Morphing tree drawings in a small 3D grid

Abstract

We study crossing-free grid morphs for planar tree drawings using 3D. A morph consists of morphing steps, where vertices move simultaneously along straight-line trajectories at constant speed. A crossingfree morph is known between two drawings of an n-vertex planar graph G with O(n) morphing steps and sometimes Ω(n) steps is necessary even when G is a path [Alamdari et al. 2017], while using 3D can reduce the number of steps to O(log n) for an n-vertex tree [Arseneva et al. 2019]. However, these morphs do not bound one practical parameter, the resolution. Recently [Barrera-Cruz et al. 2019] a 2D crossing-free morph was devised between two planar drawings of an n-vertex tree in O(n) steps with polynomial resolution, that is, all intermediate drawings are grid drawings on a grid of polynomial size. Can the number of steps in this setting be reduced substantially by using 3D? We answer this question in affirmative and present a 3D non-crossing morph between two planar grid drawings of an n-vertex tree in O( √ n log n) morphing steps. Each intermediate drawing lies in a 3D grid of polynomial volume.