525 research outputs found
Morphing Planar Graph Drawings Optimally
We provide an algorithm for computing a planar morph between any two planar
straight-line drawings of any -vertex plane graph in morphing steps,
thus improving upon the previously best known upper bound. Further, we
prove that our algorithm is optimal, that is, we show that there exist two
planar straight-line drawings and of an -vertex plane
graph such that any planar morph between and requires
morphing steps
Upward Planar Morphs
We prove that, given two topologically-equivalent upward planar straight-line
drawings of an -vertex directed graph , there always exists a morph
between them such that all the intermediate drawings of the morph are upward
planar and straight-line. Such a morph consists of morphing steps if
is a reduced planar -graph, morphing steps if is a planar
-graph, morphing steps if is a reduced upward planar graph, and
morphing steps if is a general upward planar graph. Further, we
show that morphing steps might be necessary for an upward planar
morph between two topologically-equivalent upward planar straight-line drawings
of an -vertex path.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018) The current version is the
extended on
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
Optimal Morphs of Convex Drawings
We give an algorithm to compute a morph between any two convex drawings of
the same plane graph. The morph preserves the convexity of the drawing at any
time instant and moves each vertex along a piecewise linear curve with linear
complexity. The linear bound is asymptotically optimal in the worst case.Comment: To appear in SoCG 201
Convexity-Increasing Morphs of Planar Graphs
We study the problem of convexifying drawings of planar graphs. Given any
planar straight-line drawing of an internally 3-connected graph, we show how to
morph the drawing to one with strictly convex faces while maintaining planarity
at all times. Our morph is convexity-increasing, meaning that once an angle is
convex, it remains convex. We give an efficient algorithm that constructs such
a morph as a composition of a linear number of steps where each step either
moves vertices along horizontal lines or moves vertices along vertical lines.
Moreover, we show that a linear number of steps is worst-case optimal.
To obtain our result, we use a well-known technique by Hong and Nagamochi for
finding redrawings with convex faces while preserving y-coordinates. Using a
variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and
Nagamochi's result which comes with a better running time. This is of
independent interest, as Hong and Nagamochi's technique serves as a building
block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201
Pole Dancing: 3D Morphs for Tree Drawings
We study the question whether a crossing-free 3D morph between two
straight-line drawings of an -vertex tree can be constructed consisting of a
small number of linear morphing steps. We look both at the case in which the
two given drawings are two-dimensional and at the one in which they are
three-dimensional. In the former setting we prove that a crossing-free 3D morph
always exists with steps, while for the latter steps
are always sufficient and sometimes necessary.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Morphing Schnyder drawings of planar triangulations
We consider the problem of morphing between two planar drawings of the same
triangulated graph, maintaining straight-line planarity. A paper in SODA 2013
gave a morph that consists of steps where each step is a linear morph
that moves each of the vertices in a straight line at uniform speed.
However, their method imitates edge contractions so the grid size of the
intermediate drawings is not bounded and the morphs are not good for
visualization purposes. Using Schnyder embeddings, we are able to morph in
linear morphing steps and improve the grid size to
for a significant class of drawings of triangulations, namely the class of
weighted Schnyder drawings. The morphs are visually attractive. Our method
involves implementing the basic "flip" operations of Schnyder woods as linear
morphs.Comment: 23 pages, 8 figure
- …