Small sets supporting Fáry embeddings of planar graphs

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 2n - 4 by n - 2 grid and provide an O(n) space, O(nlogn) time algorithm to efl'ect 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-(-(1-0(1))√N which settles a problem of Mohar. © 1988 ACM.