A-differentials and total domination in graphs

Abstract

Let G = (V,E) be an arbitrary graph. For any subset X of V let B(X) be the set of all vertices in V – X that have a neigbor in a set X. J.L. Mashburn et al.,, defined the differential of a set X, to be ∂ (X) = | B(X) | − | X | and the differential of a graph ∂ (G) = max{∂ (X)}, for any subset X of V. The A-differential of a set X is defi ned as ∂ A (X) = | B(X) | − | A(X) | , where A(X) = X ⋂ N(X), the non isolates in < x >, the vertices in X having a neighbor in X. The A -differential of a graph is ∂ A (G) = max{∂A(X)}, for any subset X of V. For any graph G, it is observed that ∂ A (G) + 2γ t (G) ≥ n and ∂ A (G) + i (G) ≥ n, where γ t (G) is the total domination number of G and i (G) is the independent domination number of G. In this paper, we characterize certain classes of graphs for which ∂ A (G) + 2γ t (G) = n and ∂ A (G) + i (G) = n.