Abstract
We 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.
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Cabello, S., Mohar, B. (2005). Finding Shortest Non-separating and Non-contractible Cycles for Topologically Embedded Graphs. In: Brodal, G.S., Leonardi, S. (eds) Algorithms – ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol 3669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11561071_14
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DOI: https://doi.org/10.1007/11561071_14
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