Abstract
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.
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Notes
Some authors refer to a Reeb graph without cycles as a contour tree.
A graph admitting multiple edges and loops, i.e., edges that connect a vertex with itself.
The number of independent cycles in the graph, also called cyclomatic number or nullity.
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Gelbukh, I. Loops in Reeb Graphs of n-Manifolds. Discrete Comput Geom 59, 843–863 (2018). https://doi.org/10.1007/s00454-017-9957-9
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DOI: https://doi.org/10.1007/s00454-017-9957-9