{"paper":{"title":"Plane and Planarity Thresholds for Random Geometric Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.CO"],"primary_cat":"cs.DM","authors_text":"Ahmad Biniaz, Anil Maheshwari, Evangelos Kranakis, Michiel Smid","submitted_at":"2018-09-27T19:36:25Z","abstract_excerpt":"A random geometric graph, $G(n,r)$, is formed by choosing $n$ points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most $r$. For a given constant $k$, we show that $n^{\\frac{-k}{2k-2}}$ is a distance threshold function for $G(n,r)$ to have a connected subgraph on $k$ points. Based on this, we show that $n^{-2/3}$ is a distance threshold for $G(n,r)$ to be plane, and $n^{-5/8}$ is a distance threshold to be planar. We also investigate distance thresholds for $G(n,r)$ to have a non-crossing edge, a cl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.10737","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":""},"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"}