Abstract
For a point set P on the plane, a four element subset S ⊂ P is called a 4-hole of P if the convex hull of S is a quadrilateral and contains no point of P in its interior. Let R be a point set on the plane. We say that a point set B covers all the 4-holes of R if any 4-hole of R contains an element of B in its interior. We show that if |R|≥ 2|B| + 5 then B cannot cover all the 4-holes of R. A similar result is shown for a point set R in convex position. We also show a point set R for which any point set B that covers all the 4-holes of R has approximately 2|R| points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Czyzowicz, J., Kranakis, E., Urrutia, J.: Dissections, cuts and triangulations. Proceedings of the 11th Canadian Conference on Computational Geometry, CCCG-99, pp. 154–157, Vancouver August 15–18, 1999
Devillers, O., Hurtado, F., Károlyi, G., Seara, C.: Chromatic variants of the Erdős-Szekeres theorem on points in convex position. Comput. Geom. Theory Appl. 26, 193–208 (2003)
Heineken, H.: Regelmassige Vielecke und ihre Diagonalen. Enseignement Math. 2(8), 275–278 (1962)
Kalbfleisch J.D., Kalbfleisch, J.G., Stanton, R.G.: A combinatorial problem on convex n-gons. Proc. Louisiana Conf. on Combinatorics, Graph Theory and Computing, pp. 180–188, 1970
Katchalski, M., Meir, A.: On empty triangles determined by points in the plane. Acta Math. Hungar. 51, 323–328 (1988)
Moser, W.: Research Problems in Discrete Geometry, McGill Univ., 1981
Author information
Authors and Affiliations
Additional information
Supported by CONACYT of Mexico, Proyecto SEP-2004-Co1-45876, and PAPIIT (UNAM), Proyecto IN110802.
Rights and permissions
About this article
Cite this article
Sakai, T., Urrutia, J. Covering the Convex Quadrilaterals of Point Sets. Graphs and Combinatorics 23 (Suppl 1), 343–357 (2007). https://doi.org/10.1007/s00373-007-0717-0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00373-007-0717-0