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Strictly convex drawings of planar graphs

Imre Barany·Guenter Rote
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Abstract

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.

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Imre Barany, Guenter Rote. 2006-06-21. Strictly convex drawings of planar graphs. https://arxiv.org/abs/cs/0507030

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