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 2 ^a(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 ^a(G) is a greedoid for every forest G. The bipartite graphs and the triangle-free 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 ^a(G) forms a greedoid on its vertex set. In particular, we infer that ^a(G) forms a greedoid for every well-covered graph G of girth at least 6, non-isomorphic to C ₇.