How to Morph Graphs on the Torus

TL;DR

This paper introduces the first algorithm for morphing graphs on the torus, enabling continuous deformation between isotopic embeddings while maintaining geodesic edges, with applications in topological graph visualization.

Contribution

It presents a novel algorithm for morphing torus graphs that guarantees geodesic edges throughout the deformation, extending planar morphing techniques to the torus.

Findings

Algorithm runs in O(n^{1+ω/2}) time.

Morph consists of O(n) linear steps.

First known method for geodesic-preserving torus graph morphing.

Abstract

We present the first algorithm to morph graphs on the torus. Given two isotopic essentially 3-connected embeddings of the same graph on the Euclidean flat torus, where the edges in both drawings are geodesics, our algorithm computes a continuous deformation from one drawing to the other, such that all edges are geodesics at all times. Previously even the existence of such a morph was not known. Our algorithm runs in $O(n^{1+\omega/2})$ time, where $\omega$ is the matrix multiplication exponent, and the computed morph consists of $O(n)$ parallel linear morphing steps. Existing techniques for morphing planar straight-line graphs do not immediately generalize to graphs on the torus; in particular, Cairns' original 1944 proof and its more recent improvements rely on the fact that every planar graph contains a vertex of degree at most 5. Our proof relies on a subtle geometric analysis of 6-regular triangulations of the torus. We also make heavy use of a natural extension of Tutte's spring embedding theorem to torus graphs.