On the number of edges in graphs with a given 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(s). When G is connected, we say V′ is a connected dominating set if the subgraph of G induced by V′ is connected. The connected domination number of G is the size of its smallest connected dominating set, and is denoted by γc(G). In this paper we determine the maximum number of edges that a connected graph with a given number of vertices and a given connected domination number can have. We also characterize the extremal graphs achieving the bound.