Critical sets, crowns and local maximum independent sets

Abstract

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