How to morph planar graph drawings

TL;DR

This paper presents an optimal linear-step method for morphing between two planar graph drawings while preserving planarity, improving upon previous exponential-step approaches.

Contribution

It introduces an efficient linear-step morphing algorithm for planar graphs that preserves planarity and matches the theoretical lower bound on steps needed.

Findings

Morphs can be achieved in O(n) steps, which is optimal.

Each step is a unidirectional linear morph with parallel lines.

The method improves upon Cairns' exponential-step proof for triangulated graphs.

Abstract

Given an $n$-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of $O(n)$ steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns' 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps.