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Title: How to morph graphs on the torus
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(n1+ω/2) time, where ω 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.  more » « less
Award ID(s):
1907612
PAR ID:
10297730
Author(s) / Creator(s):
Date Published:
Journal Name:
Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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