Holes in 2-convex point sets

Oswin Aichholzer, Martin Balko, Thomas Hackl, Alexander Pilz, Pedro Ramos, Pavel Valtr, Birgit Vogtenhuber*

*Korrespondierende/r Autor/-in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikelBegutachtung

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(log⁡n)-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(log⁡n).

Originalspracheenglisch
Seiten (von - bis)38-49
Seitenumfang12
FachzeitschriftComputational Geometry
Jahrgang74
DOIs
PublikationsstatusVeröffentlicht - 1 Okt. 2018

ASJC Scopus subject areas

  • Angewandte Informatik
  • Geometrie und Topologie
  • Steuerung und Optimierung
  • Theoretische Informatik und Mathematik
  • Computational Mathematics

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