Dominating direct products of graphs

An upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H, γ(G × H) ≤ 3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G × H) ≥ Γ(G)Γ(H), thus confirming a conjecture from [16]. Finally, for paired-domination of direct products we prove that γpr(G × H) ≤ γpr(G)γpr(H) for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound.