Determining unit distance graphs with coordinates in $\mathbb{Z}^2$ is NP-complete

Eric Binnendyk

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arXiv · 2510.15002

Determining unit distance graphs with coordinates in $\mathbb{Z}^2$ is NP-complete

Eric Binnendyk

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Abstract

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.

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Eric Binnendyk. 2025-10-16. Determining unit distance graphs with coordinates in $\mathbb{Z}^2$ is NP-complete. https://arxiv.org/abs/2510.15002

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