The problem of determining whether a graph $G$ can be realized as a unit-distance graph in $\mathbb{Z}^2$ is NP-complete. As far as we can tell, a proof of this result has never been written up. We prove NP-completeness of this problem by implementing Eades and Whitesides' logic engine in this setting, and construct a graph that is realizable if and only if an arbitrary NA3SAT formula is satisfiable.

翻译：判定一个图$G$能否在$\mathbb{Z}^2$中实现为单位距离图的问题是NP完全的。据我们所知，这一结果的证明从未被正式发表。我们通过在此场景下实现Eades和Whitesides的逻辑引擎，并构造一个当且仅当任意NA3SAT公式可满足时才可实现的图，从而证明了该问题的NP完全性。