Perfect domination in rectangular grid graphs

A dominating set S in a graph G is said to be perfect if every vertex of G not in S is adjacent to just one vertex of S. Given a vertex subset S of a side Pm of an m×n grid graph G, the perfect dominating sets S in G with S = S ∩ V (Pm) can be determined via an exhaustive algorithm Θ of running time O(2m+n). Extending Θ to infinite grid graphs of width m − 1, periodicity makes the binary decision tree of Θ prunable into a finite threaded tree, a closed walk of which yields all such sets S. The graphs induced by the complements of such sets S can be codified by arrays of ordered pairs of positive integers via Θ, for the growth and determination of which a speedier algorithm exists. A recent characterization of grid graphs having total perfect codes S (with just 1-cubes as induced components), due to Klostermeyer and Goldwasser, is given in terms of Θ, which allows to show that these sets S are restrictions of only one total perfect code S1 in the integer lattice graph Λ of R. Moreover, the complement Λ− S1 yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in Λ are in 1-1 correspondence with the doubly infinite {0, 1}-sequences.