Computation of Cycle Bases in Surface Embedded Graphs

Authors Kyle Fox, Thomas Stanley

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Author Details

Kyle Fox
  • University of Texas at Dallas, Richardson, TX, USA
Thomas Stanley
  • Unaffiliated, Dallas, TX, USA


Research by T. Stanley was partially performed while this author was a student at the University of Texas at Dallas.

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Kyle Fox and Thomas Stanley. Computation of Cycle Bases in Surface Embedded Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We present an O(n³ g²log g + m) + Õ(n^{ω + 1}) time deterministic algorithm to find the minimum cycle basis of a directed graph embedded on an orientable surface of genus g. This result improves upon the previous fastest known running time of O(m³n + m²n² log n) applicable to general directed graphs. While an O(n^ω + 2^{2g}n² + m) time deterministic algorithm was known for undirected graphs, the use of the underlying field ℚ in the directed case (as opposed to ℤ₂ for the undirected case) presents extra challenges. It turns out that some of our new observations are useful for both variants of the problem, so we present an O(n^ω + n² g² log g + m) time deterministic algorithm for undirected graphs as well.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graphs and surfaces
  • cycle basis
  • surface embedded graphs
  • homology


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