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Finding shortest nonseparating and noncontractible cycles for topologically embedded graphs
 Discrete Comput. Geom
, 2005
"... We present an algorithm for finding shortest surface nonseparating 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−ε), this represents a considerable improvement over p ..."
Abstract

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We present an algorithm for finding shortest surface nonseparating 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−ε), this represents a considerable improvement over previous results by Thomassen, and Erickson and HarPeled. We also give algorithms to find a shortest noncontractible 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 (nonseparating) facewidth of embedded graphs. Using similar ideas we provide the first nearlinear running time algorithm for computing the facewidth of a graph embedded on the projective plane, and an algorithm to find the facewidth of embedded toroidal graphs in O(V 5/4 log V) time. 1