We show that every planar graph has a monotone topological 2-page book embedding where at most (4n-10)/5 (of potentially 3n-6) edges cross the spine, and every edge crosses the spine at most once; such an edge is called a biarc. We can also guarantee that all edges that cross the spine cross it in the same direction (e.g., from bottom to top). For planar 3-trees we can further improve the bound to (3n-9)/4, and for so-called Kleetopes we obtain a bound of at most (n-8)/3 edges that cross the spine. The bound for Kleetopes is tight, even if the drawing is not required to be monotone. A Kleetope is a plane triangulation that is derived from another plane triangulation T by inserting a new vertex v_f into each face f of T and then connecting v_f to the three vertices of f.
@InProceedings{chaplick_et_al:LIPIcs.GD.2024.11, author = {Chaplick, Steven and F\"{o}rster, Henry and Hoffmann, Michael and Kaufmann, Michael}, title = {{Monotone Arc Diagrams with Few Biarcs}}, booktitle = {32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)}, pages = {11:1--11:16}, 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.2024.11}, URN = {urn:nbn:de:0030-drops-212955}, doi = {10.4230/LIPIcs.GD.2024.11}, annote = {Keywords: planar graph, topological book embedding, monotone drawing, linear layout} }
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