Dominating direct products of graphs Original Research Article

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, image. Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that image, thus confirming a conjecture from [R. Nowakowski, D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53–79]. Finally, for paired-domination of direct products we prove that image for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound.