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.