A \(\{2\}\) -dominating function of a graph G is a function \(f:V(G)\longrightarrow \{0,1,2\}\) such that \(f(N[v])\ge 2\) for all \(v\in V(G),\) where N[v] stands for the set of neighbors of v plus v. If in addition no two vertices assigned 0 under f are adjacent, then f is called an outer independent \(\{2\}\) -dominating function (OI \({\{2\}}\) D-function). The weight of an OI \(\{2\}\) D-function is the value \(\omega (f)=\Sigma _{u\in V(G)}f(u)\) , and the minimum weight of an OI \(\{2\}\) D-function of G is called the outer independent \(\{2\}\) -domination number \(\gamma _{oi\{2\}}(G)\) of G. In this paper, we study the outer independent \(\{2\}\) -domination number. We first show that the problem of computing this parameter is NP-complete, even when restricted to bipartite graphs. Then various bounds on this parameter are established. Moreover, for the class of trees, lower and upper bounds are provided in terms of the order, the number of stems (support vertices) and the number of leaves.
