21 research outputs found
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 Planar Orthogonal Drawings
We describe an algorithm that morphs between two planar orthogonal drawings Gamma_I and Gamma_O of a connected graph G, while preserving planarity and orthogonality. Necessarily Gamma_I and Gamma_O share the same combinatorial embedding. Our morph uses a linear number of linear morphs (linear interpolations between two drawings) and preserves linear complexity throughout the process, thereby answering an open question from Biedl et al. [Biedl et al., 2013].
Our algorithm first unifies the two drawings to ensure an equal number of (virtual) bends on each edge. We then interpret bends as vertices which form obstacles for so-called wires: horizontal and vertical lines separating the vertices of Gamma_O. We can find corresponding wires in Gamma_I that share topological properties with the wires in Gamma_O. The structural difference between the two drawings can be captured by the spirality of the wires in Gamma_I, which guides our morph from Gamma_I to Gamma_O
Morphing Contact Representations of Graphs
We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type.
We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs.
We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set
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