Graph products and integer domination

We consider the { k } -domination number γ { k } ( G ) of a graph G and the Cartesian product G □ H and the strong direct product G ⊠ H of graphs G and H . We prove that for integers k , m ≥ 1 , γ { k } ( G ⊠ H ) ≥ γ { γ { k } ( H ) } ( G ) and γ { k m } ( G ⊠ H ) ≤ γ { k } ( G ) γ { m } ( H ) , from which earlier results obtained by Fisher on γ ( G ⊠ H ) and Fisher et al. on the fractional domination number γ f ( G ⊠ H ) were derived. We extend a result from Brešar et al. on γ ( G □ H ) for claw-free graphs G . We also point out some sufficient conditions for graphs to satisfy the generalized form of Vizing’s conjecture suggested by Hou and Lu.