On the $\{2\}$-domination number of graphs

Let $G$ be a nontrivial graph and $k\ge 1$ an integer. Given a vector of nonnegative integers $w=(w_0,\dots ,w_k)$, a function $f:V(G)\to \{0,\dots ,k\}$ is a $w$-dominating function on $G$ if $f(N(v))\ge w_i$ for every $v\in V(G)$ such that $f(v)=i$. The $w$-domination number of $G$, denoted by ${\gamma }_w(G)$, is the minimum weight $\omega (f)=\sum _{v\in V(G)}f(v)$ among all $w$-dominating functions on $G$. In particular, the $\{2\}$-domination number of a graph $G$ is defined as ${\gamma }_{\{2\}}(G)={\gamma }_{(2,1,0)}(G)$. In this paper we continue with the study of the $\{2\}$-domination number of graphs. In particular, we obtain new tight bounds on this parameter and provide closed formulas for some specific families of graphs.