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arXiv · 2301.05550

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

Nicholas Bieker·Thomas Bläsius·Emil Dohse·Paul Jungeblut
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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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Nicholas Bieker, Thomas Bläsius, Emil Dohse, Paul Jungeblut. 2023-01-13. Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\exists\mathbb{R}$-Complete. https://arxiv.org/abs/2301.05550

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