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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)

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

Let cr(Kn) be the minimum number of crossings over all rectilinear drawings of the complete graph on n vertices on the plane. In this paper we prove that cr(Kn) < 0.380473 (

- Ruy Fabila Monroy, David R. Wood
- Electr. J. Comb.
- 2013

Let a, b, c, d be four vertices in a graph G. A K4-minor rooted at a, b, c, d consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of G, respectively containing a, b, c, d. We characterise precisely when G contains a K4-minor rooted at a, b, c, d by describing six classes of obstructions, which are the edge-maximal graphs containing no… (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.

- 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 pointobject 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)

- 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)

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 Erdős-type question on the least number hk(n) of convex k-holes in S, and give improved lower bounds on hk(n), for 3 ≤ k ≤ 5. Specifically, we show that h3(n) ≥ n − 32n 7 + 22 7 , h4(n) ≥ n 2 2 − 9n 4 − o(n), and h5(n) ≥ 3n 4 − o(n).

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

We consider a variation of the classical Erdős-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)

Let P be a simple polygon on the plane. Two vertices of P are visible if the open line segment joining them is contained in the interior of P . In this paper we study the following questions posed in [5, 6]: (1) Is it true that every non-convex simple polygon has a vertex that can be continuously 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)