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

TL;DR

This work studies the $d$-distance $p$-packing domination number $\\gamma_d^p(G)$ under the strong product of graphs. It proves the general bound $\\gamma_d^p(G\\boxtimes H) \\le \\gamma_d^p(G)\\gamma_d^p(H)$ and establishes sharp exact values for strong grids and prisms, namely $\\gamma_d^p(P_m\\boxtimes P_n)=\\left\\lceil \\frac{m}{2d+1}\\right\\lceil \\frac{n}{2d+1}\\right\\rceil$ (for $p\\le 2d$) and similar results with cycles when finite. It also presents non-sharpness phenomena for strong toruses, showing that the upper bound can be strictly exceeded and even have arbitrarily large gaps, along with constructions illustrating the limits of the bound. The paper proposes a central conjecture: if $\\gamma_d^p(G)=\\infty$, then $\\gamma_d^p(G\\boxtimes H)=\\infty$ for all $H$, and provides partial results supporting it, especially when $H$ contains a $(d,p)$-close vertex or when cycle factors exhibit infinity, linking to perfect codes and pendant-path arguments. These results clarify the landscape of $d$-distance $p$-packing domination under strong products and raise open problems about exact equality regimes and broader product behavior.

Abstract

The $d$-distance $p$-packing domination number $γ_d^p(G)$ 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 $γ_d^p(G) = \infty$. For an arbitrary strong product $G\boxtimes H$ it is proved that $γ_d^p(G\boxtimes H) \le γ_d^p(G) γ_d^p(H)$. By proving that $γ_d^p(P_m \boxtimes P_n) = \left \lceil \frac{m}{2d+1} \right \rceil \left \lceil \frac{n}{2d+1} \right \rceil$, and that if $γ_d^p(C_n) < \infty$, then $γ_d^p(P_m \boxtimes C_n) = \left \lceil \frac{m}{2d+1} \right \rceil \left \lceil \frac{n}{2d+1} \right \rceil$, 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 $γ_d^p(G) = \infty$, then $γ_d^p(G\boxtimes H) = \infty$ for every graph $H$. Several results are proved which support the conjecture, in particular, if $γ_d^p(C_m)= \infty$, then $γ_d^p(C_m \boxtimes C_n)=\infty$.

Figures (4)

• Figure 1: Sets $R_i$, $A_i$, and $B_i$ in $C_m\boxtimes C_n$ with respect to given $d$ and $p$, where $d < p <2d$.
• Figure 2: A schematic representation of $C_{11} \boxtimes C_{11}$; vertices from $X$ are marked with black squares. For clarity, not all edges are drawn. From left to right, the vertices which are $2$-distance dominated by the first three vertices from $X$ are marked with vertical black strips, the vertices which are $2$-distance dominated by the next three with horizontal black strips, and the vertices which are $2$-distance dominated by the final one with color gray.
• Figure 3: A schematic representation of $C_N \boxtimes C_{11}$; vertices from $S$ are marked with black squares. For clarity, not all edges are drawn. Vertices which are $2$-distance dominated by the first triple are marked with horizontal black strips, vertices which are $2$-distance dominated by the last triple with vertical black strips, and vertices which are $2$-distance dominated by the final vertex with gray color. The remaining white vertices are $2$-distance dominated by the remaining triples.
• Figure 4: A schematic representation of $C_{16} \boxtimes C_{16}$, i.e. $m = n = 16$, $d = p = 3$, $q = r = 2$ and $m_1 = n_1 = 2$. Vertices from $Z \cup A \cup B \cup \{z^*, c^*\}$ are marked with black squares and their labels are written on the side of the graph. For clarity, not all edges are drawn. The vertices $d$-distance dominated by $a(1)$ and by $c^*$ are marked with color gray.

Theorems & Definitions (24)

• Proposition 2.1
• proof
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• proof
• Conjecture 3.1
• Remark 3.2
• Theorem 3.3
• proof