A note on the Roman domatic number of a digraph

Document Type : Original paper


RWTH Aachen University


Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling
$fcolon V(D)to {0, 1, 2}$
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 $vin V(D)$,
is called a {em Roman dominating family} (of functions) on $D$. The maximum number of functions in a
Roman dominating family on $D$ is the {em 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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