Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More generally, a strictly convex drawing exists on a grid of size O(W) x O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.
@article{arxiv.cs/0507030,
title = {Strictly convex drawings of planar graphs},
author = {Imre Barany and Guenter Rote},
journal= {arXiv preprint arXiv:cs/0507030},
year = {2007}
}
Comments
20 pages, 13 figures. to be published in Documenta Mathematica. The revision includes numerous small additions, corrections, and improvements, in particular: - a discussion of the constants in the O-notation, after the statement of thm.1. - a different set-up and clarification of the case distinction for Lemma 1