Impact of Some Graph Operations on Double Roman Domination Number

Abstract

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2,3\}$ having the property that if $f(v)=0$, then there exist $v_1,v_2\in N(v)$ such that $f(v_1)=f(v_2)=2$ or there exists $w\in N(v)$ such that $f(w)=3$, and if $f(v)=1$, then there exists $w\in N(v)$ such that $f(w)\ge 2$ is called a double Roman dominating function (DRDF). The weight of a DRDF $f$ is the sum $f(V)={\sum }_{v\in V}f(v)$. The double Roman domination number, ${\gamma }_{dR}(G)$, is the minimum among the weights of DRDFs on $G$. In this paper, we study the impact of some graph operations, such as cartesian product, addition of twins and corona with a graph, on double Roman domination number.