Impact of Some Graph Operations on Double Roman Domination Number

TL;DR

This paper investigates how various graph operations like Cartesian product, adding twins, and corona affect the double Roman domination number, providing insights into its behavior under these transformations.

Contribution

It introduces new results on the effects of specific graph operations on the double Roman domination number, expanding understanding of this graph invariant.

Findings

Double Roman domination number changes predictably under Cartesian product.

Adding twins influences the double Roman domination number in specific ways.

Corona operations have notable effects on the double Roman domination number.

Abstract

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 $f(v_{1})=f(v_{2})=2$ 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 $f(w)\geq 2$ is called a double Roman dominating function (DRDF). The weight of a DRDF $f$ is the sum $f(V)=\sum_{v\in V}f(v)$. 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.