Article

Combinatorica

, Volume 10, Issue 1, pp 41-51

How to draw a planar graph on a grid

  • H. De FraysseixAffiliated withCNRS
  • , J. PachAffiliated withMathematical Institute of the Hungarian Academy of SciencesCourant Institute, NYU
  • , R. PollackAffiliated withCourant Institute, NYU

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Abstract

Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fáry embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) 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 setF, which can support a Fáry embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

AMS subject classification (1980)

05C10 05C35 68E10