Double Roman domination and domatic numbers of graphs
Lutz Volkmann · DOAJ (DOAJ: Directory of Open Access Journals) · 2018
A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in \cite{bhh} as a function $f:V(G)\rightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must have at least one neighbor $u$ with $f(u)\ge 2$. The weight of a double Roman dominating function $f$ is the sum $\sum_{v\in V(G)}f(v)$, and the minimum weight of a double Roman dominating function on $G$ is the double Roman domination number $\gamma_{dR}(G)$ of $G$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct double Roman dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 3$ for each $v\in V(G)$ is called in \cite{v} a double Roman dominating family (of functions) on $G$. The maximum number of functions in a double Roman dominating family on $G$ is the double Roman domatic number of $G$. In this note we continue the study of the double Roman domination and domatic numbers. In particular, we present a sharp lower bound on $\gamma_{dR}(G)$, and we determine the double Roman domination and domatic numbers of some classes of graphs.