, Volume 45, Issue 1, pp 65-87

Small Grid Embeddings of 3-Polytopes

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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(27.55n )=O(188 n ). If the graph contains a triangle we can bound the integer coordinates by O(24.82n ). If the graph contains a quadrilateral we can bound the integer coordinates by O(25.46n ). 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.

Research of A. Ribó Mor was partially supported by the Deutsche Forschungsgemeinschaft within the European Research Training Network Combinatorics, Geometry and Computation (No. GRK 588/2).
Research of A. Schulz was partially supported by the German Research Foundation (DFG) under grant SCHU 2458/1-1.