Abstract We let γ ( G ) and i ( G ) denote the domination number and the independent domination number of G , respectively. Recently, Rad and Volkmann conjectured that i ( G ) / γ ( G ) ≤ Δ ( G ) / 2 for every graph G , where Δ ( G ) is the maximum degree of G . In this note, we construct counterexamples of the conjecture for Δ ( G ) ≥ 6 and give a sharp upper bound of the ratio i ( G ) / γ ( G ) by using the maximum degree of G .