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2011

Small Grid Embeddings of 3-Polytopes

9 years 1 months ago
Small Grid Embeddings of 3-Polytopes
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(188n ). 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(25.54n ). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such 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.
Ares Ribó Mor, Günter Rote, Andr&eacut
Added 14 May 2011
Updated 14 May 2011
Type Journal
Year 2011
Where DCG
Authors Ares Ribó Mor, Günter Rote, André Schulz
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