Double Roman Domination in Cartesian Product

Anu, V. and Aparna, Lakshmanan S.

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creative_2024_33_1_01_06

DOI: https://doi.org/10.37193/CMI.2024.01.01

Issue no: Vol 33/2024 no. 1 Tags: Cartesian product, Double Roman Dominating Function, Double Roman Domination Number

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Given a graph $G=(V,E)$ , a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$ , then there exist $v_{1},v_{2}\in N(v)$ such that $f(v_{1})=2=f(v_{2})$ 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 $f(w)\geq 2$ is called a double Roman dominating function (DRDF). The weight of a DRDF $f$ is
the sum $f(V)=\sum_{v\in V}f(v)$ , and the minimum among the weights of DRDFs on $G$ is the double Roman domination number, $\gamma_{dR}(G)$ , of $G$ . In this paper, we study the impact of cartesian product on the double Roman domination number.