Finding one tight cycle

Sergio Cabello, Matt Devos, Jeff Erickson, Bojan Mohar

Research output: Contribution to journalArticlepeer-review


A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class. We describe an algorithm to compute a single tight, noncontractible, essentially simple cycle on a given orientable combinatorial surface in O(n log n) time. The only method previously known for this problem was to compute the globally shortest noncontractible or nonseparating cycle in O(min{g3, n} n log n) time, where g is the genus of the surface. As a consequence, we can compute the shortest cycle freely homotopic to a chosen boundary cycle in O (n log n) time, a tight octagonal decomposition in O(gn log n) time, and a shortest contractible cycle enclosing a nonempty set of faces in O(n log2 n) time.

Original languageEnglish (US)
Article number61
JournalACM Transactions on Algorithms
Issue number4
StatePublished - Aug 2010


  • Combinatorial surface
  • Noncontractible cycle
  • Nonseparating cycle
  • Topological graph theory

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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