Pole Dancing: 3D Morphs for Tree Drawings

Abstract

We study the question whether a crossing-free 3D morph between two straight-line drawings of an n-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with O(log n) steps, while for the latter T(n) steps are always sufficient and sometimes necessary.