Graphs and Combinatorics (2012) 28:123–131 DOI 10.1007/s00373-011-1028-z ORIGINAL PAPER Connected Domination Number of a Graph and its Complement H. Karami · S. M. Sheikholeslami · Abdollah Khodkar · Douglas B. West Received: 20 July 2009 / Revised: 7 January 2011 / Published online: 8 February 2011 © Springer 2011 Abstract 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 γ c (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, γ c (G) + γ c ( G) ≤ δ ∗ (G) + 4 - (γ c (G) - 3)(γ c ( G) - 3). As a corollary, γ c (G) + γ c ( G) ≤ 3n 4 when δ ∗ (G) ≥ 3 and n ≥ 14, where G has n vertices. We also prove that γ c (G) + γ c ( G) ≤ δ ∗ (G) + 2 when γ c (G), γ c ( G) ≥ 4. This bound is sharp when δ ∗ (G) = 6, and equality can only hold when δ ∗ (G) = 6. Finally, we prove that γ c (G)γ c ( G) ≤ 2n - 4 when n ≥ 7, with equality only for paths and cycles. This research was in part supported by a grant from IPM (No. 89050042). This research partially supported by the National Security Agency under Awards H98230-06-1-0065 and H98230-10-1-0363. H. Karami · S. M. Sheikholeslami (B ) Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, Islamic Republic of Iran e-mail: s.m.sheikholeslami@azaruniv.edu S. M. Sheikholeslami School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Islamic Republic of Iran A. Khodkar Department of Mathematics, University of West Georgia, Carrollton, GA 30118, USA e-mail: akhodkar@westga.edu D. B. West Department of Mathematics, University of Illinois, Urbana, IL 61801, USA e-mail: west@math.uiuc.edu 123