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.