Tightness of domination inequalities for direct product graphs

A set D of vertices in a graph G is called dominating if every vertex of G is either in D or adjacent to a vertex of D. The domination number gamma(G) is the minimum size of a dominating set in G, the paired domination number gamma(pr)(G) is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination number Gamma(G) is the maximum size of a minimal dominating set. In this paper, we investigate the sharpness of multiplicative inequalities involving the domination number and these variants, where the graph product is the direct product x. We show that for every positive constant c, there exist graphs G and H of arbitrarily large diameter such that gamma(G x H) <= c gamma(G)gamma(H), thus answering two questions of Paulraja and Sampath Kumar involving the paired domination number. We then study when the inequality gamma(pr)(G x H) <= 1/2 gamma(pr)(G)gamma(pr)(H) is satisfied, in particular proving that it holds whenever G and Hare trees. Finally, we demonstrate that the bound Gamma(G x H) >= Gamma(G)Gamma(H), due to Bresar, Klavzar, and Rall, is tight. (C) 2021 Elsevier B.V. All rights reserved.