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

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.