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Loops in Reeb Graphs of 2-Manifolds


Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(n log n), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.

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Correspondence to Kree Cole-McLaughlin or Herbert Edelsbrunner or John Harer or Vijay Natarajan or Valerio Pascucci.

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Cole-McLaughlin, K., Edelsbrunner, H., Harer, J. et al. Loops in Reeb Graphs of 2-Manifolds. Discrete Comput Geom 32, 231–244 (2004). https://doi.org/10.1007/s00454-004-1122-6

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  • Lower Bound
  • Boundary Component
  • Morse Function
  • Reeb Graph