# A note on the Roman domatic number of a digraph

Document Type : Original paper

Authors

RWTH Aachen University

Abstract

A  Roman dominating function on a digraph $D$ with vertex set $V(D)$ is a labeling $f\colon V(D)\to \{0, 1, 2\}$ such that every vertex with label $0$ has an in-neighbor with label $2$. A set $\{f_1,f_2,\ldots,f_d\}$ of Roman dominating functions on $D$ with the property that $\sum_{i=1}^d f_i(v)\le 2$ for each $v\in V(D)$, is called a Roman dominating family (of functions) on $D$. The maximum number of functions in a Roman dominating family on $D$ is the  Roman domatic number of $D$, denoted by $d_{R}(D)$. In this note, we study the Roman domatic number in digraphs, and we present some sharp bounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs. Some of our results are extensions of well-known properties of the Roman domatic number of undirected graphs.

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