Inequalities of Nordhaus–gaddum Type for Connected Domination

A set S of vertices of a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γc(G) is the minimum size of a connected dominating set of G. In this paper we prove that γc(G) + γc(G) ≤ min{δ(G), δ(G)} + 4 for every n-vertex graph G such that G and G have diameter 2 and show that the bound is sharp for each value of the right side. Also, γc(G) + γc(G) ≤ 3n 4 if G and G are connected, have minimum degree at least 3 and n ≥ 14. We also prove that γc(G) + γc(G) ≤ min{δ(G), δ(G)} + 2 if γc(G), γc(G) ≥ 4 and show that the bound is sharp when min{δ(G), δ(G)} = 6.