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Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\exists\mathbb{R}$-Complete (2301.05550v1)

Published 13 Jan 2023 in cs.CG

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.

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