Strictly convex drawings of planar graphs

TL;DR

This paper proves that three-connected planar graphs can be drawn with strictly convex faces on polynomial-sized grids, improving understanding of geometric representations of such graphs.

Contribution

It introduces a method to produce strictly convex drawings of planar graphs on polynomial grids, extending previous convex drawing techniques.

Findings

Strictly convex drawings exist on grids of size O(n^2) x O(n^2).

A more general construction allows for adjustable grid sizes based on a parameter W.

Explicit bounds are provided for the number of primitive vectors in a triangle.

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.