Connected Domination Number of a Graph and its Complement

A set S of vertices in 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 such a set. Let δ(G) = min{δ(G), δ(G)}, where G is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and G are both connected, γc(G) + γc(G) ≤ δ (G) + 4 − (γc(G) − 3)(γc(G) − 3). As a corollary, γc(G) + γc(G) ≤ 3n 4 when δ(G) ≥ 3 and n ≥ 14, where G has n vertices. We also prove that γc(G) + γc(G) ≤ δ (G) + 2 when γc(G), γc(G) ≥ 4. This bound is sharp when δ(G) = 6, and equality can only hold when δ(G) = 6. Finally, we prove that γc(G)γc(G) ≤ 2n − 4 when n ≥ 7, with equality only for paths and cycles.