On the structure of the minimum critical independent set of a graph

Let G = (V,E). A set S ⊆ V is independent if no two vertices from S are adjacent, and by Ind(G) we mean the set of all independent sets of G. The number d (X) = |X| − |N(X)| is the difference of X ⊆ V , and A ∈ Ind(G) is critical if d(A) = max{d (I) : I ∈ Ind(G)} [7]. Let us recall the following definitions: ker(G) = ∩{S : S is a critical independent set} [5], core (G) = ∩{S : S is a maximum independent set} [4]. Recently, it was established that ker(G) ⊆ core(G) is true for every graph [5], while the corresponding equality holds for bipartite graphs [6]. In this paper we present various structural properties of ker(G). The main finding claims that ker(G) = ∪{S0 : S0 is an inclusion minimal independent set with d (S0) > 0} . .