Restrained {2}-domination in graphs

A restrained $\{2\}$-dominating function (R$\{2\}$-DF) on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,2\}$ such that : \textrm{(i)} $f(N[v])\geq2$ for all $v\in V,$ where $N[v]$ is the set containing $v$ and all vertices adjacent to $v;$ \textrm{(ii)} the subgraph induced by the vertices assigned 0 under $f$ has no isolated vertices. The weight of an R$\{2\}$-DF is the sum of its function values over all vertices, and the restrained $\{2\}$-domination number $\gamma_{r\{2\}}(G)$ is the minimum weight of an R$\{2\}$-DF on $G.$ In this paper, we initiate the study of the restrained $\{2\}$-domination number. We first prove that the problem of computing this parameter is NP-complete, even when restricted to bipartite graphs. Then we give various bounds on this parameter. In particular, we establish upper and lower bound on the restrained $\{2\}$-domination number of a tree $T$ in terms of the order, the numbers of leaves and support vertices.