Holes in Convex and Simple Drawings
Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erd\H{o}s--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and survey generaliz...
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| Published in: | Journal of graph algorithms and applications Vol. 29; no. 3; pp. 23 - 38 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
15.10.2025
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| ISSN: | 1526-1719, 1526-1719 |
| Online Access: | Get full text |
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| Summary: | Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erd\H{o}s--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and survey generalizations thereof, like empty $k$-cycles. 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 is the structural investigation of pseudolinear subdrawings in convex~drawings. With respect to empty $k$-cycles, we show the existence of empty $4$-cycles in every simple drawing of $K_n$ and give a construction that admits only $\Theta(n^2)$ of them. |
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| ISSN: | 1526-1719 1526-1719 |
| DOI: | 10.7155/jgaa.v29i3.2999 |