Perfect domination in regular grid graphs

TL;DR

This paper explores the existence and characterization of perfect codes in regular grid graphs and their products, revealing uncountably many in the integer lattice and unique codes in certain restricted cases.

Contribution

It provides a comprehensive characterization of perfect codes in grid graphs and their products, including new results on partitions and generalizations to higher dimensions.

Findings

Uncountably many parallel total perfect codes in the integer lattice.

Only one 1-perfect code and one total perfect code in the lattice with rectangular restrictions.

Complete characterization of cycle product graphs with parallel total perfect codes.

Abstract

We show there is an uncountable number of parallel total perfect codes in the integer lattice graph ${\Lambda}$ of $\R^2$. In contrast, there is just one 1-perfect code in ${\Lambda}$ and one total perfect code in ${\Lambda}$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of ${\Lambda}$ and $C_m\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\Z_{2d^2+2d+1}$ with generator set $\{1,2d^2\}$. For $r>1$, generalization for 1-perfect codes is provided in the integer lattice of $\R^r$ and in the products of $r$ cycles, with partition quotient graph $K_{2r+1}$ taken as the undirected Cayley graph of $\Z_{2r+1}$ with generator set $\{1,...,r\}$.