  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
$\left\vert A\right\vert -\left\vert N(A)\right\vert =\max\{\left\vert
I\right\vert -\left\vert N(I)\right\vert :I\in \mathrm{Ind}(G)\}$, 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.
