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 γt. 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 γt(G)γt(G¯)≤2n, with equality if and only if G or G¯ consists of disjoint copies of K2. When δ∗(G)∈{2,3,4}, we improve the bounds on the sum and product of the total domination numbers of G and G¯.