Extremal Problems for Roman Domination

A Roman dominating function of a graph $G$ is a labeling $f\colon\,V(G)\to\{0,1,2\}$ such that every vertex with label 0 has a neighbor with label 2. The Roman domination number $\gamma_R(G)$ of $G$ is the minimum of $\sum_{v\in V(G)}f(v)$ over such functions. Let $G$ be a connected $n$-vertex graph. We prove that $\gamma_R(G)\leq4n/5$, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for $\gamma_R(G)+\gamma_R(\overline{G})$ and $\gamma_R(G)\gamma_R(\overline{G})$, improving known results for domination number. We prove that $\gamma_R(G)\leq8n/11$ when $\delta(G)\geq2$ and $n\geq9$, and this is sharp.