Critical Sets in Bipartite Graphs

Let G = (V,E) be a graph. A set S ⊆ V is independent if no two vertices from S are adjacent, and by Ind(G) (Ω(G)) we mean the set of all (maximum) independent sets of G, while α(G) = |S| for S ∈ Ω(G), and core(G) = ∩{S : S ∈ Ω(G)} [6]. The neighborhood of A ⊆ V is denoted by N(A) = {v ∈ V : N(v) ∩ A 6= ∅}, where N(v) is the neighborhood of the vertex v. The number d (X) = |X| − |N(X)| is the difference of the set X ⊆ V , and dc(G) = max{d (I) : I ∈ Ind(G)} is called the critical difference of G. A set X is critical if d(X) = dc(G) [14]. For a graph G we define ker(G) = ∩{S : S is a critical independent set}, while diadem(G) = ∪{S : S is a critical independent set}. For a bipartite graph G = (A,B,E), with bipartition {A,B}, Ore [11] defined δ (X) = d (X) for every X ⊆ A, while δ0 (A) = max{δ (X) : X ⊆ A}. Similarly is defined δ0 (B). In this paper we prove that for every bipartite graph G = (A,B,E) the following assertions hold: • dc (G) = δ0 (A) + δ0 (B); • ker(G) = core(G); • |ker(G)|+ |diadem(G)| = 2α (G).