Perfect domination in regular grid graphs

We show there is an uncountable number of parallel total perfect codes in the integer lattice graph Λ of R. 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 Cm × Cn with parallel total perfect codes, and the d-perfect and total perfect code partitions of Λ and Cm×Cn, the former having as quotient graph the undirected Cayley graphs of Z2d2+2d+1 with generator set {1, 2d}. For r > 1, generalization for 1-perfect codes is provided in the integer lattice of R and in the products of r cycles, with partition quotient graph K2r+1 taken as the undirected Cayley graph of Z2r+1 with generator set {1, . . . , r}.

[1]  Italo J. Dejter,et al.  Perfect domination in rectangular grid graphs , 2007, 0711.4345.

[2]  Jan Kratochvíl,et al.  On the Computational Complexity of Codes in Graphs , 1988, MFCS.

[3]  Sueli I. Rodrigues Costa,et al.  Graphs, tessellations, and perfect codes on flat tori , 2004, IEEE Transactions on Information Theory.

[4]  Paul M. Weichsel,et al.  Dominating sets in n-cubes , 1994, J. Graph Theory.

[5]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[6]  Bader F. AlBdaiwi,et al.  Perfect Distance-d Placements in 2D Toroidal Networks , 2004, The Journal of Supercomputing.

[7]  Michael R. Fellows,et al.  Perfect domination , 1991, Australas. J Comb..

[8]  Miquel Àngel Fiol Mora,et al.  The Diameter of undirected graphs associated to plane tessellations , 1985 .

[9]  Quentin F. Stout,et al.  PERFECT DOMINATING SETS , 1990 .