Discrete & Computational Geometry

, Volume 37, Issue 2, pp 213–235 | Cite as

Finding Shortest Non-Separating and Non-Contractible Cycles for Topologically Embedded Graphs

  • Sergio CabelloEmail author
  • Bojan MoharEmail author


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.


Short Path Planar Graph Universal Cover Short Cycle Short Path Tree 
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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of Mathematics, Institute for Mathematics, Physics and Mechanics, University of Ljubljana, 1000LjubljanaSlovenia
  2. 2.Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, 1000LjubljanaSlovenia

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