On Roman balanced domination of graphs

Let $G$ be a graph with vertex set $V$. A function $f$ : $V\to \{-1,0,2\}$ is called a Roman balanced dominating function (RBDF) of $G$ if $\sum _{u\in N_G[v]}f(u)=0$ for each vertex $v\in V$. The maximum (resp. minimum) Roman balanced domination number ${\gamma }_{Rb}^M(G)$ (resp. ${\gamma }_{Rb}^m(G)$) is the maximum (resp. minimum) value of $\sum _{v\in V}f(v)$ among all Roman balanced dominating functions $f$. A graph $G$ is called $Rd$-balanced if ${\gamma }_{Rb}^M(G)={\gamma }_{Rb}^m(G)=0$. In this paper, we obtain several upper and lower bounds on ${\gamma }_{Rb}^M(G)$ and ${\gamma }_{Rb}^m(G)$ and further determine several classes of $Rd$-balanced graphs.