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

Let G=(V,E)G=(V,E). A set S⊆VS⊆V is independent   if no two vertices from SS are adjacent, and by Ind(G) we mean the family of all the independent sets of GG. The number d(X)=|X|−|N(X)|d(X)=|X|−|N(X)| is the difference   of X⊆VX⊆V, and A∈Ind(G) is critical   if d(A)=max{d(I):I∈Ind(G)}[18].Let us recall the following definitions: core(G)=⋂{S:S is a maximum independent set}[10] ,ker(G)=⋂{S:S is a critical independent set}[12].ker(G)=⋂{S:S is a critical independent set}[12].Recently, it was established that ker(G)⊆core(G) is true for every graph [12], while the corresponding equality holds for bipartite graphs [13].In this paper, we present various structural properties of ker(G)ker(G). The main finding claims that ker(G)=⋃{S0:S0 is an inclusion minimal independent set with d(S0)=1}=⋃{S0:S0 is an inclusion minimal independent set with d(S0)>0}.ker(G)=⋃{S0:S0 is an inclusion minimal independent set with d(S0)=1}=⋃{S0:S0 is an inclusion minimal independent set with d(S0)>0}.