N (S) is a König-Egerváry graph.