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arXiv · 2007.07927

How to Morph Graphs on the Torus

Erin Wolf Chambers·Jeff Erickson·Patrick Lin·Salman Parsa
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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.

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Erin Wolf Chambers, Jeff Erickson, Patrick Lin, Salman Parsa. 2020-07-15. How to Morph Graphs on the Torus. https://arxiv.org/abs/2007.07927

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