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We study crossing-free grid morphs for planar tree drawings using 3D. A morph consists of morphing steps, where vertices move simultaneously along straight-line trajectories at constant speeds. A crossing-free morph is known between two…

Computational Geometry · Computer Science 2021-10-07 Elena Arseneva , Rahul Gangopadhyay , Aleksandra Istomina

We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an $n$-vertex planar graph and two planar straight-line drawings of the graph…

Computational Geometry · Computer Science 2024-09-09 Therese Biedl , Anna Lubiw , Jack Spalding-Jamieson

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 $O(n^2)$ steps where each step is a linear morph that…

Computational Geometry · Computer Science 2014-11-25 Fidel Barrera-Cruz , Penny Haxell , Anna Lubiw

A morph between two straight-line planar drawings of the same graph is a continuous transformation from the first to the second drawing such that planarity is preserved at all times. Each step of the morph moves each vertex at constant…

Computational Geometry · Computer Science 2013-08-21 Patrizio Angelini , Fabrizio Frati , Maurizio Patrignani , Vincenzo Roselli

In this paper, we investigate crossing-free 3D morphs between planar straight-line drawings. We show that, for any two (not necessarily topologically equivalent) planar straight-line drawings of an $n$-vertex planar graph, there exists a…

Computational Geometry · Computer Science 2025-03-03 Kevin Buchin , Will Evans , Fabrizio Frati , Irina Kostitsyna , Maarten Löffler , Tim Ophelders , Alexander Wolff

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…

Computational Geometry · Computer Science 2014-11-25 Fidel Barrera-Cruz , Penny Haxell , Anna Lubiw

Given an $n$-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves…

We study the question whether a crossing-free 3D morph between two straight-line drawings of an $n$-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…

Computational Geometry · Computer Science 2018-09-05 Elena Arseneva , Prosenjit Bose , Pilar Cano , Anthony D'Angelo , Vida Dujmovic , Fabrizio Frati , Stefan Langerman , Alessandra Tappini

A method for creating a forest of model trees to fit samples of a function defined on images is described in several steps: down-sampling the images, determining a tree's hyperplanes, applying convolutions to the hyperplanes to handle small…

Machine Learning · Computer Science 2026-01-28 William Ward Armstrong , Hongyi Li , Jun Xu

We provide an algorithm for computing a planar morph between any two planar straight-line drawings of any $n$-vertex plane graph in $O(n)$ morphing steps, thus improving upon the previously best known $O(n^2)$ upper bound. Further, we prove…

Data Structures and Algorithms · Computer Science 2014-02-20 Patrizio Angelini , Giordano Da Lozzo , Giuseppe Di Battista , Fabrizio Frati , Maurizio Patrignani , Vincenzo Roselli

We prove that, given two topologically-equivalent upward planar straight-line drawings of an $n$-vertex directed graph $G$, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and…

Data Structures and Algorithms · Computer Science 2018-10-15 Giordano Da Lozzo , Giuseppe Di Battista , Fabrizio Frati , Maurizio Patrignani , Vincenzo Roselli

A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction. Trees, as a special class of graphs, have been the focus of several…

Data Structures and Algorithms · Computer Science 2025-05-19 Anargyros Oikonomou , Antonios Symvonis

We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic…

Data Structures and Algorithms · Computer Science 2007-05-23 David Eppstein

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…

Computational Geometry · Computer Science 2015-04-01 Patrizio Angelini , Giordano Da Lozzo , Fabrizio Frati , Anna Lubiw , Maurizio Patrignani , Vincenzo Roselli

We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to…

Discrete Mathematics · Computer Science 2024-02-29 Julia Katheder , Stephen G. Kobourov , Axel Kuckuk , Maximilian Pfister , Johannes Zink

In this work we study the problem of network morphism, an effective learning scheme to morph a well-trained neural network to a new one with the network function completely preserved. Different from existing work where basic morphing types…

Machine Learning · Computer Science 2017-01-13 Tao Wei , Changhu Wang , Chang Wen Chen

We consider the problem of morphing between contact representations of a plane graph. In an $\mathcal F$-contact representation of a plane graph $G$, vertices are realized by internally disjoint elements from a family $\mathcal F$ of…

Computational Geometry · Computer Science 2019-03-19 Patrizio Angelini , Steven Chaplick , Sabine Cornelsen , Giordano Da Lozzo , Vincenzo Roselli

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…

Computational Geometry · Computer Science 2018-03-20 Arthur van Goethem , Kevin Verbeek

A partial edge drawing (PED) of a graph is a variation of a node-link diagram. PED draws a link, which is a partial visual representation of an edge, and reduces visual clutter of the node-link diagram. However, more time is required to…

Data Structures and Algorithms · Computer Science 2019-10-03 Kazuo Misue , Katsuya Akasaka

Computing a morph between two drawings of a graph is a classical problem in computational geometry and graph drawing. While this problem has been widely studied in the context of planar graphs, very little is known about the existence of…

Computational Geometry · Computer Science 2021-05-28 Patrizio Angelini , Michael A. Bekos , Fabrizio Montecchiani , Maximilian Pfister
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