Extremal digraphs for an upper bound on the Roman domination number

Abstract

Let $D=(V,A)$ be a digraph. A Roman dominating function of a digraph D is a function f :$V\longrightarrow \{0,1,2\}$ such that every vertex u for which $f(u)=0$ has an in-neighbor v for which $f(v)=2$. The weight of a Roman dominating function is the value $f(V)=\sum _{u\in V}f(u)$. The minimum weight of a Roman dominating function of a digraph D is called the Roman domination number of D, denoted by $\gamma _{R}(D)$. In this paper, we characterize some special classes of oriented graphs, namely out-regular oriented graphs and tournaments satisfying $\gamma _{R}(D)=n-\Delta ^{+}(D)+1$. Moreover, we characterize digraphs D for which the equality $\gamma _{R}(D)+\gamma _{R}(\overline{D})=n+3$ holds, where $\overline{D}$ is the complement of D. Finally, we prove that the problem of deciding whether an oriented graph D satisfies $\gamma _{R}(D)=n-\Delta ^{+}(D)+1$ is CO-$\mathcal {NP}$-complete.