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 ) = max { d ( I ) : I ∈ Ind ( G ) } [18]. recall the following definitions: core ( G ) = ⋂ { S : S  is a maximum independent set } [10] , ker ( G ) = ⋂ { S : S  is a critical independent set } [12]. ly, it was established that ker ( G ) ⊆ core ( G ) is true for every graph [12], while the corresponding equality holds for bipartite graphs [13]. s paper, we present various structural properties of ker ( G ) . The main finding claims that ker ( G ) = ⋃ { S 0 : S 0  is an inclusion minimal independent set with  d ( S 0 ) = 1 } = ⋃ { S 0 : S 0  is an inclusion minimal independent set with  d ( S 0 ) > 0 } .