Small grid embeddings of 3-polytopes

TL;DR

This paper presents an algorithm to embed 3-connected planar graphs as convex 3-polytopes with integer coordinates, providing bounds on coordinate size and extending Tutte's method to boundary vertices.

Contribution

The authors introduce a novel algorithm that guarantees convex 3-polytope embeddings with bounded integer coordinates, extending Tutte's technique to boundary vertices.

Findings

Coordinates bounded by exponential functions of n

Algorithm works for graphs with triangles and quadrilaterals

Extends Tutte's method to boundary vertices

Abstract

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face.