Abstract
We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in \(O(g^{3/2}V^{3/2}\log V+g^{5/2}V^{1/2})\) time, where V is the number of vertices in the graph and g is the genus of the surface. If \(g=o(V^{1/3})\), this represents an improvement over previous results by Thomassen, and Erickson and Har-Peled. We also give algorithms to find a shortest non-contractible cycle in \(O(g^{O(g)}V^{3/2})\) time, which improves previous results for fixed genus. This result can be applied for computing the face-width and 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 \(O(V^{5/4}\log V)\) time.
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