  Let G=(V,E) be a graph. A set S is independent if no two vertices from S are
adjacent. The independence number alpha(G) is the cardinality of a maximum
independent set, and mu(G) is the size of a maximum matching. The number
id_{c}(G)=max{|I|-|N(I)|:I is an independent set} is called the critical
independence difference of G, and A is critical if |A|-|N(A)|=id_{c}(G). We
define core(G) as the intersection of all maximum independent sets, and
ker(G)as the intersection of all critical independent sets. In this paper we
prove that if a graph G is non-quasi-regularizable (i.e., there exists some
independent set A, such that |A|>|N(A)|), then: ker(G) is a subset of core(G),
and |ker(G)|> id_{c}(G) >= alpha(G)-mu(G) > 0.
