Very Well-Covered Graphs of Girth at least Four and Local Maximum Stable Set Greedoids

A maximum stable set in a graph G is a stable set of maximum cardinality. 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 induced by S ∪N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [20], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [12] 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 [14], [15], [17], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥ 4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.