Tightness of domination inequalities for direct product graphs

Abstract

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 γ(G) is the minimum size of a dominating set in G, the paired domination numberγpr(G) is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination numberΓ(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 ×.We show that for every positive constant c, there exist graphs G and H of arbitrarily large diameter such that γ(G×H)≤cγ(G)γ(H), thus answering two questions of Paulraja and Sampath Kumar involving the paired domination number. We then study when the inequality γpr(G×H)≤12γpr(G)γpr(H) is satisfied, in particular proving that it holds whenever G and H are trees. Finally, we demonstrate that the bound Γ(G×H)≥Γ(G)Γ(H), due to Brešar, Klavžar, and Rall, is tight.