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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)

A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph… (More)

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)

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)

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)

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)

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

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)

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)

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)