TY - JOUR

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

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.A. and B.V. supported by ESF EUROCORES programme EuroGIGA, CRP ComPoSe, Austrian Science Fund (FWF): I648-N18. M.B. and P.V. supported by grant GAUK 690214, by project CE-ITI no. P202/12/G061 of the Czech Science Foundation GAČR, and by ERC Advanced Research Grant no. 267165 (DISCONV). T.H. supported by Austrian Science Fund (FWF): P23629-N18. A.P. supported by an Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J-3847-N35. P.R. supported by MINECO project MTM2014-54207, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain. An extended abstract of this paper appeared in the proceedings of the 28th International Workshop on Combinatorial Algorithms (IWOCA 2017).
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.A. and B.V. are supported by ESF EUROCORES programme Euro-GIGA – ComPoSe, Austrian Science Fund (FWF): I648-N18. M.B. and P.V. are supported by grant GAUK 690214, by project CE-ITI no. P202/12/G061 of the Czech Science Foundation GAČR, and by ERC Advanced Research Grant no. 267165 (DISCONV). T.H. is supported by Austrian Science Fund (FWF): P23629-N18. A.P. is supported by an Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J-3847-N35. P.R. is supported by MINECO project MTM2014-54207, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain. An extended abstract of this paper appeared in the proceedings of the 28th International Workshop on Combinatorial Algorithms (IWOCA 2017).
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/10/1

Y1 - 2018/10/1

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(logn)-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 in which every hole has size O(logn).

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(logn)-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 in which every hole has size O(logn).

UR - http://www.scopus.com/inward/record.url?scp=85048744397&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2018.06.002

DO - 10.1016/j.comgeo.2018.06.002

M3 - Article

AN - SCOPUS:85048744397

SN - 0925-7721

VL - 74

SP - 38

EP - 49

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

ER -