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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)

- 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)

- 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)

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)

- Ruy Fabila Monroy, Jorge López
- J. Graph Algorithms Appl.
- 2014

Let cr(Kn) be the minimum number of crossings over all rectilin-ear drawings of the complete graph on n vertices on the plane. In this paper we prove that cr(Kn) < 0.380473 n 4 + Θ(n 3); improving thus on the previous best known upper bound. This is done by obtaining new rectilinear drawings of Kn for small values of n, and then using known constructions to… (More)