Further Results on the Majority Roman Domination in graphs

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 $f:V\to \{-1,+1,2\}$ 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 $v\in V$, $f(N[v])\ge 1$. 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.