  Let G be a simple graph with vertex set V(G). A subset S of V(G) is
independent if no two vertices from S are adjacent. By Ind(G) we mean the
family of all independent sets of G while core(G) and corona(G) denote the
intersection and the union of all maximum independent sets, respectively. The
number d(X)= |X|-|N(X)| is the difference of the set of vertices X, and an
independent set A is critical if d(A)=max{d(I):I belongs to Ind(G)} (Zhang,
1990). Let ker(G) and diadem(G) be the intersection and union, respectively, of
all critical independent sets of G (Levit and Mandrescu, 2012). In this paper,
we present various connections between critical unions and intersections of
maximum independent sets of a graph. These relations give birth to new
characterizations of Koenig-Egervary graphs, some of them involving ker(G),
core(G), corona(G), and diadem(G).
