Further Results on the Majority Roman Domination in graphs

TL;DR

This paper investigates the properties of the Majority Roman Domination number in graphs, establishing bounds and analyzing its behavior across different graph classes.

Contribution

It introduces new bounds and properties for the Majority Roman Domination number, extending understanding of this graph parameter.

Findings

Derived lower and upper bounds for the Majority Roman Domination number

Analyzed the parameter for specific classes of graphs

Extended previous results on Roman domination concepts

Abstract

Let $G=(V,E)$ be a simple graph of order $n$. A Majority Roman Dominating Function (MRDF) on a graph G is a function $f: V\rightarrow\{-1, +1, 2\}$ if the sum of its function values over at least half the closed neighborhoods is at least one , this is , for at least half of the vertices $v\in V$, $f(N[v])\geq 1$. Moreover, every vertex u with $f(u)=-1$ is adjacent to at least one vertex $w$ with $f(w)=2$. The Majority Roman Domination number of a graph $G$, denoted by $\gamma_{MR}(G)$ , is the minimum value of $\sum_{v\in{V(G)}}f(v)$ over all Majority Roman Dominating Function $f$ of $G$. In this paper we study properties of the Majority Roman Domination in graphs and obtain lower and upper bounds the Majority Roman Domination number of some graphs.