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Vertices Belonging to All Critical Independent Sets of a Graph

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.