We study algorithms for drawing planar graphs and 1-planar graphs using cubic Bézier curves with bounded curvature. We show that any n-vertex 1-planar graph has a 1-planar RAC drawing using a single cubic Bézier curve per edge, and this drawing can be computed in O(n) time given a combinatorial 1-planar drawing. We also show that any n-vertex planar graph G can be drawn in O(n) time with a single cubic Bézier curve per edge, in an O(n)× O(n) bounding box, such that the edges have Θ(1/degree(v)) angular resolution, for each v ∈ G, and O(√n) curvature.
@InProceedings{eppstein_et_al:LIPIcs.GD.2024.39, author = {Eppstein, David and Goodrich, Michael T. and Illickan, Abraham M.}, title = {{Drawing Planar Graphs and 1-Planar Graphs Using Cubic B\'{e}zier Curves with Bounded Curvature}}, booktitle = {32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)}, pages = {39:1--39:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-343-0}, ISSN = {1868-8969}, year = {2024}, volume = {320}, editor = {Felsner, Stefan and Klein, Karsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.39}, URN = {urn:nbn:de:0030-drops-213237}, doi = {10.4230/LIPIcs.GD.2024.39}, annote = {Keywords: graph drawing, planar graphs, B\'{e}zier curves, and RAC drawings} }
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