There is but one PDS in ${\mathbb{Z}}^3$ inducing just square components

Abstract

It is known that in the unit distance graph of the lattice ${\mathbb{Z}}^3\subset {\mathbb{R}}^3$ there exists a dominating set $S$ with $4$-cycles as sole induced components and each vertex of ${\mathbb{Z}}^3\setminus S$ having a unique neighbor in $S$. We show $S$ is unique.

22 pages, 14 figures