On the upper Bound of double Roman dominating function

Atieh Teimourzadeh; Doost Ali Mojdeh

paper

On the upper Bound of double Roman dominating function

arXiv:1911.02394

Abstract

A double Roman Dominating function on a graph $G$ is a function $f : V \to {0, 1, 2, 3}$ such that the following conditions hold. If $f (v) = 0$ , then vertex $v$ must have at least two neighbors in $V_{2}$ or one neighbor in $V_{3}$ and if $f (v) = 1$ , then vertex $v$ must have at least one neighbor in $V_{2} ⋃ V_{3}$ . The weight of a double Roman dominating function is the sum $w_{f} = \sum_{v \in V (G)} f (v)$ . In this paper, we improve the upper bounds of $γ_{d R} (G)$ that has already obtained and we show that $γ_{d R} (G) \leq \frac{12 n}{11}$ , for any graph with $δ (G) \geq 2$ . This bound improve the bounds that have already been presented in \cite{chen} and \cite{kkcs}. Finally we prove the conjecture posed in \cite{kkcs}.