Nordhaus–Gaddum bounds for total Roman domination

A Nordhaus–Gaddum-type result is a lower or an upper bound on the sum or the product of a parameter of a graph and its complement. In this paper we continue the study of Nordhaus–Gaddum bounds for the total Roman domination number $$\gamma _{tR}$$ γ t R . Let G be a graph on n vertices and let $$\overline{G}$$ G ¯ denote the complement of G, and let $$\delta ^*(G)$$ δ ∗ ( G ) denote the minimum degree among all vertices in G and $$\overline{G}$$ G ¯ . For $$\delta ^*(G)\ge 1$$ δ ∗ ( G ) ≥ 1 , we show that (i) if G and $$\overline{G}$$ G ¯ are connected, then $$(\gamma _{tR}(G)-4)(\gamma _{tR}(\overline{G})-4)\le 4\delta ^*(G)-4$$ ( γ t R ( G ) - 4 ) ( γ t R ( G ¯ ) - 4 ) ≤ 4 δ ∗ ( G ) - 4 , (ii) if $$\gamma _{tR}(G), \gamma _{tR}(\overline{G})\ge 8$$ γ t R ( G ) , γ t R ( G ¯ ) ≥ 8 , then $$\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le 2\delta ^*(G)+5$$ γ t R ( G ) + γ t R ( G ¯ ) ≤ 2 δ ∗ ( G ) + 5 and (iii) $$\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le n+5$$ γ t R ( G ) + γ t R ( G ¯ ) ≤ n + 5 and $$\gamma _{tR}(G)\gamma _{tR}(\overline{G})\le 6n-5$$ γ t R ( G ) γ t R ( G ¯ ) ≤ 6 n - 5 .