{"paper":{"title":"Universal Slope Sets for 1-Bend Planar Drawings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Fabrizio Montecchiani, Giuseppe Liotta, Michael A. Bekos, Patrizio Angelini","submitted_at":"2017-03-13T08:00:31Z","abstract_excerpt":"We describe a set of $\\Delta -1$ slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree $\\Delta \\geq 4$; this establishes a new upper bound of $\\Delta-1$ on the 1-bend planar slope number. By universal we mean that every planar graph of degree $\\Delta$ has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was $\\frac{3}{2} (\\Delta -1)$ (the kn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.04283","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"}