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

TL;DR

This paper proves the uniqueness of a specific dominating set in the lattice  whose induced components are exclusively 4-cycles, highlighting a unique structural property of the lattice's unit distance graph.

Contribution

The paper establishes the uniqueness of a dominating set with 4-cycle components in the  lattice's unit distance graph, a previously unknown structural result.

Findings

The dominating set with 4-cycle components is unique in .

Each vertex outside the dominating set has a unique neighbor in the set.

The structure of the dominating set is rigid and uniquely determined.

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.