Jesús Valenzuela

Compact Grid Representation of Graphs

A graph G is said to be grid locatable if it admits a representation such that vertices are mapped to grid points and edges to line segments that avoid grid points but the extremes. Additionally G is said to be properly embeddable in the grid if it is grid locatable and the segments representing edges do not cross each other. We study the area needed to obtain those representations for some graph families.

Transforming Triangulations on Nonplanar Surfaces

We consider whether any two triangulations of a polygon or a point set on a nonplanar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder.

Monochromatic geometric k-factors in red-blue sets with white and Steiner points

Abstract We study the existence of monochromatic planar geometric k-factors on sets of red and blue points. When it is not possible to find a k-factor we make use of auxiliary points: white points, whose position is given as a datum and which color is free; and Steiner points whose position and color is free. We present bounds on the number of white and/or Steiner points necessary and/or sufficient to draw a monochromatic planar geometric k-factor.

Euclidean position in Euclidean 2-orbifolds

Intuitively, a set of sites on a surface is in Euclidean position if points are so close to each other that planar algorithms can be easily adapted in order to solve most of the classical problems in Computational Geometry. In this work we formalize a definition of the term Euclidean position for a relevant class of metric spaces, the Euclidean 2-orbifolds, and present methods to compute whether a set of sites has this property. We also show the relation between the convex hull of a point set in Euclidean position on a Euclidean 2-orbifold and the planar convex hull of the inverse image (via the quotient map) of the set.

Monochromatic geometric k-factors for bicolored point sets with auxiliary points

Given a bicolored point set S, it is not always possible to construct a monochromatic geometric planar k-factor of S. We consider the problem of finding such a k-factor of S by using auxiliary points. Two types are considered: white points whose position is fixed, and Steiner points which have no fixed position. Our approach provides algorithms for constructing those k-factors, and gives bounds on the number of auxiliary points needed to draw a monochromatic geometric planar k-factor of S. Draw monochromatic geometric k-factors of bicolored point sets using auxiliary points.We use two types of auxiliary points: Steiner points and white points.We provide algorithms for constructing those k-factors.We give bounds on the number of auxiliary points needed.

Reporting Bichromatic Segment Intersections from Point Sets

In this paper, we introduce a natural variation of the problem of computing all bichromatic intersections between two sets of segments. Given two sets R and B of n points in the plane defining two sets of segments, say red and blue, we present an O(n2) time and space algorithm for solving the problem of reporting the set of segments of each color intersected by segments of the other color. We also prove that this problem is 3-Sum hard and provide some illustrative examples of several point configurations.

Cover contact graphs

We study problems that arise in the context of covering certain geometric objects (so-called seeds, e.g., points or disks) by a set of other geometric objects (a so-called cover, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds).