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 ${\gamma }_{\mathrm{pr}}(G)$ is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination number $\mathrm{\Gamma }(G)$ 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 ×.

We show that for every positive constant c, there exist graphs G and H of arbitrarily large diameter such that $\gamma (G\times H)\le c\gamma (G)\gamma (H)$, thus answering two questions of Paulraja and Sampath Kumar involving the paired domination number. We then study when the inequality ${\gamma }_{\mathrm{pr}}(G\times H)\le \frac{1}{2}{\gamma }_{\mathrm{pr}}(G){\gamma }_{\mathrm{pr}}(H)$ is satisfied, in particular proving that it holds whenever G and H are trees. Finally, we demonstrate that the bound $\mathrm{\Gamma }(G\times H)\ge \mathrm{\Gamma }(G)\mathrm{\Gamma }(H)$, due to Brešar, Klavžar, and Rall, is tight.

一组d中的曲线图的顶点ģ称为主导如果每个顶点ģ是无论是在d或邻近的顶点d。统治数$\gamma (G)$是G 中支配集的最小大小，成对支配数 ${\gamma }_{\mathrm{公关}}(G)$是其诱导子图允许完美匹配的支配集的最小大小，以及上支配数 $\mathrm{\Gamma }(G)$是最小支配集的最大大小。在本文中，我们研究了涉及支配数和这些变体的乘法不等式的锐度，其中图积是直接积 ×。

我们证明对于每个正常数c，存在任意大直径的图G和H使得$\gamma (G\times H)\le C\gamma (G)\gamma (H)$，从而回答了 Paulraja 和 Sampath Kumar 的两个涉及配对统治号码的问题。然后我们研究什么时候不等式${\gamma }_{\mathrm{公关}}(G\times H)\le \frac{1}{2}{\gamma }_{\mathrm{公关}}(G){\gamma }_{\mathrm{公关}}(H)$满足，特别是证明当G和H是树时它成立。最后，我们证明了界$\mathrm{\Gamma }(G\times H)\ge \mathrm{\Gamma }(G)\mathrm{\Gamma }(H)$，由于 Brešar、Klavžar 和 Rall，很紧。