How to Morph Planar Graph Drawings

Alamdari, Soroush; Angelini, Patrizio; Barrera-Cruz, Fidel; Chan, Timothy M.; Lozzo, Giordano Da; Battista, Giuseppe Di; Frati, Fabrizio; Haxell, Penny; Lubiw, Anna; Patrignani, Maurizio; Roselli, Vincenzo; Singla, Sahil; Wilkinson, Bryan T.

doi:10.1137/16m1069171

Cited by 31 publications

(82 citation statements)

References 39 publications

(53 reference statements)

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“…Suppose that q i is to the left (to the right) of the line through r i and s i , for each i = 0, 1. Then Corollary 7.2 in [1] ensures that q t is to the left (resp. to the right) of the line through r t and s t , for any t ∈ [0, 1].…”

Section: Remarkmentioning

confidence: 99%

“…The problem of morphing combinatorial structures is a consolidated research topic with important applications in several areas of Computer Science such as Computational Geometry, Computer Graphics, Modeling, and Animation. The structures of interest typically are drawings of graphs; a morph between two drawings Γ 0 and Γ 1 of the same graph G is defined as a continuously changing family of drawings {Γ t } of G indexed by time t ∈ [0, 1], such that the drawing at time t = 0 is Γ 0 and the drawing at time t = 1 is Γ 1 . A morph is usually required to preserve a certain drawing standard and pursues certain qualities.…”

Section: Introductionmentioning

confidence: 99%

“…Following the pioneeristic works of Cairns and Thomassen [9,13], most of the literature focused on the straight-line planar drawing standard. A sequence of recent results in [1,2,3,4,5] proved that a linear number of morphing steps suffices, and is sometimes necessary, to construct a morph between any two straight-line planar drawings of a graph.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

How to Morph a Tree on a Small Grid

Barrera-Cruz

Borrazzo

Lozzo

et al. 2019

Lecture Notes in Computer Science

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In this paper we study planar morphs between straight-line planar grid drawings of trees. A morph consists of a sequence of morphing steps, where in a morphing step vertices move along straight-line trajectories at constant speed. We show how to construct planar morphs that simultaneously achieve a reduced number of morphing steps and a polynomially-bounded resolution. We assume that both the initial and final drawings lie on the grid and we ensure that each morphing step produces a grid drawing; further, we consider both upward drawings of rooted trees and drawings of arbitrary trees.Upward planar morphs between strictly-upward drawings of rooted ordered trees maintain the drawing order-preserving at all times, as proved in the following.Lemma 2. Let Γ 0 and Γ 1 be two order-preserving strictly-upward straight-line planar drawings of a rooted ordered tree T . Let M be any upward planar morph between Γ 0 and Γ 1 . Then any intermediate drawing of M is order-preserving.Proof. Assume that the morph M happens between the time instants t = 0 and t = 1. For any t ∈ [0, 1], denote by Γ t the drawing of T in M at time t. Since M is upward, the drawing Γ t is strictly-upward, for any t ∈ [0, 1]. Hence, it suffices to prove that, for any internal node v of T , the edges from v to its children enter v in Γ t in the left-to-right order associated with v; this is indeed true for t = 0 and t = 1. Suppose that, for some t ∈ (0, 1), the left-to-right order in which two edges (u 1 , v) and (u 2 , v) enter v in Γ t is different than in Γ 0 . Since such edges are represented by curves monotonically increasing in the y-direction from u 1 and u 2 to v throughout M, it follows that there is a time t * ∈ (0, t) such that the edges (u 1 , v) and (u 2 , v) overlap in Γ t * . However, this cannot happen due to the planarity of M. 4

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

How to Morph a Tree on a Small Grid

Barrera-Cruz

Borrazzo

Lozzo

et al. 2019

Lecture Notes in Computer Science

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show abstract

“…Reconfiguration arises in countless problems that involve movement and change, including problems in computational geometry such as morphing graph drawings [1] and polygons [2], and problems relating to games and puzzles, such as the 15-puzzle (Figure 1a), a topic of research since 1879 [3]. Work on related problems was unified into a common framework in 2011 [4], leading to a widening interest in the area and the types of problems and approaches considered.…”

Section: Introductionmentioning

confidence: 99%

Introduction to Reconfiguration

2018

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Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

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“…The area of reconfiguration considers both structural and algorithmic problems on the space of solutions, under various definitions of feasibility and adjacency. Reconfiguration arises in countless problems that involve movement and change, including problems in computational geometry such as morphing graph drawings [1] and polygons [2], and problems relating to games and puzzles, such as the 15-puzzle ( Figure 1, part (a)), a topic of research since 1879 [3]. Work on related problems was unified into a common framework in 2011 [4], leading to a widening interest in the area and the types of problems and approaches considered.…”

Section: Introductionmentioning

confidence: 99%