Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erdős-Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of k-holes for simple drawings and study their existence with respect to the convexity hierarchy. We present a family of simple drawings without 4-holes and prove a generalization of Gerken’s empty hexagon theorem for convex drawings. A crucial intermediate step will be the structural investigation of pseudolinear subdrawings in convex drawings.
@InProceedings{bergold_et_al:LIPIcs.GD.2024.5, author = {Bergold, Helena and Orthaber, Joachim and Scheucher, Manfred and Schr\"{o}der, Felix}, title = {{Holes in Convex and Simple Drawings}}, booktitle = {32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)}, pages = {5:1--5:9}, 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.2024.5}, URN = {urn:nbn:de:0030-drops-212895}, doi = {10.4230/LIPIcs.GD.2024.5}, annote = {Keywords: simple topological graph, convexity hierarchy, k-gon, k-hole, empty k-cycle, Erd\H{o}s-Szekeres theorem, Empty Hexagon theorem, planar point set, pseudolinear drawing} }
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