How to draw a planar graph on a grid

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.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.