### 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 |
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Title of host publication | Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers |

Publisher | Springer Verlag Heidelberg |

Pages | 169-181 |

Number of pages | 13 |

Volume | 10765 |

ISBN (Print) | 9783319788241 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

Event | 28th International Workshop on Combinational Algorithms, IWOCA 2017 - Newcastle, NSW, Australia Duration: 17 Jul 2017 → 21 Jul 2017 |

### Publication series

Name | Lecture Notes in Computer Science |
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Volume | 10765 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 28th International Workshop on Combinational Algorithms, IWOCA 2017 |
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Country | Australia |

City | Newcastle, NSW |

Period | 17/07/17 → 21/07/17 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review

*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

}

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 -