Morphing Planar Graph Drawings Through 3D

TL;DR

This paper demonstrates that any two planar straight-line drawings of an n-vertex planar graph can be transformed into each other through a crossing-free 3D morph with quadratic steps, highlighting differences from 2D morphs.

Contribution

It establishes the existence of a quadratic-step crossing-free 3D morph between any two planar drawings, extending morphing theory into three dimensions.

Findings

Existence of O(n^2) step crossing-free 3D morphs

Difficulty in achieving linear bounds in 3D

Comparison with 2D morphing bounds

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.