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Flexible Placements of Periodic Graphs in the Plane

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Abstract

Given a periodic graph, we wish to determine via combinatorial methods whether it has periodic embeddings in the plane that—via motions that preserve edge-lengths and periodicity—can be continuously deformed into another non-congruent embedding of the graph. By introducing NBAC-colourings for the corresponding quotient gain graphs, we identify which periodic graphs have flexible embeddings in the plane when the lattice of periodicity is fixed. We further characterise with NBAC-colourings which 1-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity, and characterise in special cases which 2-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity.

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Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. Although (G, p) is the standard notation for a framework, we shall instead reserve this for the quotient frameworks that we use throughout the majority of this paper.

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Acknowledgements

I would like to thank the anonymous referees for their valuable corrections and comments, all of which helped greatly to improve the paper.

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  1. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, 4040, Linz, Austria

    Sean Dewar

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  1. Sean Dewar
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Correspondence to Sean Dewar.

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Dewar, S. Flexible Placements of Periodic Graphs in the Plane. Discrete Comput Geom 66, 1286–1329 (2021). https://doi.org/10.1007/s00454-021-00328-x

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