Now showing items 1-20 of 45

• #### 3-colorability of pseudo-triangulations ﻿

(2015)
Article
Open Access
Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-tri ...
• #### 4-Holes in point sets ﻿

(2014-08-01)
Article
Open Access
We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and ...
• #### 4-labelings and grid embeddings of plane quadrangulations ﻿

(2009-06-03)
External research report
Open Access
We show that each quadrangulation on $n$ vertices has a closed rectangle of influence drawing on the $(n-2) \times (n-2)$ grid. Further, we present a simple algorithm to obtain a straight-line drawing of a quadrangulation ...
• #### A new lower bound on the maximum number of plane graphs using production matrices ﻿

(2018)
Conference lecture
Restricted access - publisher's policy
We use the concept of production matrices to show that there exist sets of n points in the plane that admit ¿(41.77n) crossing-free geometric graphs. This improves the previously best known bound of ¿(41.18n) by Aichholzer ...
• #### A new lower bound on the maximum number of plane graphs using production matrices ﻿

(2019-11-01)
Article
Open Access
We use the concept of production matrices to show that there exist sets of n points in the plane that admit ¿(42.11n ) crossing-free geometric graphs. This improves the previously best known bound of ¿(41.18n ) by Aichholzer ...
• #### Binary labelings for plane quadrangulations and their relatives ﻿

(2010)
Article
Open Access
Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two ...
• #### CAL-A: Examen FQ: primavera 2019 ﻿

(Universitat Politècnica de Catalunya, 2019-06-07)
Exam
• #### Carathodory's theorem in depth ﻿

(2017-07-01)
Article
Open Access
Let X be a finite set of points in RdRd . The Tukey depth of a point q with respect to X is the minimum number tX(q)tX(q) of points of X in a halfspace containing q. In this paper we prove a depth version of Carathéodory’s ...
• #### Characteristic polynomials of production matrices for geometric graphs ﻿

(2017-08-01)
Article
Open Access
An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production ...
• #### Coloring decompositions of complete geometric graphs ﻿

(2019-06-25)
Article
Open Access
A decomposition of a non-empty simple graph G is a pair [G,P] such that P is a set of non-empty induced subgraphs of G, and every edge of G belongs to exactly one subgraph in P. The chromatic index ¿'([G,P]) of a decomposition ...
• #### Compatible spanning trees ﻿

(2014-07-01)
Article
Open Access
Two plane geometric graphs are said to be compatible when their union is a plane geometric graph. Let S be a set of n points in the Euclidean plane in general position and let T be any given plane geometric spanning tree ...
• #### DISCRETE AND ALGORITHMIC GEOMETRY | PARCIAL ﻿

(Universitat Politècnica de Catalunya, 2019-10-31)
Exam
• #### Edge-Removal and Non-Crossing Configurations in Geometric Graphs ﻿

(2010)
Article
Open Access
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 ...
• #### Empty monochromatic simplices ﻿

(2014-03-01)
Article
Open Access
Let S be a k-colored (finite) set of n points in , da parts per thousand yen3, in general position, that is, no (d+1) points of S lie in a common (d-1)-dimensional hyperplane. We count the number of empty monochromatic ...
• #### Empty non-convex and convex four-gons in random point sets ﻿

(2015-03-01)
Article
Open Access
Let S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty ...
• #### Large bichromatic point sets admit empty monochromatic 4-gons ﻿

(2009)
Conference report
Open Access
We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that ...
• #### Lower bounds for the number of small convex k-holes ﻿

(2014-07-01)
Article
Open Access
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 Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds ...
• #### Lower bounds on the maximum number of non-crossing acyclic graphs ﻿

(2015-08-01)
Article
Open Access
This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain ...
• #### Matching points with diametral disks ﻿

(2017)
Conference report
Open Access
We consider matchings between a set R of red points and a set B of blue points with diametral disks. In other words, for each pair of matched points p ¿ R and q ¿ B, we consider the diametral disk defined by p and q. We ...
• #### Matching points with disks with a common intersection ﻿

(2019-07-01)
Article
Restricted access - publisher's policy
We consider matchings with diametral disks between two sets of points R and B. More precisely, for each pair of matched points p ¿ R and q ¿ B, we consider the disk through p and q with the smallest diameter. We prove that ...