Discrete & Computational Geometry

, Volume 59, Issue 4, pp 843–863 | Cite as

Loops in Reeb Graphs of n-Manifolds



The Reeb graph of a smooth function on a connected smooth closed orientable n-manifold is obtained by contracting the connected components of the level sets to points. The number of loops in the Reeb graph is defined as its first Betti number. We describe the set of possible values of the number of loops in the Reeb graph in terms of the co-rank of the fundamental group of the manifold and show that all such values are realized for Morse functions and, except on surfaces, even for simple Morse functions. For surfaces, we describe the set of Morse functions with the number of loops in the Reeb graph equal to the genus of the surface.


Reeb graph Contour tree Number of loops Morse function Co-rank of the fundamental group 

Mathematics Subject Classification

05C38 05E45 58C05 


Compliance with Ethical Standards

Conflict of interest

The author declares that she has no conflict of interest.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.CIC, IPNMexico CityMexico

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