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Covering islands in plane point sets. (English) Zbl 1374.52021
Márquez, Alberto (ed.) et al., Computational geometry. XIV Spanish meeting on computational geometry, EGC 2011, dedicated to Ferran Hurtado on the occasion of his 60th birthday, Alcalá de Henares, Spain, June 27–30, 2011. Revised selected papers. Berlin: Springer (ISBN 978-3-642-34190-8/pbk). Lecture Notes in Computer Science 7579, 220-225 (2012).
Summary: 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 $$\mathrm{Conv}(I) \cap S = I$$. We show that, for an arbitrary but fixed number $$k \geq 2$$, the minimum number of $$k$$-islands among all sets $$S$$ of $$n$$ points is $$\Theta (n^{2})$$. 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 $$C_{k,l}(S)$$ be the minimum number of $$l$$-islands needed to cover all the $$k$$-islands of $$S$$ and let $$C _{k,l }(n)$$ be the minimum of $$C_{k,l }(S)$$ among all sets $$S$$ of $$n$$ points. We show asymptotic bounds for $$C _{k,l }(n)$$.
For the entire collection see [Zbl 1253.68016].
##### MSC:
 52C10 Erdős problems and related topics of discrete geometry
##### Keywords:
planar point sets; islands; Horton sets
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##### References:
 [1] Aronov, B., Dulieu, M., Hurtado, F.: Witness (Delaunay) Graphs. Comput. Geom. 44, 329–344 (2011) · Zbl 1232.05190 · doi:10.1016/j.comgeo.2011.01.001 [2] Bárány, I., Füredi, Z.: Empty simplices in Euclidean space. Canad. Math. Bull. 30, 436–445 (1987) · Zbl 0639.52006 · doi:10.4153/CMB-1987-064-1 [3] Bárány, I., Valtr, P.: Planar point sets with a small number of empty convex polygons. Stud. Sci. Math. Hung. 41, 243–269 (2004) · Zbl 1075.52008 [4] 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) · Zbl 1283.05147 [5] 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) · Zbl 1242.90183 [6] 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) · Zbl 1290.05064 [7] Devillers, O., Hurtado, F., Kàrolyi, G., Seara, C.: Chromatic variants of the Erdos-Szekeres Theorem. Comput. Geom. 26, 193–208 (2003) · Zbl 1034.52014 · doi:10.1016/S0925-7721(03)00013-0 [8] Dumitrescu, A.: Planar sets with few empty convex polygons. Stud. Sci. Math. Hung. 36, 93–109 (2000) · Zbl 0980.52007 [9] Gerken, T.: Empty convex hexagons in planar point sets. Discrete Comput. Geom. 39, 239–272 (2008) · Zbl 1184.52016 · doi:10.1007/s00454-007-9018-x [10] Harborth, H.: Konvexe Fünfecke in ebenen Punktmengen. Elem. Math. 33, 116–118 (1978) · Zbl 0397.52005 [11] Horton, J.D.: Sets with no empty convex 7-gons. Canad. Math. Bull. 26, 482–484 (1983) · Zbl 0521.52010 · doi:10.4153/CMB-1983-077-8 [12] 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) · Zbl 1079.52505 · doi:10.1007/978-3-642-55566-4_25 [13] Katchalski, M., Meir, A.: On empty triangles determined by points in the plane. Acta Math. Hung. 51, 323–328 (1988) · Zbl 0655.52007 · doi:10.1007/BF01903339 [14] Nicolás, C.M.: The empty hexagon theorem. Discrete Comput. Geom. 38, 389–397 (2007) · Zbl 1146.52010 · doi:10.1007/s00454-007-1343-6 [15] Sakai, T., Urrutia, J.: Covering the convex quadrilaterals of point sets. Graphs Combin. 23, 343–357 (2007) · Zbl 1118.52021 · doi:10.1007/s00373-007-0717-0 [16] Valtr, P.: On the minimum number of empty polygons in planar point sets. Stud. Sci. Math. Hungar. 30, 155–163 (1995) · Zbl 0867.52004 [17] Valtr, P.: On empty hexagons. In: Surveys on Discrete and Computational Geometry, Twenty Years Later, pp. 433–441. AMS (2008) · Zbl 1147.52301 · doi:10.1090/conm/453/08809
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