16 research outputs found
On Schnyder's Theorm
The central topic of this thesis is Schnyder's Theorem. Schnyder's Theorem provides
a characterization of planar graphs in terms of their poset dimension, as follows: a graph
G is planar if and only if the dimension of the incidence poset of G is at most three. One
of the implications of the theorem is proved by giving an explicit mapping of the vertices
to R^2 that defines a straightline embedding of the graph. The other implication is proved
by introducing the concept of normal labelling. Normal labellings of plane triangulations
can be used to obtain a realizer of the incidence poset. We present an exposition of
Schnyder’s theorem with his original proof, using normal labellings. An alternate proof
of Schnyder’s Theorem is also presented. This alternate proof does not use normal
labellings, instead we use some structural properties of a realizer of the incidence poset
to deduce the result.
Some applications and a generalization of one implication of Schnyder’s Theorem
are also presented in this work. Normal labellings of plane triangulations can be used to
obtain a barycentric embedding of a plane triangulation, and they also induce a partition
of the edge set of a plane triangulation into edge disjoint trees. These two applications
of Schnyder’s Theorem and a third one, relating realizers of the incidence poset and
canonical orderings to obtain a compact drawing of a graph, are also presented. A
generalization, to abstract simplicial complexes, of one of the implications of Schnyder’s
Theorem was proved by Ossona de Mendez. This generalization is also presented in this
work. The concept of order labelling is also introduced and we show some similarities of
the order labelling and the normal labelling. Finally, we conclude this work by showing
the source code of some implementations done in Sage
Morphing planar triangulations
A morph between two drawings of the same graph can be thought of as a continuous deformation between the two given drawings. A morph is linear if every vertex moves along a straight line segment from its initial position to its final position. In this thesis we study algorithms for morphing, in which the morphs are given by sequences of linear morphing steps.
In 1944, Cairns proved that it is possible to morph between any two planar drawings of a planar triangulation while preserving planarity during the morph. However this morph may require exponentially many steps. It was not until 2013 that Alamdari et al. proved that the morphing problem for planar triangulations can be solved using polynomially many steps.
In 1990 it was shown by Schnyder that using special drawings that we call Schnyder drawings it is possible to draw a planar graph on a O(n)×O(n) grid, and moreover such drawings can be found in O(n) time (here n denotes the number of vertices of the graph). It still remains unknown whether there is an efficient algorithm for morphing in which all drawings are on a polynomially sized grid.
In this thesis we give two different new solutions to the morphing problem for planar triangulations. Our first solution gives a strengthening of the result of Alamdari et al. where each step is a unidirectional morph. This also leads to a simpler proof of their result.
Our second morphing algorithm finds a planar morph consisting of O(n²) steps between any two Schnyder drawings while remaining in an O(n)×O(n) grid. However, there are drawings of planar triangulations which are not Schnyder drawings, and for these drawings we show that a unidirectional morph consisting of O(n) steps that ends at a Schnyder drawing can be found. We conclude this work by showing that the basic steps from our morphs can be implemented using a Schnyder wood and weight shifts on the set of interior faces
Morphing Planar Graph Drawings with Unidirectional Moves
Alamdari et al. showed that given two straight-line planar drawings of a
graph, there is a morph between them that preserves planarity and consists of a
polynomial number of steps where each step is a \emph{linear morph} that moves
each vertex at constant speed along a straight line. An important step in their
proof consists of converting a \emph{pseudo-morph} (in which contractions are
allowed) to a true morph. Here we introduce the notion of \emph{unidirectional
morphing} step, where the vertices move along lines that all have the same
direction. Our main result is to show that any planarity preserving
pseudo-morph consisting of unidirectional steps and contraction of low degree
vertices can be turned into a true morph without increasing the number of
steps. Using this, we strengthen Alamdari et al.'s result to use only
unidirectional morphs, and in the process we simplify the proof.Comment: 13 pages, 9 figure
Analogies between the crossing number and the tangle crossing number
Tanglegrams are special graphs that consist of a pair of rooted binary trees
with the same number of leaves, and a perfect matching between the two
leaf-sets. These objects are of use in phylogenetics and are represented with
straightline drawings where the leaves of the two plane binary trees are on two
parallel lines and only the matching edges can cross. The tangle crossing
number of a tanglegram is the minimum crossing number over all such drawings
and is related to biologically relevant quantities, such as the number of times
a parasite switched hosts.
Our main results for tanglegrams which parallel known theorems for crossing
numbers are as follows. The removal of a single matching edge in a tanglegram
with leaves decreases the tangle crossing number by at most , and this
is sharp. Additionally, if is the maximum tangle crossing number of
a tanglegram with leaves, we prove
. Further,
we provide an algorithm for computing non-trivial lower bounds on the tangle
crossing number in time. This lower bound may be tight, even for
tanglegrams with tangle crossing number .Comment: 13 pages, 6 figure
Implementación de algoritmos en teorÃa de graficas
En este trabajo de tesis se abordan algunos problemas clásicos de la teorÃa de gráficas como determinación de circuitos eulerianos, determinación de circuitos hamiltonianos, planaridad y coloración; también se presentan algoritmos para resolver dichos problemas, en el caso de los circuitos hamiltonianos sólo se aborda cuando la gráfica cumple ciertas hipótesis. Los algoritmos fueron implementados en el programa computacional QGraphs, el cual fue producto de nuestro trabajo de tesis