Restrained {2}-domination in graphs.

A restrained {2}-dominating function (R{2}DF) on a graph G = ( V , E ) is a function f : V → {0, 1, 2} such that : (i) f ( N [ v ]) ≥ 2 for all v ∈ V , where N [ v ] is the set containing v and all vertices adjacent to v ; (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 γ 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 bounds on the restrained {2}-domination number of a tree T in terms of the order, the numbers of leaves and support vertices.