A Note on Roman Domination of Digraphs

A vertex subset $S$ of a digraph $D$ is called a dominating set of $D$ if every vertex not in $S$ is adjacent from at least one vertex in $S$. The domination number of a digraph $D$, denoted by $ \gamma(D) $, is the minimum cardinality of a dominating set of $D$. A Roman dominating function (RDF) on a digraph $D$ is a function $ f : V (D) \rightarrow {0, 1, 2} $ satisfying the condition that every vertex $v$ with $f(v) = 0$ has an in-neighbor $u$ with $f(u) = 2$. The weight of an RDF $f$ is the value $ \omega (f) = \Sigma_{ v \in V(D) } f(v) $. The Roman domination number of a digraph $D$, denoted by $ \gamma_R (D) $, is the minimum weight of an RDF on $D$. In this paper, for any integer $k$ with $ 2 \le k \le \gamma(D) $, we characterize the digraphs $D$ of order $ n \ge 4 $ with $ \delta − (D) \ge 1 $ for which $ \gamma_R(D) = (D) + k $ holds. We also characterize the digraphs $D$ of order $ n \ge k $ with $ \gamma_R(D) = k $ for any positive integer $k$. In addition, we present a Nordhaus-Gaddum bound on the Roman domination number of digraphs.