Critical Sets in Bipartite Graphs

Let G = (V, E) be a graph. A set I ⊆ V is independent if no two vertices from I are adjacent, and by Ind(G) we mean the family of all independent sets of G, while core(G) is the intersection of all maximum independent sets [4]. The number d_c(G) = max{{pipe}I{pipe} - {pipe}N(I){pipe}: I ∈ Ind(G)} is called the critical difference of G. A set X is critical if {pipe}X{pipe} - {pipe}N(X){pipe} = d_c(G) [10]. For a bipartite graph G = (A, B, E), Ore [7] defined δ₀(A) = max{{pipe}X{pipe} - {pipe}N(X){pipe}: X ⊆ A}. In this paper, we prove that d_c(G) = δ₀(A)+δ₀(B) and ker(G) = core(G) hold for every bipartite graph G = (A, B, E), where ker(G) denotes the intersection of all critical independent sets.