Morphing Planar Graph Drawings Through 3D - Buchin, Kevin; Evans, Will; Frati, Fabrizio; Kostitsyna, Irina; Löffler, Maarten; Ophelders, Tim; Wolff, Alexander (Computing in Geometry and Topology, 2023)
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.