Morphing Tree Drawings in a Small 3D Grid

Abstract

AbstractWe study crossing-free grid morphs for planar tree drawings using the third dimension. A morph consists of morphing steps, where vertices move simultaneously along straight-line trajectories at constant speeds. There is a crossing-free morph between two drawings of an n-vertex planar graph G with \(\mathcal {O}(n)\) morphing steps, and using the third dimension the number of steps can be reduced to \(\mathcal {O}(\log n)\) for an n-vertex tree [Arseneva et al. 2019]. However, these morphs do not bound one practical parameter, the resolution. Can the number of steps be reduced substantially by using the third dimension while keeping the resolution bounded throughout the morph? We present a 3D crossing-free morph between two planar grid drawings of an n-vertex tree in \(\mathcal {O}(\sqrt{n} \log n)\) morphing steps. Each intermediate drawing lies in a 3D grid of polynomial volume.Keywordsmorphing grid drawingsbounded resolution3D morphing