Modern Roman Dominating Functions In Graphs

Abstract

Let $G=(V(G), E(G))$ be any connected graph. A function $f:V(G) \to \{0,1,2,3\}$ is a modern Roman dominating function of $G$ if for each $v\in V(G)$ with $f(v)=0$, there exist $u,w \in N_G (v)$ such that $f(u)=2$ and $f(w)=3$; and for each $v\in V(G)$ with $f(v)=1$, there exists $u \in N_G (v)$ such that $f(u)=2$ or $f(w)=3$. The weight  of a modern Roman dominating function $f$ of $G$ is the sum $\omega_G^{mR}(f)=\sum_{v\in V(G)}f(v)$ and its minimum weight is called the modern Roman domination number $\gamma_{mR}(G)$ of $G$. In this paper, we characterize graphs with smaller modern Roman domination number  and obtain the $\gamma_{mR}(G)$ of some special graphs. Moreover, we investigate and characterize the modern Roman domination of the join and corona of graphs.