{"paper":{"title":"Towards Tight Bounds on Theta-Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Andr\\'e van Renssen, Jean-Lou De Carufel, Pat Morin, Prosenjit Bose, Sander Verdonschot","submitted_at":"2014-04-24T19:24:30Z","abstract_excerpt":"We present improved upper and lower bounds on the spanning ratio of $\\theta$-graphs with at least six cones. Given a set of points in the plane, a $\\theta$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $\\theta=2\\pi/m$, and adds an edge to the `closest' vertex in each cone. We show that for any integer $k \\geq 1$, $\\theta$-graphs with $4k+2$ cones have a spanning ratio of $1+2\\sin(\\theta/2)$ and we provide a matching lower bound, showing that this spanning ratio tight.\n  Next, we show that for any integer $k \\geq 1$, $\\theta$-graphs with $4k+4$ cone"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.6233","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"}