On computational complexity of length embeddability of graphs

TL;DR

This paper proves that determining whether a graph can be embedded in Euclidean space of dimension greater than two with unit distances is an NP-hard problem, highlighting its computational difficulty.

Contribution

It establishes the NP-hardness of the graph embeddability problem in dimensions greater than two for various notions of embeddability.

Findings

NP-hardness of embeddability verification in $\

for all reasonable notions of embeddability in $\

dimensions greater than 2.

Abstract

A graph $G$ is embeddable in $\mathbb{R}^d$ if vertices of $G$ can be assigned with points of $\mathbb{R}^d$ in such a way that all pairs of adjacent vertices are at the distance 1. We show that verifying embeddability of a given graph in $\mathbb{R}^d$ is NP-hard in the case $d > 2$ for all reasonable notions of embeddability.