%0 Journal Article
%T The Roman domination and domatic numbers of a digraph
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Xie, Zhihong
%A Hao, Guoliang
%A Wei, Shouliu
%D 2019
%\ 06/01/2019
%V 4
%N 1
%P 47-59
%! The Roman domination and domatic numbers of a digraph
%K Roman dominating function
%K Roman domination number
%K Roman domatic number
%K digraph
%R 10.22049/cco.2019.26356.1101
%X 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 $sum_{vin V(D)}f(v)$. The Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$. A set ${f_1,f_2,dots,f_d}$ of Roman dominating functions on $D$ with the property that $sum_{i=1}^df_i(v)le2$ for each $vin V(D)$, 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 paper we continue the investigation of the Roman domination number, and we initiate the study of the Roman domatic number in digraphs. We present some bounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs.
%U http://comb-opt.azaruniv.ac.ir/article_13841_9bb2c1e1cb7db1916a327f1866d037b2.pdf