  Let $G$ be a simple graph with vertex set $V\left( G\right) $. A set
$S\subseteq V\left( G\right) $ is independent if no two vertices from $S$ are
adjacent, and by $\mathrm{Ind}(G)$ we mean the family of all independent sets
of $G$.
  The number $d\left( X\right) =$ $\left\vert X\right\vert -\left\vert
N(X)\right\vert $ is the difference of $X\subseteq V\left( G\right) $, and a
set $A\in\mathrm{Ind}(G)$ is critical if $d(A)=\max \{d\left( I\right)
:I\in\mathrm{Ind}(G)\}$ (Zhang, 1990).
  Let us recall the following definitions:
  $\mathrm{core}\left( G\right) $ = $\bigcap$ {S : S is a maximum independent
set}.
  $\mathrm{corona}\left( G\right)$ = $\bigcup$ {S :S is a maximum independent
set}.
  $\mathrm{\ker}(G)$ = $\bigcap$ {S : S is a critical independent set}.
  $\mathrm{diadem}(G)$ = $\bigcup$ {S : S is a critical independent set}.
  In this paper we present various structural properties of $\mathrm{\ker}(G)$,
in relation with $\mathrm{core}\left( G\right) $, $\mathrm{corona}\left(
G\right) $, and $\mathrm{diadem}(G)$.
