Properties of disjunctive domination in product graphs

Abstract

In this paper properties of disjunctive domination in some graph products are studied. We examine whether disjunctive domination number is multiplicative with respect to different graph products, that is, $\gamma_2^d\left(G_1 * G_2\right) \geq$ $\gamma_2^d\left(G_1\right) \gamma_2^d\left(G_2\right)$ for all graphs $G_1$ and $G_2$ or $\gamma_2^d\left(G_1 * G_2\right) \leq \gamma_2^d\left(G_1\right) \gamma_2^d\left(G_2\right)$ for all graphs $G_1$ and $G_2$ where $*$ denotes lexicographic, tensor, strong or Cartesian product of graphs. Some other inequalities involving disjunctive domination number of product graphs and the graphs attaining these inequalities are also given.