  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 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 its
family of local maximum stable sets coinsides with (V,F). It has been shown
that the family local maximum stable sets of a forest T forms a greedoid on its
vertex set. In this paper we demonstrate that if G is a very well-covered
graph, then its family of local maximum stable sets is a greedoid if and only
if G has a unique perfect matching.
