Critical sets, crowns and local maximum independent sets

A set $$S\subseteq V(G)$$ S ⊆ V ( G ) is independent if no two vertices from S are adjacent, and by $$\mathrm {Ind}(G)$$ Ind ( G ) we mean the set of all independent sets of G. A set $$A\in \mathrm {Ind}(G)$$ A ∈ Ind ( G ) is critical (and we write $$A\in CritIndep(G)$$ A ∈ C r i t I n d e p ( G ) ) if $$\left| A\right| -\left| N(A)\right| =\max \{\left| I\right| -\left| N(I)\right| :I\in \mathrm {Ind}(G)\}$$ A - N ( A ) = max { I - N ( I ) : I ∈ Ind ( G ) } [37], where N(I) denotes the neighborhood of I. If $$S\in \mathrm {Ind}(G)$$ S ∈ Ind ( G ) and there is a matching from N(S) into S, then S is a crown [1], and we write $$S\in Crown(G)$$ S ∈ C r o w n ( G ) . Let $$\Psi (G)$$ Ψ ( G ) be the family of all local maximum independent sets of graph G, i.e., $$S\in \Psi (G)$$ S ∈ Ψ ( G ) if S is a maximum independent set in the subgraph induced by $$S\cup N(S)$$ S ∪ N ( S ) [22]. In this paper, we present some classes of graphs where the families CritIndep(G), Crown(G), and $$\Psi (G)$$ Ψ ( G ) coincide and form greedoids or even more general set systems that we call augmentoids.