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 family of all the 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)=maxd(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]. 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). 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.