Holes in 2-convex point sets

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

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review


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 with holes of size at most O(log n).

Original languageEnglish
Title of host publicationCombinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers
PublisherSpringer Verlag Heidelberg
Number of pages13
ISBN (Print)9783319788241
Publication statusPublished - 1 Jan 2018
Event28th International Workshop on Combinational Algorithms, IWOCA 2017 - Newcastle, NSW, Australia
Duration: 17 Jul 201721 Jul 2017

Publication series

NameLecture Notes in Computer Science
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference28th International Workshop on Combinational Algorithms, IWOCA 2017
CityNewcastle, NSW


  • 2-convex set
  • Convex position
  • Hole
  • Horton set
  • Point set

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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