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 Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandBegutachtung


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).

TitelCombinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers
Herausgeber (Verlag)Springer Verlag Heidelberg
ISBN (Print)9783319788241
PublikationsstatusVeröffentlicht - 1 Jan. 2018
Veranstaltung28th International Workshop on Combinational Algorithms, IWOCA 2017 - Newcastle, NSW, Australien
Dauer: 17 Juli 201721 Juli 2017


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


Konferenz28th International Workshop on Combinational Algorithms, IWOCA 2017
OrtNewcastle, NSW

ASJC Scopus subject areas

  • Theoretische Informatik
  • Informatik (insg.)


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