A note on the Roman domatic number of a digraph

Communications in Combinatorics and Optimization

卷:5

Lutz Volkmann¹  D. Meierling¹ 

[1] RWTH Aachen University;

关键词: digraphs;    roman dominating function;    roman domination number;    roman domatic number;   

DOI  :  10.22049/CCO.2019.26419.1107

来源: DOAJ

【 摘 要 】

A {\em 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 $\{f_1,f_2,\ldots,f_d\}$ ofRoman 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 {\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, westudy the Roman domatic number in digraphs, and wepresent 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.

【 授权许可】

Unknown