A note on Nordhaus–Gaddum inequalities for domination

Abstract

For a graph G of order n, let γ( G), γ 2( G) and γ t( G) be the domination, double domination and total domination numbers of G, respectively. The minimum degree of the vertices of G is denoted by δ( G) and the maximum degree by Δ( G). In this note we prove a conjecture due to Harary and Haynes saying that if a graph G has γ(G),γ( G ̄ )⩾4 , then γ 2(G)+γ 2( G ̄ )⩽n−Δ(G)+δ(G)−1⩽n−1 and γ t (G)+γ t ( G ̄ )⩽n−Δ(G)+δ(G)−1⩽n−1, where G ̄ is the complement of G.