Morphing Schnyder Drawings of Planar Triangulations

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

doi:10.1007/s00454-018-0018-9

Cited by 16 publications

(14 citation statements)

References 20 publications

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“…Other combinatorial structures of graph representations also form lattices; see the work by Felsner and colleagues [19][20][21]. In the context of morphs, Barrera-Cruz et al [6] exploited the lattice structure of Schnyder woods of a plane triangulation to obtain a sequence on linear morphs for planar straight-line drawings. While their morphs require O(n 2 ) steps (compared to the optimum of O(n)), their morphs have the advantage that they are "visually pleasing" and that they maintain a quadratic-size drawing area before or after any step that consists of a linear morph.…”

Section: Important Related Workmentioning

confidence: 99%

Morphing Rectangular Duals

Chaplick¹,

Kindermann²,

Klawitter³

et al. 2021

Preprint

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We consider the problem of morphing between rectangular duals of a plane graph G, that is, contact representations of G by axis-aligned rectangles such that (i) no four rectangles share a point and (ii) the union of all rectangles is a rectangle. Combinatorially, a rectangular dual can be described by a regular edge labeling (REL), which determines the orientations of the rectangle contacts.If we require that we have a rectangular dual continuously throughout the morph, then a morph only exists if the source and target rectangular duals realize the same REL. Hence, we are less strict and allow intermediate contact representations of non-rectangular polygons of constant complexity (at most 8-gons). We show how to compute a morph consisting of O(n 2 ) linear morphs in O(n 3 ) time. We implement the rotations of RELs as linear morphs to traverse the lattice structure of RELs.

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Section: Important Related Workmentioning

confidence: 99%

Morphing Rectangular Duals

Chaplick¹,

Kindermann²,

Klawitter³

et al. 2021

Preprint

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“…The main challenge is to design morphing algorithms that maintain some additional geometric properties of the input drawings throughout the transformation, such as planarity with straight-line edges (see, e.g., [1,16,24]), convexity [6,33], orthogonality [12,25,25], and upwardness [20]. We point the interested reader to [4,8,10,11] for additional related work.…”

Section: Introductionmentioning

confidence: 99%

On Morphing 1-Planar Drawings

Angelini¹,

Bekos²,

Fabrizio³

et al. 2021

Preprint

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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 topology-preserving morphs for pairs of non-planar graph drawings. We make a step towards this problem by showing that a topology-preserving morph always exists for drawings of a meaningful family of 1-planar graphs. While our proof is constructive, the vertices may follow trajectories of unbounded complexity.

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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 16 publications

References 20 publications

Morphing Rectangular Duals

Morphing Rectangular Duals

On Morphing 1-Planar Drawings

How to Morph a Tree on a Small Grid

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