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 R 2 . Recognizing them is known to be ∃R-complete, i.e., as hard as solving a system of polynomial inequalities. In this note we describe a simple framework to translate ∃R-hardness reductions from the Euclidean plane R 2 to the hyperbolic plane H 2 . We apply our framework to prove that the recognition of unit disk graphs in the hyperbolic plane is also ∃R-complete.