Well-covered Graphs and Greedoids

G is a well-covered graph provided all its maximal stable sets are of the same size (Plummer, 1970). S is a local maximum stable set of G, and we denote by S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. In 2002 we have proved that Ψ(G) is a greedoid for every forest G. The bipartite graphs and the trianglefree graphs, whose families of local maximum stable sets form greedoids were characterized by Levit and Mandrescu (2003, 2007a). In this paper we demonstrate that if a graph G has a perfect matching consisting of only pendant edges, then Ψ(G) forms a greedoid on its vertex set. In particular, we infer that Ψ(G) forms a greedoid for every well-covered graph G of girth at least 6, nonisomorphic to C7.