A note on Nordhaus–Gaddum inequalities for domination Original Research Article

For a graph G of order n, let γ(G), γ2(G) and γt(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(Ḡ)⩾4, thenγ2(G)+γ2(Ḡ)⩽n−Δ(G)+δ(G)−1⩽n−1andγt(G)+γt(Ḡ)⩽n−Δ(G)+δ(G)−1⩽n−1,where Ḡ is the complement of G.