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

Let $ G $ be a nontrivial graph and $ k\geq 1 $ an integer. Given a vector of nonnegative integers $ w = (w_0, \ldots, w_k) $, a function $ f: V(G)\rightarrow \{0, \ldots, k\} $ is a $ w $-dominating function on $ G $ if $ f(N(v))\geq 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.