# 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 contributionResearchpeer-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 Springer Verlag Heidelberg 169-181 13 10765 9783319788241 https://doi.org/10.1007/978-3-319-78825-8_14 Published - 1 Jan 2018 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 10765 0302-9743 1611-3349

### Conference

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

### Fingerprint

Point Sets
Convex Sets
Interior
Simple Polygon
Integer
Convex polygon
Collinear
Intersect
Connected Components
Straight Line
Large Set

### Keywords

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

### ASJC Scopus subject areas

• Theoretical Computer Science
• Computer Science(all)

### Cite this

Aichholzer, O., Balko, M., Hackl, T., Pilz, A., Ramos, P., Valtr, P., & Vogtenhuber, B. (2018). Holes in 2-convex point sets. In Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers (Vol. 10765, pp. 169-181). (Lecture Notes in Computer Science ; Vol. 10765 ). Springer Verlag Heidelberg. https://doi.org/10.1007/978-3-319-78825-8_14

Holes in 2-convex point sets. / Aichholzer, Oswin; Balko, Martin; Hackl, Thomas; Pilz, Alexander; Ramos, Pedro; Valtr, Pavel; Vogtenhuber, Birgit.

Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers. Vol. 10765 Springer Verlag Heidelberg, 2018. p. 169-181 (Lecture Notes in Computer Science ; Vol. 10765 ).

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

Aichholzer, O, Balko, M, Hackl, T, Pilz, A, Ramos, P, Valtr, P & Vogtenhuber, B 2018, Holes in 2-convex point sets. in Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers. vol. 10765, Lecture Notes in Computer Science , vol. 10765 , Springer Verlag Heidelberg, pp. 169-181, 28th International Workshop on Combinational Algorithms, IWOCA 2017, Newcastle, NSW, Australia, 17/07/17. https://doi.org/10.1007/978-3-319-78825-8_14
Aichholzer O, Balko M, Hackl T, Pilz A, Ramos P, Valtr P et al. Holes in 2-convex point sets. In Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers. Vol. 10765. Springer Verlag Heidelberg. 2018. p. 169-181. (Lecture Notes in Computer Science ). https://doi.org/10.1007/978-3-319-78825-8_14
Aichholzer, Oswin ; Balko, Martin ; Hackl, Thomas ; Pilz, Alexander ; Ramos, Pedro ; Valtr, Pavel ; Vogtenhuber, Birgit. / Holes in 2-convex point sets. Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers. Vol. 10765 Springer Verlag Heidelberg, 2018. pp. 169-181 (Lecture Notes in Computer Science ).
@inproceedings{d973006f2a9a484fbdb9f3d3e5ec126b,
title = "Holes in 2-convex point sets",
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).",
keywords = "2-convex set, Convex position, Hole, Horton set, Point set",
author = "Oswin Aichholzer and Martin Balko and Thomas Hackl and Alexander Pilz and Pedro Ramos and Pavel Valtr and Birgit Vogtenhuber",
year = "2018",
month = "1",
day = "1",
doi = "10.1007/978-3-319-78825-8_14",
language = "English",
isbn = "9783319788241",
volume = "10765",
series = "Lecture Notes in Computer Science",
publisher = "Springer Verlag Heidelberg",
pages = "169--181",
booktitle = "Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers",

}

TY - GEN

T1 - Holes in 2-convex point sets

AU - Aichholzer, Oswin

AU - Balko, Martin

AU - Hackl, Thomas

AU - Pilz, Alexander

AU - Ramos, Pedro

AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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).

AB - 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).

KW - 2-convex set

KW - Convex position

KW - Hole

KW - Horton set

KW - Point set

UR - http://www.scopus.com/inward/record.url?scp=85045984952&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-78825-8_14

DO - 10.1007/978-3-319-78825-8_14

M3 - Conference contribution

SN - 9783319788241

VL - 10765

T3 - Lecture Notes in Computer Science

SP - 169

EP - 181

BT - Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers

PB - Springer Verlag Heidelberg

ER -