{"paper":{"title":"Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\\exists\\mathbb{R}$-Complete","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Emil Dohse, Nicholas Bieker, Paul Jungeblut, Thomas Bl\\\"asius","submitted_at":"2023-01-13T13:55:03Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.05550","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2301.05550/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}