# Holes in 2-convex point sets

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

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

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

Original language English Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers William F. Smyth, Ljiljana Brankovic, Joe Ryan Springer Verlag 169-181 13 9783319788241 https://doi.org/10.1007/978-3-319-78825-8_14 Published - 1 Jan 2018 Yes 28th International Workshop on Combinational Algorithms, IWOCA 2017 - Newcastle, NSW, AustraliaDuration: 17 Jul 2017 → 21 Jul 2017

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 10765 LNCS 0302-9743 1611-3349

### Conference

Conference 28th International Workshop on Combinational Algorithms, IWOCA 2017 Australia Newcastle, NSW 17/07/17 → 21/07/17

## Keywords

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

## ASJC Scopus subject areas

• Theoretical Computer Science
• Computer Science (all)

## Fingerprint

Dive into the research topics of 'Holes in 2-convex point sets'. Together they form a unique fingerprint.