RESTRAINED WEAK ROMAN DOMINATION IN GRAPHS

 Let $G = (V, E)$ be a graph and $f : V \to \{0, 1, 2\}$ be a weak Roman dominating function on $G$. $f$ is called a restrained weak Roman dominating function, if each vertex $u \in V$ with $f(u) = 0$ is adjacent to another vertex $v \in V$ such that $f(v) = 0$. The weight of a restrained weak Roman dominating function $f$ is defined as $\ds w(f) = f(V) = \sum_{v \in V} f(v)$. The minimum weight of a restrained weak Roman dominating function on $G$ is called the restrained weak Roman domination number of $G$ and is denoted by $\gamma_{rr}(G)$.