On Double Roman Dominating Functions in Graphs

Abstract

Let $G$ be a connected graph. A function $f:V(G)\to \{0,1,2,3\}$ is a double Roman dominating function of $G$ if for each $v\in V(G)$ with $f(v)=0$, $v$ has two adjacent vertices $u$ and $w$ for which $f(u)=f(w)=2$ or $v$ has an adjacent vertex $u$ for which $f(u)=3$, and for each $v\in V(G)$ with $f(v)=1$, $v$ is adjacent to a vertex $u$ for which either $f(u)=2$ or $f(u)=3$. The minimum weight ${\omega }_G(f)=\sum _{v\in V(G)}f(v)$ of a double Roman dominating function $f$ of $G$ is the double Roman domination number of $G$. In this paper, we continue the study of double Roman domination introduced and studied by R.A. Beeler et al. in [2]. First, we characterize some double Roman domination numbers with small values in terms of the domination numbers and 2-domination numbers. Then we determine the double Roman domination numbers of the join, corona, complementary prism and lexicographic product of graphs.