How to draw a planar graph on a grid (original) (raw)

Abstract

Answering a question of Rosenstiehl and Tarjan, we show that every plane graph with_n_ vertices has a Fáry embedding (i.e., straight-line embedding) on the 2_n_−4 by_n_−2 grid and provide an_O(n)_ space,O(n log_n_) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any set_F_, which can support a Fáry embedding of every planar graph of size_n_, has cardinality at least_n_+(1−o(1))√n which settles a problem of Mohar.

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Authors and Affiliations

  1. CNRS, Paris, France
    H. De Fraysseix
  2. Mathematical Institute of the Hungarian Academy of Sciences, Hungary
    J. Pach
  3. Courant Institute, NYU, New York, USA
    J. Pach & R. Pollack

Authors

  1. H. De Fraysseix
  2. J. Pach
  3. R. Pollack

Additional information

Supported in part by P. R. C. Mathematiques et Informatique.

Supported in part by HSF grant 1814.

Part of the work on this paper was done while this author was on sabbatical leave at école Normal Supérieure and école des Hautes études en Sciences Sociales; supported in part by NSF grant DMS-850 1947.

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De Fraysseix, H., Pach, J. & Pollack, R. How to draw a planar graph on a grid.Combinatorica 10, 41–51 (1990). https://doi.org/10.1007/BF02122694

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