Nordhaus–Gaddum bounds for total Roman domination

Abstract

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 $\gamma _{tR}$. Let G be a graph on n vertices and let $\overline{G}$ denote the complement of G, and let $\delta ^*(G)$ denote the minimum degree among all vertices in G and $\overline{G}$. For $\delta ^*(G)\ge 1$, we show that (i) if G and $\overline{G}$ are connected, then $(\gamma _{tR}(G)-4)(\gamma _{tR}(\overline{G})-4)\le 4\delta ^*(G)-4$, (ii) if $\gamma _{tR}(G), \gamma _{tR}(\overline{G})\ge 8$, then $\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le 2\delta ^*(G)+5$ and (iii) $\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le n+5$ and $\gamma _{tR}(G)\gamma _{tR}(\overline{G})\le 6n-5$.