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(n2) 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.
@article{arxiv.2210.05384,
title = {Morphing Planar Graph Drawings Through 3D},
author = {Kevin Buchin and Will Evans and Fabrizio Frati and Irina Kostitsyna and Maarten Löffler and Tim Ophelders and Alexander Wolff},
journal= {arXiv preprint arXiv:2210.05384},
year = {2025}
}