Morphing Schnyder Drawings of Planar Triangulations

Barrera-Cruz, Fidel; Haxell, Penny; Lubiw, Anna

doi:10.1007/978-3-662-45803-7_25

Cited by 3 publications

(5 citation statements)

References 19 publications

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“…Techniques from our paper have been used in algorithms to morph Schnyder drawings [10], and algorithms to morph convex drawings while preserving convexity [5].…”

Section: Resultsmentioning

confidence: 99%

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How to Morph Planar Graph Drawings

et al. 2017

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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 straight-line planarity and consists of O(n) steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns’ 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps

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“…Techniques from our paper have been used in algorithms to morph Schnyder drawings [10], and algorithms to morph convex drawings while preserving convexity [5].…”

Section: Resultsmentioning

confidence: 99%

“…We leave as an open problem to find a morph that uses a polynomial number of discrete morphing steps and uses only a logarithmic number of bits per coordinate. Barrera-Cruz et al [10] made a first step in this direction by solving the case where the two drawings are Schnyder drawings.…”

Section: Discussionmentioning

confidence: 99%

How to Morph Planar Graph Drawings

et al. 2017

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“…Our motivation in examining this question partly lies in the elegance of the mathematics, but it was also posed to us by Veronika Irvine (see [2,9]), who needed the characterisation to to create and classify "grounds" for bobbin lace drawings; this paper is a first step in this direction. Further, we note that our result relates to the problem of morphing from one planar graph drawing to another (see [1,8]). Previous work has characterised drawings that arise from the Schnyder algorithm (see [3]) in this context.…”

Section: Introductionmentioning

confidence: 83%

The Weighted Barycenter Drawing Recognition Problem

Eades

Healy

Nikolov

2018

Lecture Notes in Computer Science

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“…The only paper we are aware of where the resolution problem has been successfully addressed is the one by Barrera-Cruz et al [7], who showed how to construct a morph with polynomially-bounded resolution between two Schnyder drawings Γ 0 and Γ 1 of the same planar triangulation. The model they use in order to ensure a bound on the resolution requires that Γ 0 = ∆ 0 , ∆ 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we show how to construct morphs of tree drawings that simultaneously achieve a reduced number of morphing steps and a polynomially-bounded resolution. Adopting the setting of [7], we assume that Γ 0 and Γ 1 are grid drawings and we ensure that each morphing step produces a grid drawing.…”

Section: Introductionmentioning

confidence: 99%

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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Morphing Schnyder Drawings of Planar Triangulations

Cited by 3 publications

References 19 publications

How to Morph Planar Graph Drawings

How to Morph Planar Graph Drawings

The Weighted Barycenter Drawing Recognition Problem

How to Morph a Tree on a Small Grid

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