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A geometric graph is ;I g~.aph 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 I,'. 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(More)
We study a generalization of the classical problem of the 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 sufficient and sometimes necessary to illuminate monotone and monotone orthogonal(More)
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 rec-tilinear 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,(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)
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 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)
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)