Now showing items 1-20 of 35

• #### A polynomial bound for untangling geometric planar graphs ﻿

(2009-12)
Article
Open Access
To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002) asked if every n-vertex geometric planar ...
• #### Blocking the k-holes of point sets in the plane ﻿

(2015-09)
Article
Open Access
Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a k-hole of P, if there is no element of P in the interior of its ...
• #### Cell-paths in mono- and bichromatic line arrangements in the plane ﻿

(2014)
Conference report
Open Access
We show that in every arrangement of n red and blue lines | in general position and not all of the same color | there is a path through a linear number of cells where red and blue lines are crossed alternatingly (and no ...
• #### Colored spanning graphs for set visualization ﻿

(Springer, 2013)
Conference report
Open Access
We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the ...
• #### Compatible matchings in geometric graphs ﻿

(Centre de Recerca Matemàtica, 2011)
Conference report
Open Access
Two non-crossing geometric graphs on the same set of points are compatible if their union is also non-crossing. In this paper, we prove that every graph G that has an outerplanar embedding admits a non-crossing perfect ...
• #### 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 ...
• #### Connectivity-preserving transformations of binary images ﻿

(2009-10)
Article
Open Access
A binary image $\emph{I}$ is $B_a$, $W_b$-connected, where $\emph{a,b}$ ∊ {4,8}, if its foreground is $\emph{a}$-connected and its background is $\emph{b}$-connected. We consider a local modification of a $B_a$, $W_b$-connected ...
• #### Coverage with k-transmitters in the presence of obstacles ﻿

(Springer Verlag, 2010)
Conference report
Restricted access - publisher's policy
For a fixed integer k ≥ 0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k “walls”, represented as line segments in the plane. We develop lower ...
• #### 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 ...
• #### Every large point set contains many collinear points or an empty pentagon ﻿

(2011-01)
Article
Restricted access - publisher's policy
We prove the following generalised empty pentagon theorem for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next ...
• #### Geometric biplane graphs I: maximal graphs ﻿

(2013)
Conference report
Open Access
• #### Geometric Biplane Graphs II: Graph Augmentation ﻿

(2013)
Conference report
Open Access
We study biplane graphs drawn on a nite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every ...
• #### Geometric tree graphs of points in convex position ﻿

(1997)
Article
Open Access
Given a set $P$ of points in the plane, the geometric tree graph of $P$ is defined as the graph $T(P)$ whose vertices are non-crossing rectilinear spanning trees of $P$, and where two trees $T_1$ and $T_2$ are adjacent ...
• #### Graphs of non-crossing perfect matchings ﻿

(2001)
Article
Open Access
Let Pn be a set of n = 2m points that are the vertices of a convex polygon, and let Mm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross, ...
• #### Improving shortest paths in the Delaunay triangulation ﻿

(2012)
Article
Open Access
We study a problem about shortest paths in Delaunay triangulations. Given two nodes s, t in the Delaunay triangulation of a point set P, we look for a new point p that can be added, such that the shortest path from s to ...
• #### 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 ...
• #### Matching points with things ﻿

(Springer Verlag, 2010)
Conference report
Restricted access - publisher's policy
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. We show that when the objects we match the points to ...
• #### Moving rectangles ﻿

(2010)
Conference report
Open Access
Consider a set of n pairwise disjoint axis-parallel rectangles in the plane. We call this set the source rectangles S. The aim is to move S to a set of (pairwise disjoint) target rectangles T . A move consists in a ...
• #### Necklaces, convolutions, and X plus Y ﻿

(2014-06-01)
Article
Restricted access - publisher's policy
We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓ p ...
• #### On k-enclosing objects in a coloured point set ﻿

(2014)
Conference report
Open Access
We introduce the exact coloured k -enclosing object problem: given a set P of n points in R 2 , each of which has an associated colour in f 1 ;:::;t g , and a vec- tor c = ( c 1 ;:::;c t ), ...