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Perfect domination in regular grid graphs. (English)
Australas. J. Comb. 42, 99-114 (2008).
Summary: We show there is an uncountable number of parallel total perfect codes in the integer lattice graph $Λ$ of $\Bbb R^2$ In contrast, there is just one 1-perfect code in $Λ$ and one total perfect code in $Λ$ 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 $Λ$ and $C_m\times C_n$ the former having as quotient graph the undirected Cayley graphs of $\Bbb Z_{2d^2+2d+1}$ with generator set $\{1,2d^2\}$. For $r>1$, generalization for 4-perfect codes is provided in the integer lattice of $\Bbb R^r$ and in the products of $r$ cycles, with partition quotient graph $K_{2r+1}$ taken as the undirected Cayley graph of $\Bbb Z_{2r+1}$ with generator set $\{1,\dots,r\}$.