Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 8, 2021

A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function

Irina Gelbukh
From the journal Mathematica Slovaca


We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion ℝ3.

  1. (Communicated by Július Korbaš)


We thank D. Panov for finding, by our request, the immersion [16] of the Klein bottle, Figure 8. We also thank the anonymous reviewer for valuable suggestions.


[1] Biasotti, S.—Giorgi, D.—Spagnuolo, M.—Falcidieno, B.: Reeb Graphs for shape analysis and applications, Theoret. Comput. Sci. 392 (2008), 5–22.10.1016/j.tcs.2007.10.018Search in Google Scholar

[2] Bolsinov, A. V.—Fomenko, A. T.: Integrable Hamiltonian Systems: Geometry, Topology, Classification, CRC Press, USA, 2004.10.1201/9780203643426Search in Google Scholar

[3] Franks, J.: Nonsingular Smale flows on S3, Topology 24(3) (1985), 265–282.10.1016/0040-9383(85)90002-3Search in Google Scholar

[4] Fraysseix, H.—de Mendez, O.—Rosenstiehl, P.: Bipolar orientations revisited, Discrete Appl. Math. 56(2–3) (1995), 157–179.10.1016/0166-218X(94)00085-RSearch in Google Scholar

[5] Gelbukh, I.: The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations, Math. Slovaca 67(3) (2017), 645–656.10.1515/ms-2016-0298Search in Google Scholar

[6] Gelbukh, I.: Loops in Reeb graphs of {n}-manifolds, Discrete Comput. Geom. 59(4) (2018), 843–863.10.1007/s00454-017-9957-9Search in Google Scholar

[7] Gelbukh, I.: Approximation of metric spaces by Reeb graphs: Cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifolds, Filomat 33(7) (2019), 2031–2049.10.2298/FIL1907031GSearch in Google Scholar

[8] Gelbukh, I.: Morse–Bott functions with two critical values on a surface, Czech. Math. J. (2021), 10.21136/CMJ.2021.0125-20.Search in Google Scholar

[9] Kaluba, M.—Marzantowicz, W.—Silva, N.: On representation of the Reeb graph as a sub-complex of manifold, Topol. Meth. Nonl. Anal. 45(1) (2015), 287–305.10.12775/TMNA.2015.015Search in Google Scholar

[10] Kudryavtseva, E. A.: Realization of smooth functions on surfaces as height functions, Sb. Math. 190(3) (1999), 349–405.10.1070/SM1999v190n03ABEH000392Search in Google Scholar

[11] Leininger, C. J.—Reid, A. W. The Co-rank conjecture for 3-manifold groups, Alg. Geom. Topol. 2 (2002), 37–50.10.2140/agt.2002.2.37Search in Google Scholar

[12] Martínez-Alfaro, J.—Meza-Sarmiento, I. S.—Oliveira, R.: Topological classification of simple Morse Bott functions on surfaces. In: Real and Complex Singularities, Contemporary Mathematics, Vol. 675, AMS, 2016, pp. 165–179.10.1090/conm/675/13590Search in Google Scholar

[13] Masumoto, Y.—Saeki, O.: Smooth function on a manifold with given Reeb graph, Kyushu J. of Math. 65(1) (2011), 75–84.10.2206/kyushujm.65.75Search in Google Scholar

[14] Michalak, Ł. P.: Combinatorial modifications of Reeb graphs and the realization problem, Discrete Comput. Geom. (2021), in Google Scholar

[15] Michalak, Ł. P.: Realization of a graph as the Reeb graph of a Morse function on a manifold, Topol. Methods Nonlinear Anal. 52(2) (2018), 749–76210.12775/TMNA.2018.029Search in Google Scholar

[16] Panov, D.: Immersion in ℝ3 of a Klein bottle with Morse-Bott height function without centers,, (version: 2019-10-13).Search in Google Scholar

[17] Reeb, G: Sur les points singuliers d'une forme de Pfaff complétement intégrable ou d'une fonction numérique, C.R.A.S. Paris 222 (1946), 847–849.Search in Google Scholar

[18] Osamu Saeki, O.: Reeb spaces of smooth functions on manifolds, IMRN (2021), 10.1093/imrn/rnaa301.Search in Google Scholar

[19] Sharko, V. V.: About Kronrod-Reeb graph of a function on a manifold, Methods Funct. Anal. Topol. 12(4) (2006), 389–396.Search in Google Scholar

Received: 2020-03-08
Accepted: 2020-07-20
Published Online: 2021-06-08
Published in Print: 2021-06-25

© 2021 Mathematical Institute Slovak Academy of Sciences