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A Simpler and Faster Torus Embedding Algorithm

A graph is embeddable on a surface S if it can be drawn on S with no crossing edges. A topological obstruction for a surface S is a graph G that does not embed on S, but for all edges e in G, G − e embeds on S. A minor order obstruction has the additional property that, for all edges e, G · e (G contract e) also embeds on S. Solving the well-studied problem of finding a complete set of obstructions for the torus is facilitated by having a large database of torus obstructions. With this in mind, we have designed a new exponential torus embedding algorithm inspired by Demoucron’s O(n) planar embedding algorithm. Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented. Our implementation of our new algorithm is faster than implementations of previous exponential algorithms that have been used to find torus obstructions.