How to morph graphs on the torus

Erin Wolf Chambers, Jeff Erickson, Patrick Lin, Salman Parsa

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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(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.

Original languageEnglish (US)
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Pages2759-2778
Number of pages20
ISBN (Electronic)9781611976465
StatePublished - 2021
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: Jan 10 2021Jan 13 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual
Period1/10/211/13/21

ASJC Scopus subject areas

  • Software
  • General Mathematics

Fingerprint

Dive into the research topics of 'How to morph graphs on the torus'. Together they form a unique fingerprint.

Cite this