The restrained double Roman domination and graph operations

Abstract

Let $G=(V(G),E(G))$ be a simple graph. A restrained double Roman dominating function (RDRD-function) of $G$ is a function $f: V(G) \rightarrow \{0,1,2,3\}$ satisfying the following properties: if $f(v)=0$, then the vertex $v$ has at least two neighbours assigned 2 under $f$ or one neighbour $u$ with $f(u)=3$; and if $f(v)=1$, then the vertex $v$ must have one neighbor $u$ with $f(u) \geq 2$; the induced graph by vertices assigned 0 under $f$ contains no isolated vertex. The weight of a RDRD-function $f$ is the sum $f(V)=\sum_{v \in V(G)} f(v)$, and the minimum weight of a RDRD-function on $G$ is the restrained double Roman domination number (RDRD-number) of $G$, denoted by $\gamma_{rdR}(G)$. In this paper, we first prove that the problem of computing RDRD-number is NP-hard even for chordal graphs. And then we study the impact of some graph operations, such as strong product, cardinal product and corona with a graph, on restrained double Roman domination number.