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We study a generalization of the classical problem of illumination of polygons. Instead of modeling a light source we model a wireless device whose radio signal can penetrate a given number k of walls. We call these objects k-modems and study the minimum number of k-modems necessary to illuminate monotone and monotone orthogonal polygons. We show that every… (More)

- Jorge L. Arocha, Imre Bárány, Javier Bracho, Ruy Fabila Monroy, Luis Pedro Montejano
- Discrete & Computational Geometry
- 2009

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger's theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

- Oswin Aichholzer, Ruy Fabila Monroy, David Flores-Peñaloza, Thomas Hackl, Clemens Huemer, Jorge Urrutia
- Comput. Geom.
- 2008

We consider a variation of a problem stated by Erdös and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points… (More)

- Greg Aloupis, Jean Cardinal, +10 authors Perouz Taslakian
- Comput. Geom.
- 2013

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. In this paper, we address the algorithmic problem of determining whether a non-crossing matching exists between a given point-object pair. We show that when the objects we match the points to… (More)

Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erd˝ os-type question on the least number h k (n) of convex k-holes in S, and give improved lower bounds on

- Oswin Aichholzer, Ruy Fabila Monroy, +4 authors Birgit Vogtenhuber
- CCCG
- 2010

Given a set B of n blue points in general position, we say that a set of red points R blocks B if in the Delaunay triangulation of B ∪ R there is no edge connecting two blue points. We give the following bounds for the size of the smallest set R blocking B: (i) 3n/2 red points are always sufficient to block a set of n blue points, (ii) if B is in convex… (More)

- Birgit Vogtenhuber, Oswin Aichholzer, +6 authors Pavel Valtr
- CCCG
- 2011

We consider a variation of the classical Erd˝ os-Szekeres problems on the existence and number of convex k-gons and k-holes (empty k-gons) in a set of n points in the plane. Allowing the k-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any k and sufficiently large n, we give a… (More)

- Ruy Fabila Monroy, David R. Wood
- EGC
- 2011

Let P be a set of n points in general and convex position in the plane. Let Dn be the graph whose vertex set is the set of all line segments with endpoints in P , where disjoint segments are adjacent. The chromatic number of this graph was first studied by Araujo et al. [CGTA, 2005]. The previous best bounds are 3n 4 ≤ χ(Dn) < n − n 2 (ignoring lower order… (More)

1 Let P be a simple polygon on the plane. Two vertices of P are visible if 2 the open line segment joining them is contained in the interior of P. In this 3 paper we study the following questions posed in [5, 6]: (1) Is it true that 4 every non-convex simple polygon has a vertex that can be continuously 5 moved such that during the process no vertex-vertex… (More)

- Bernardo M. Ábrego, Ruy Fabila Monroy, +4 authors Maria Saumell
- Comput. Geom.
- 2011

Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straight-line segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the k-nearest neighbor graph, the k-relative neighborhood graph, the k-Gabriel graph and the… (More)