Determining unit distance graphs with coordinates in $\mathbb{Z}^2$ is NP-complete
Computational Complexity
2026-05-25 v3
Abstract
The problem of determining whether a graph can be realized as a unit-distance graph in 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.
Keywords
Cite
@article{arxiv.2510.15002,
title = {Determining unit distance graphs with coordinates in $\mathbb{Z}^2$ is NP-complete},
author = {Eric Binnendyk},
journal= {arXiv preprint arXiv:2510.15002},
year = {2026}
}
Comments
It turns out that this result is already known. A stronger result (about unit-distance trees instead of arbitrary graphs) is: Sandeep N. Bhatt, Stavros S. Cosmadakis, The complexity of minimizing wire lengths in VLSI layouts, 1987