@inproceedings{f64b7d1c9da449f39bd9228cc63880b1,

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).",

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",

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",

year = "2018",

month = jan,

day = "1",

doi = "10.1007/978-3-319-78825-8_14",

language = "English",

isbn = "9783319788241",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "169--181",

editor = "Smyth, {William F.} and Ljiljana Brankovic and Joe Ryan",

booktitle = "Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers",

address = "Germany",

}