Holes in 2-convex point sets

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

doi:10.1007/978-3-319-78825-8_14

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 proceeding › Conference contribution › peer-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
Title of host publication	Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers
Editors	William F. Smyth, Ljiljana Brankovic, Joe Ryan
Publisher	Springer Verlag
Pages	169-181
Number of pages	13
ISBN (Print)	9783319788241
DOIs	https://doi.org/10.1007/978-3-319-78825-8_14
State	Published - 1 Jan 2018
Externally published	Yes
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 (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	10765 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	28th International Workshop on Combinational Algorithms, IWOCA 2017
Country/Territory	Australia
City	Newcastle, NSW
Period	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)

Access to Document

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

Cite this

Aichholzer, O., Balko, M., Hackl, T., Pilz, A., Ramos, P., Valtr, P., & Vogtenhuber, B. (2018). Holes in 2-convex point sets. In W. F. Smyth, L. Brankovic, & J. Ryan (Eds.), Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers (pp. 169-181). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10765 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-78825-8_14

Aichholzer, Oswin ; Balko, Martin ; Hackl, Thomas et al. / Holes in 2-convex point sets. Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers. editor / William F. Smyth ; Ljiljana Brankovic ; Joe Ryan. Springer Verlag, 2018. pp. 169-181 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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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 Erd{\H o}s-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).",

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author = "Oswin Aichholzer and Martin Balko and Thomas Hackl and Alexander Pilz and Pedro Ramos and Pavel Valtr and Birgit Vogtenhuber",

note = "Funding Information: Research supported by OEAD project CZ 18/2015 and by project no. 7AMB15A T023 of the Ministry of Education of the Czech Republic. O. Aich-holzer and B. Vogtenhuber supported by ESF EUROCORES programme Euro-GIGA - ComPoSe, Austrian Science Fund (FWF): I648-N18. M. Balko and P. Valtr supported by grant GAUK 690214, by project CE-ITI no. P202/12/G061 of the Czech Science Foundation GACˇR, and by ERC Advanced Research Grant no 267165 (DISCONV). T. Hackl supported by Austrian Science Fund (FWF): P23629-N18. A. Pilz supported by an Erwin Schr{\"o}dinger fellowship, Austrian Science Fund (FWF): J-3847-N35. P. Ramos supported by MINECO project MTM2014-54207, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain. Publisher Copyright: {\textcopyright} Springer International Publishing AG, part of Springer Nature 2018.; 28th International Workshop on Combinational Algorithms, IWOCA 2017 ; Conference date: 17-07-2017 Through 21-07-2017",

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Aichholzer, O, Balko, M, Hackl, T, Pilz, A, Ramos, P, Valtr, P & Vogtenhuber, B 2018, Holes in 2-convex point sets. in WF Smyth, L Brankovic & J Ryan (eds), Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10765 LNCS, Springer Verlag, 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

Holes in 2-convex point sets. / Aichholzer, Oswin; Balko, Martin; Hackl, Thomas et al.

Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers. ed. / William F. Smyth; Ljiljana Brankovic; Joe Ryan. Springer Verlag, 2018. p. 169-181 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10765 LNCS).

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

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AU - Aichholzer, Oswin

AU - Balko, Martin

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AU - Pilz, Alexander

AU - Ramos, Pedro

AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

N1 - Funding Information: Research supported by OEAD project CZ 18/2015 and by project no. 7AMB15A T023 of the Ministry of Education of the Czech Republic. O. Aich-holzer and B. Vogtenhuber supported by ESF EUROCORES programme Euro-GIGA - ComPoSe, Austrian Science Fund (FWF): I648-N18. M. Balko and P. Valtr supported by grant GAUK 690214, by project CE-ITI no. P202/12/G061 of the Czech Science Foundation GACˇR, and by ERC Advanced Research Grant no 267165 (DISCONV). T. Hackl supported by Austrian Science Fund (FWF): P23629-N18. A. Pilz supported by an Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J-3847-N35. P. Ramos supported by MINECO project MTM2014-54207, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain. Publisher Copyright: © Springer International Publishing AG, part of Springer Nature 2018.

PY - 2018/1/1

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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).

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KW - Convex position

KW - Hole

KW - Horton set

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Aichholzer O, Balko M, Hackl T, Pilz A, Ramos P, Valtr P et al. Holes in 2-convex point sets. In Smyth WF, Brankovic L, Ryan J, editors, Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers. Springer Verlag. 2018. p. 169-181. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-319-78825-8_14

Holes in 2-convex point sets

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