Title: On $d$-distance $p$-packing domination number in strong products Show Chinese title

Title: 关于强积图中$d$-距离$p$-包装控制数

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 $\gamma_d^p(C_m \boxtimes C_n)=\infty$. Show Chinese abstract

Abstract: 图 $G$ 的 $d$-距离 $p$-打包支配数 $\gamma_d^p(G)$ 是 $G$ 中一个最小顶点集合的基数，该集合既是 $d$-距离支配集，又是 $p$-打包。 如果不存在这样的集合，那么我们设置$\gamma_d^p(G) = \infty$。 对于任意的强积$G\boxtimes H$，证明了$\gamma_d^p(G\boxtimes H) \le \gamma_d^p(G) \gamma_d^p(H)$。 通过证明$\gamma_d^p(P_m \boxtimes P_n) = \left \lceil \frac{m}{2d+1} \right \rceil \left \lceil \frac{n}{2d+1} \right \rceil$，并且如果$\gamma_d^p(C_n) < \infty$，则$\gamma_d^p(P_m \boxtimes C_n) = \left \lceil \frac{m}{2d+1} \right \rceil \left \lceil \frac{n}{2d+1} \right \rceil$，从而展示了上界的有效性。 另一方面，给出了无限多个强环面族，其中严格不等式成立。 例如，我们给出了差值为$2$的强环面，并证明当只有一个因子是环时，差值可以无限大。 如果$\gamma_d^p(G) = \infty$，则对于每个图$H$，$\gamma_d^p(G\boxtimes H) = \infty$。 几个结果被证明支持该猜想，特别是如果$\gamma_d^p(C_m)= \infty$，则$\gamma_d^p(C_m \boxtimes C_n)=\infty$。