{"paper":{"title":"The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bernard Lidick\\'y, Eva Jel\\'inkov\\'a, Jan Kratochv\\'il, Marek Tesa\\v{r}, Tom\\v{s} Vysko\\v{c}il, V\\'it Jel\\'inek","submitted_at":"2010-12-19T00:46:05Z","abstract_excerpt":"It is known that every planar graph has a planar embedding where edges are represented by non-crossing straight-line segments. We study the planar slope number, i.e., the minimum number of distinct edge-slopes in such a drawing of a planar graph with maximum degree $\\Delta$. We show that the planar slope number of every planar partial 3-tree and also every plane partial 3-tree is at most $O(\\Delta^5)$. In particular, we answer the question of Dujmovi\\'c et al. [Computational Geometry 38 (3), pp. 194--212 (2007)] whether there is a function $f$ such that plane maximal outerplanar graphs can be "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.4137","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"}