Nordhaus–Gaddum bounds for total domination

A Nordhaus–Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we continue the study of Nordhaus–Gaddum bounds for the total domination number γ t . Let G be a graph on n vertices and let G ¯ denote the complement of G , and let δ ∗ ( G ) denote the minimum degree among all vertices in G and G ¯ . For δ ∗ ( G ) ≥ 1 , we show that γ t ( G ) γ t ( G ¯ ) ≤ 2 n , with equality if and only if G or G ¯ consists of disjoint copies of K 2 . When δ ∗ ( G ) ∈ { 2 , 3 , 4 } , we improve the bounds on the sum and product of the total domination numbers of G and G ¯ .