We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with 3+3 vertices where no two vertices coincide or are antipodal.


  1. graph
  2. sphere
  3. Dixon
  4. flexibility
  5. moduli space

MSC codes

  1. 05C99
  2. 53A17
  3. 70B99
  4. 51F99

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Information & Authors


Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 325 - 361
ISSN (online): 1095-7146


Submitted: 24 September 2019
Accepted: 16 November 2020
Published online: 11 March 2021


  1. graph
  2. sphere
  3. Dixon
  4. flexibility
  5. moduli space

MSC codes

  1. 05C99
  2. 53A17
  3. 70B99
  4. 51F99



Funding Information

Austrian Science Fund https://doi.org/10.13039/501100002428 : W1214-N15
H2020 Marie Skłodowska-Curie Actions https://doi.org/10.13039/100010665 : 675789
Erwin Schrödinger International Institute for Mathematics and Physics https://doi.org/10.13039/501100003066 : J425

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