Critical sets, crowns, and local maximum independent sets

A set $S\subseteq V(G)$ is independent (or stable) if no two vertices from $S$ are adjacent, and by $\mathrm{I}\mathrm{n}\mathrm{d}(G)$ we mean the set of all independent sets of $G$. A set $A\in \mathrm{I}\mathrm{n}\mathrm{d}(G)$ is critical (and we write $A\in CritIndep(G)$) if $\left\vert A\right\vert -\left\vert N(A)\right\vert =\max{\left\vert I\right\vert -\left\vert N(I)\right\vert :I\in $\mathrm{I}\mathrm{n}\mathrm{d}$(G)}$,where$N(I)$denotestheneighborhoodof$I$. If $S\in \mathrm{I}\mathrm{n}\mathrm{d}(G)$ and there is a matching from $N(S)$ into $S$, then $S$ is a crown, and we write $S\in Crown(G)$. Let $\mathrm{\Psi }(G)$ be the family of all local maximum independent sets of graph $G$, i.e., $S\in \mathrm{\Psi }(G)$ if $S$ is a maximum independent set in the subgraph induced by $S\cup N(S)$. In this paper we show that $CritIndep(G)\subseteq Crown(G)$ $\subseteq \mathrm{\Psi }(G)$ 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.