﻿ A note on the Roman domatic number of a digraph

# A note on the Roman domatic number of a digraph

##### چکیده

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}\$ ofRoman 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 aRoman 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 sharpbounds 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 ofundirected graphs.

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عنوان ژورنال:

دوره 5  شماره 1

صفحات  19- 26

تاریخ انتشار 2020-06-01

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