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Morphing Planar Graph Drawings Through 3D

Kevin Buchin, Will Evans, Fabrizio Frati, Irina Kostitsyna, Maarten Löffler, Tim Ophelders, Alexander Wolff

TL;DR

This work proves that allowing a third dimension enables crossing-free morphs between any two planar straight-line drawings of the same $n$-vertex planar graph with $O(n^2)$ steps. The authors introduce four 3D morph operations (embedding-changing maneuvers) and show how to compose them to transform one drawing into another, first for biconnected graphs and then for general planar graphs via augmentation. The main contribution is a quadratic upper bound on the number of steps, contrasting with known linear upper bounds in 2D for topologically equivalent drawings and the constant-step morphs possible for certain trees in 3D. They also discuss lower-bound challenges, noting that no nontrivial general lower bound is known and highlighting open problems about sub-quadratic bounds and broader graph families. Overall, the paper provides a structured 3D morph framework with practical implications for graph visualization and reconfiguration in three dimensions.

Abstract

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 piecewise-linear crossing-free 3D morph with $O(n^2)$ steps that transforms one drawing into the other. We also give some evidence why it is difficult to obtain a linear lower bound (which exists in 2D) for the number of steps of a crossing-free 3D morph.

Morphing Planar Graph Drawings Through 3D

TL;DR

This work proves that allowing a third dimension enables crossing-free morphs between any two planar straight-line drawings of the same -vertex planar graph with steps. The authors introduce four 3D morph operations (embedding-changing maneuvers) and show how to compose them to transform one drawing into another, first for biconnected graphs and then for general planar graphs via augmentation. The main contribution is a quadratic upper bound on the number of steps, contrasting with known linear upper bounds in 2D for topologically equivalent drawings and the constant-step morphs possible for certain trees in 3D. They also discuss lower-bound challenges, noting that no nontrivial general lower bound is known and highlighting open problems about sub-quadratic bounds and broader graph families. Overall, the paper provides a structured 3D morph framework with practical implications for graph visualization and reconfiguration in three dimensions.

Abstract

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 -vertex planar graph, there exists a piecewise-linear crossing-free 3D morph with steps that transforms one drawing into the other. We also give some evidence why it is difficult to obtain a linear lower bound (which exists in 2D) for the number of steps of a crossing-free 3D morph.
Paper Structure (9 sections, 5 theorems, 10 figures)

This paper contains 9 sections, 5 theorems, 10 figures.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. An Upper Bound
  4. 3D Morph Operations
  5. 3D Morphs for Biconnected Planar Graphs
  6. 3D Morphs for General Planar Graphs
  7. Discussion: Lower Bounds
  8. Open Problems
  9. Acknowledgements.

Key Result

theorem 1

For any two planar straight-line drawings (not necessarily with the same embedding) of an $n$-vertex planar graph, there exists a crossing-free piecewise-linear 3D morph between them with $O(n^2)$ steps.

Figures (10)

  • Figure 1: The four operations that are the building blocks for our piecewise-linear morphs.
  • Figure 2: Illustration for Operation 3 with $i=2$ and $j=4$: Construction of $\Psi$ from $\Gamma$.
  • Figure 3: Illustration for Operation 3.
  • Figure 4: Illustration for Operation 4: $P_{\textrm{in}}$ is blue, $P_{\textrm{out}}$ is red, and $P_{\textrm{ext}}$ is purple.
  • Figure 5: Illustration for Operation 4: Construction of $\Gamma'$ from the restriction of $\Lambda$ to $G$.
  • ...and 5 more figures

Theorems & Definitions (5)

  • theorem 1
  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4