Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\exists\mathbb{R}$-Complete

TL;DR

This paper demonstrates that recognizing unit disk graphs in hyperbolic geometry is computationally as hard as in Euclidean geometry, establishing its $orall ext{R}$-completeness through a new translation framework.

Contribution

The authors introduce a framework to transfer $orall ext{R}$-hardness results from Euclidean to hyperbolic geometry, proving the recognition problem's complexity in hyperbolic space.

Findings

Recognition of unit disk graphs in hyperbolic space is $orall ext{R}$-complete.

Framework enables transferring hardness results between geometries.

Extends understanding of geometric graph recognition problems.

Abstract

A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the Euclidean plane $\mathbb{R}^2$. Recognizing them is known to be $\exists\mathbb{R}$-complete, i.e., as hard as solving a system of polynomial inequalities. In this note we describe a simple framework to translate $\exists\mathbb{R}$-hardness reductions from the Euclidean plane $\mathbb{R}^2$ to the hyperbolic plane $\mathbb{H}^2$. We apply our framework to prove that the recognition of unit disk graphs in the hyperbolic plane is also $\exists\mathbb{R}$-complete.