Local maximum stable set greedoids stemming from very well-covered graphs

A maximum stable setin a graph G is a stable set of maximum cardinality. The set S is called a local maximum stable set of G, and we write S∈Ψ(G), if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that F=Ψ(G). Nemhauser and Trotter Jr. (1975) [28] proved that any S∈Ψ(G) is a subset of a maximum stable set of G. In Levit and Mandrescu (2002) [16] we showed that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, and well-covered while Ψ(G) is a greedoid were analyzed in Levit and Mandrescu (2004) [18], Levit and Mandrescu (2007) [20], and Levit and Mandrescu (2008) [23], respectively. In this paper we demonstrate that if G is a very well-covered graph, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.