Hunting for torus obstructions




Chambers, John

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A torus is a sphere with one handle. A graph is toroidal if it embeds in the torus with no crossing edges. A topological obstruction for the torus is a graph G with minimum vertex degree three that is not embeddable in the torus but for all edges e, G - e embeds in the torus. A minor order obstruction has the additional property that for all edges e, G contract e embeds in the torus. Two algorithms to find torus obstructions are presented. Using these algorithms, a new lower bound on the number of torus obstructions is established at 239,451 topological obstructions, 16,682 of which are minor order. Also, an alternate approach to finding the torus obstructions is discussed. Implementation of this approach requires knowing the projective planar minor order torus obstructions. An additional contribution of this thesis is the determination of the complete set of 270 projective planar minor order obstructions for the torus.