Perfect domination in regular grid graphs
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 1perfect 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 1perfect 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 $\{1,...,r\}$.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.4343
 Bibcode:
 2007arXiv0711.4343D
 Keywords:

 Mathematics  Combinatorics;
 05C69
 EPrint:
 16 pages