Critical sets, crowns, and local maximum independent sets

Abstract

A set $S\subseteq V(G)$ is independent (or stable) 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 $\left|A\right|-\left|N(A)\right|=\max \{\left|I\right|-\left|N(I)\right|:I\in \mathrm{Ind}(G)\}$, 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, 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 $S\cup N(S)$. In this paper we show that $CritIndep(G)\subseteq Crown(G)$ $\subseteq \Psi (G)$ are true for every graph. In addition, we present some classes of graphs where these families coincide and form greedoids or even more general set systems that we call augmentoids.

19 pages, 11 figures