Perfect domination in rectangular grid graphs

TL;DR

This paper investigates perfect dominating sets in grid graphs, providing algorithms for their enumeration, characterizing their structure, and exploring their periodicity and tiling properties.

Contribution

It introduces an exhaustive algorithm for perfect dominating sets in finite grids, extends it to infinite grids with periodic structures, and characterizes total perfect codes using these methods.

Findings

An $O(2^{m+n})$ algorithm for perfect dominating sets in finite grids.

Periodic structures of perfect dominating sets relate to aperiodic tilings like Penrose tiling.

Total perfect codes in the lattice are characterized by binary sequences and relate to known tilings.

Abstract

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 $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$ in $G$ with $S'=S\cap V(P_m)$ can be determined via an exhaustive algorithm $\Theta$ of running time $O(2^{m+n})$. Extending $\Theta$ to infinite grid graphs of width $m-1$, periodicity makes the binary decision tree of $\Theta$ 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 $\Theta$, for the growth and determination of which a speedier %greedy algorithm exists. %and their periodic structure, further studied. 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 $\Theta$, which allows to show that these sets $S$ are restrictions of only one total perfect code $S_1$ in the integer lattice graph ${\Lambda}$ of $\R^2$. Moreover, the complement ${\Lambda}-S_1$ yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in ${\Lambda}$ are in 1-1 correspondence with the doubly infinite $\{0,1\}$-sequences.