On the number of edges in graphs with a given weakly connected domination number Original Research Article

A dominating set for a graph G=(V,E) is a subset of vertices V′⊆V such that for all v∈V−V′ there exists some u∈V′ adjacent to v. The domination number of G, denoted by γ(G), is the size of its smallest dominating set. A dominating set is weakly connected if the edges not incident on any vertices of the dominating set do not separate the graph (Discrete Math. 167–168 (1997) 261). The weakly connected domination number of G is the size of its smallest weakly connected dominating set. We show in this paper that the maximum number of edges that a graph with n vertices and weakly connected domination number equal to d⩾3 can have is (n−d+12). We also characterize the extremal graphs attaining this bound.