Abstract
AbstractWe present an algorithm for finding shortest surface non-separating cycles in graphs with given edge-lengths that are embedded on surfaces. The time complexity is O(g 3/2 V 3/2log V + g 5/2 V 1/2), where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3 − ε), this represents a considerable improvement over previous results by Thomassen, and Erickson and Har-Peled. We also give algorithms to find a shortest non-contractible cycle in O(g \(^{O({\it g})}\) V 3/2) time, improving previous results for fixed genus.This result can be applied for computing the (non-separating) face-width of embedded graphs. Using similar ideas we provide the first near-linear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the face-width of embedded toroidal graphs in O(V 5/4log V) time.KeywordsPlanar GraphFundamental GroupHomology GroupShort CycleEmbed GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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