Recognizing Unit Disk Graphs in Hyperbolic Geometry is ∃ℝ-Complete

01/13/2023

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by   Nicholas Bieker, et al.

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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.