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Graphs with Flexible Labelings

Abstract

For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings, possibly non-generic. The characterization is based on colorings of the edges with restrictions on the cycles. Furthermore, we give necessary criteria and sufficient ones for the existence of such colorings.

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Acknowledgements

This Project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement No. 675789. Partially supported by the Austrian Science Fund (FWF): P26607, W1214-N15 (Project DK9); and by the Upper Austrian Government.

Author information

Correspondence to Jan Legerský.

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Editor in Charge: Kenneth Clarkson

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Cite this article

Grasegger, G., Legerský, J. & Schicho, J. Graphs with Flexible Labelings. Discrete Comput Geom 62, 461–480 (2019). https://doi.org/10.1007/s00454-018-0026-9

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Keywords

  • Graph realization
  • Flexibility
  • Rigidity
  • Linkage
  • Laman graph

Mathematics Subject Classification

  • 51K99
  • 70B99
  • 05C78