Maximum number of edges in connected graphs with a given domination number

A dominating set for a graph G = (V,E) is a subset of vertices V' ⊆ V such that for all v ϵ V−V', there exists some u ϵ V' for which \s{v, u\s} ϵ E. The domination number of G is the size of its smallest dominating set(s). In this paper we give an upper bound on the number of edges a connected graph with a given number of vertices and a given domination number can have. We also characterize the extremal graphs attaining this upper bound.