Nordhaus-Gaddum bounds for upper total domination

A set S of vertices in an isolate-free graph G is a total dominating set if every vertex in G is adjacent to a vertex in S. A total dominating set of G is minimal if it contains no total dominating set of $\bar{G}$ as a proper subset. The upper total domination number $Γ_t(G)$ of G is the maximum cardinality of a minimal total dominating set in G. We establish Nordhaus-Gaddum bounds involving the upper total domination numbers of a graph G and its complement $\bar{G}$. We prove that if G is a graph of order n such that both G and $\bar{G}$ are isolate-free, then $Γ_t(G) + Γ_t(\bar{G}) ≤ n + 2$ and $Γ_t(G)Γ_t(\bar{G}) ≤ 1/4 (n + 2)^2$, and these bounds are tight.