Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

A maximum stable set in a graph G is a stable set of maximum cardinality. 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. [27], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [15] we have shown that the family Ψ(T ) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [17], [19], [21], 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.