On $d$-distance $p$-packing domination number in strong products

Abstract

The $d$-distance $p$-packing domination number $\gamma_d^p(G)$ of a graph $G$ is the cardinality of a smallest set of vertices of $G$ which is both a $d$-distance dominating set and a $p$-packing. If no such set exists, then we set $\gamma_d^p(G) = \infty$. For an arbitrary strong product $G\boxtimes H$ it is proved that $\gamma_d^p(G\boxtimes H) \le \gamma_d^p(G) \gamma_d^p(H)$. By proving that $\gamma_d^p(P_m \boxtimes P_n) = \left \lceil \frac{m}{2d+1} \right \rceil \left \lceil \frac{n}{2d+1} \right \rceil$, and that if $\gamma_d^p(C_n) < \infty$, then $\gamma_d^p(P_m \boxtimes C_n) = \left \lceil \frac{m}{2d+1} \right \rceil \left \lceil \frac{n}{2d+1} \right \rceil$, the sharpness of the upper bound is demonstrated. On the other hand, infinite families of strong toruses are presented for which the strict inequality holds. For instance, we present strong toruses with difference $2$ and demonstrate that the difference can be arbitrarily large if only one factor is a cycle. It is also conjectured that if $\gamma_d^p(G) = \infty$, then $\gamma_d^p(G\boxtimes H) = \infty$ for every graph $H$. Several results are proved which support the conjecture, in particular, if $\gamma_d^p(C_m)= \infty$, then $\gamma_d^p(C_m \boxtimes C_n)=\infty$.