Shortest Path Embeddings of Graphs on Surfaces

@article{Hubard2016ShortestPE,
  title={Shortest Path Embeddings of Graphs on Surfaces},
  author={Alfredo Hubard and Vojtech Kaluza and Arnaud de Mesmay and Martin Tancer},
  journal={Discrete \& Computational Geometry},
  year={2016},
  volume={58},
  pages={921-945}
}
The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry’s theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S… 
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