Abstract
Let S be a set of n points in the plane in general position (no three points from S are collinear). For a positive integer k, a k-hole in S is a convex polygon with k vertices from S and no points of S in its interior. For a positive integer l, a simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. A point set S is l-convex if there exists an l-convex polygonization of S. Considering a typical Erdős–Szekeres-type problem, we show that every 2-convex point set of size n contains an Ω(logn)-hole. In comparison, it is well known that there exist arbitrarily large point sets in general position with no 7-hole. Further, we show that our bound is tight by constructing 2-convex point sets in which every hole has size O(logn).
Originalsprache | englisch |
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Seiten (von - bis) | 38-49 |
Seitenumfang | 12 |
Fachzeitschrift | Computational Geometry |
Jahrgang | 74 |
DOIs | |
Publikationsstatus | Veröffentlicht - 1 Okt. 2018 |
ASJC Scopus subject areas
- Angewandte Informatik
- Geometrie und Topologie
- Steuerung und Optimierung
- Theoretische Informatik und Mathematik
- Computational Mathematics