  Let G=(V,E). A set S is independent if no two vertices from S are adjacent.
The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is
critical if d(A) = max{d(I):I is an independent set}. Let us recall that ker(G)
is the intersection of all critical independent sets, and core(G) is the
intersection of all maximum independent sets. Recently, it was established that
ker(G) is a subset of core(G) is true for every graph, while the corresponding
equality holds for bipartite graphs. In this paper we present various
structural properties of ker(G). The main finding claims that ker(G) is equal
to the union of all inclusion minimal independent sets with positive
difference.
