Abstract
Let S be a set of n points in general position in the plane. A k-island I of S is a subset of k points of S such that Conv(I) ∩ S = I. We show that, for an arbitrary but fixed number k ≥ 2, the minimum number of k-islands among all sets S of n points is Θ(n2). The following related counting problem is also studied: For l < k, an l-island covers a k-island if it is contained in the k-island. Let Ck,l(S) be the minimum number of l-islands needed to cover all the k-islands of S and let Ck,l(n) be the minimum of Ck,l(S) among all sets S of n points. We show asymptotic bounds for Ck,l(n).
Research partially supported by Conacyt of Mexico, grant 153984 and MEC MTM2009-07242 and Gen.Cat. DGR 2009SGR1040 and ESF EUROCORES programme EuroGIGA - ComPoSe IP04 - MICINN Project EUI-EURC-2011-4306.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aronov, B., Dulieu, M., Hurtado, F.: Witness (Delaunay) Graphs. Comput. Geom. 44, 329–344 (2011)
Bárány, I., Füredi, Z.: Empty simplices in Euclidean space. Canad. Math. Bull. 30, 436–445 (1987)
Bárány, I., Valtr, P.: Planar point sets with a small number of empty convex polygons. Stud. Sci. Math. Hung. 41, 243–269 (2004)
Bautista-Santiago, C., Cano, J., Fabila-Monroy, R., Flores-Peñaloza, D., González-Aguilar, H., Lara, D., Sarmiento, E., Urrutia, J.: On the Diameter of a Geometric Johnson Type Graph. In: 26th European Workshop on Computational Geometry, pp. 61–64 (2010)
Bautista-Santiago, C., Díaz-Báñez, J.M., Lara, D., Pérez-Lantero, P., Urrutia, J., Ventura, I.: Computing Maximal Islands. In: 25th European Workshop on Computational Geometry, pp. 333–336 (2009)
Bautista-Santiago, C., Heredia, M.A., Huemer, C., Ramírez-Vigueras, A., Seara, C., Urrutia, J.: On the number of edges in geometric graphs without empty triangles. Submitted to Journal (January 2011)
Devillers, O., Hurtado, F., Kàrolyi, G., Seara, C.: Chromatic variants of the Erdős-Szekeres Theorem. Comput. Geom. 26, 193–208 (2003)
Dumitrescu, A.: Planar sets with few empty convex polygons. Stud. Sci. Math. Hung. 36, 93–109 (2000)
Gerken, T.: Empty convex hexagons in planar point sets. Discrete Comput. Geom. 39, 239–272 (2008)
Harborth, H.: Konvexe Fünfecke in ebenen Punktmengen. Elem. Math. 33, 116–118 (1978)
Horton, J.D.: Sets with no empty convex 7-gons. Canad. Math. Bull. 26, 482–484 (1983)
Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane — a survey. In: Algorithms Combin., vol. 25, pp. 551–570. Springer (2003)
Katchalski, M., Meir, A.: On empty triangles determined by points in the plane. Acta Math. Hung. 51, 323–328 (1988)
Nicolás, C.M.: The empty hexagon theorem. Discrete Comput. Geom. 38, 389–397 (2007)
Sakai, T., Urrutia, J.: Covering the convex quadrilaterals of point sets. Graphs Combin. 23, 343–357 (2007)
Valtr, P.: On the minimum number of empty polygons in planar point sets. Stud. Sci. Math. Hungar. 30, 155–163 (1995)
Valtr, P.: On empty hexagons. In: Surveys on Discrete and Computational Geometry, Twenty Years Later, pp. 433–441. AMS (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fabila-Monroy, R., Huemer, C. (2012). Covering Islands in Plane Point Sets. In: Márquez, A., Ramos, P., Urrutia, J. (eds) Computational Geometry. EGC 2011. Lecture Notes in Computer Science, vol 7579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34191-5_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-34191-5_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34190-8
Online ISBN: 978-3-642-34191-5
eBook Packages: Computer ScienceComputer Science (R0)