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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)

We prove that a graph is 4-colorable if and only if it can be drawn with vertices in the integer lattice, using as edges only line segments not containing a third point of the lattice.

- 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)

- 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)

In 1926, Jarńık introduced the problem of drawing a convex ngon with vertices having integer coordinates. He constructed such a drawing in the grid [1, c · n] for some constant c > 0, and showed that this grid size is optimal up to a constant factor. We consider the analogous problem for drawing the double circle, and prove that it can be done within the… (More)

- José Miguel Díaz-Báñez, Ruy Fabila Monroy, David Flores-Peñaloza, Marco A. Heredia, Jorge Urrutia
- J. Comb. Optim.
- 2012

This paper concerns about energy-efficient broadcasts in mobile ad hoc networks, yet in a model where each station moves on the plane with uniform rectilinear motion. Such restriction is imposed to discern which issues arise from the introduction of movement in the wireless ad hoc networks. Given a transmission range assignment for a set of n stations S, we… (More)

- Crevel Bautista-Santiago, Javier Cano, +5 authors Jorge Urrutia
- Discrete Mathematics & Theoretical Computer…
- 2013

Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P ∩C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l… (More)

- Bernardo M. Ábrego, Ruy Fabila Monroy, +5 authors Maria Saumell
- Networks
- 2013

Given a large weighted graph G = (V, E) and a subset U of V , we define several graphs with vertex set U in which two vertices are adjacent if they satisfy some prescribed proximity rule. These rules use the shortest path distance in G and generalize the proximity rules that generate some of the most common proximity graphs in Euclidean spaces. We prove… (More)

- Oswin Aichholzer, Nieves Atienza, +5 authors Jorge Urrutia
- ArXiv
- 2015

Let P be a set of n points in general position in the plane, r of which are red and b of which are blue. In this paper we prove that there exist: for every α ∈ [ 0, 1 2 ] , a convex set containing exactly dαre red points and exactly dαbe blue points of P ; a convex set containing exactly ⌈ r+1 2 ⌉ red points and exactly ⌈ b+1 2 ⌉ blue points of P .… (More)

- Bernardo M. Ábrego, Ruy Fabila-Monroy, +4 authors Maria Saumell
- 2009

We study the number of crossings among edges of some higher order proximity graphs of the family of the Delaunay graph. That is, given a set P of n points in the Euclidean plane, we give lower and upper bounds on the minimum and the maximum number of crossings that these geometric graphs defined on P have.