Author:
Irina GelbukhCIC, Instituto Politécnico Nacional, 07738, Mexico City, Mexico

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We prove criteria for a graph to be the Reeb graph of a function of a given class on a closed manifold: Morse–Bott, round, and in general smooth functions whose critical set consists of a finite number of submanifolds. The criteria are given in terms of whether the graph admits an orientation, which we call S-good orientation, with certain conditions on the degree of sources and sinks, similar to the known notion of good orientation in the context of Morse functions. We also study when such a function is the height function associated with an immersion of the manifold. The condition for a graph to admit an S-good orientation can be expressed in terms of the leaf blocks of the graph.

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    Erica Boizan Batista, João Carlos Ferreira Costa, and Ingrid Sofia Meza-Sarmiento. Topo-logical classification of circle-valued simple Morse-Bott functions. Journal of Singularities, 17:388402, 2018.

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    Erica Boizan Batista, João Carlos Ferreira Costa, and Juan J. Nuño-Ballesteros. Loops in generalized Reeb graphs associated to stable circle-valued functions. Journal of Singular-ities, 22:104113, 2020.

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    Irina Gelbukh. Loops in Reeb graphs of n-manifolds. Discrete Comput. Geom., 59(4):843863, 2018.

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    Irina Gelbukh. 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):20312049, 2019.

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    Irina Gelbukh. A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function. Math. Slovaca, 71(3):757772, 2021.

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    Irina Gelbukh. Morse–Bott functions with two critical values on a surface. Czech. Math. J., 71(3):865880, 2021.

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    Irina Gelbukh. Criterion for a graph to admit a good orientation in terms of leaf blocks. To appear in Monatsh. Math., 2022.

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    Marek Kaluba, Wacław Marzantowicz, and Nelson Silva. On representation of the Reeb graph as a sub-complex of manifold. Topol. Methods Nonlinear Anal., 45(1):287305, 2015.

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    Olexandra Khohliyk and Sergiy Maksymenko. Diffeomorphisms preserving Morse–Bott functions. Indag. Math., 31(2):185203, 2020.

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    Naoki Kitazawa. Maps on manifolds onto graphs locally regarded as a quotient map onto a Reeb space and construction problem, 2019. pre-print, 12 pages, arXiv:1909.10315 [math.GT].

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    Naoki Kitazawa. On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely many singular values, 2019. pre-print, 9 pages, arXiv:1902.08841 [math.GT].

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    Naoki Kitazawa. On Reeb graphs induced from smooth functions on closed or open mani- folds, 2019. pre-print, 18 pages, arXiv:1908.04340 [math.GT].

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    Anna Kravchenko and Sergiy Maksymenko. Automorphisms of Kronrod–Reeb graphs of Morse functions on compact surfaces. Eur. J. Math., 6(1):114131, 2020.

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    Dahisy V.S. Lima, Oziride Manzoli Neto, and Ketty A. de Rezende. On handle theory for Morse–Bott critical manifolds. Geom. Dedicata, 202:265309, 2019.

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    Sergiy Maksymenko. Homotopy types of stabilizers and orbits of Morse functions on surfaces. Ann. Glob. Anal. Geom., 29:241285, 2006.

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    Jose Martínez-Alfaro, Ingrid Sofia Meza-Sarmiento, and Regilene Oliveira. Topological classification of simple Morse Bott functions on surfaces. In Real and Complex Singularities, number 675 in Contemporary Mathematics, pages 165–179. AMS, 2016.

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    Łukasz Patryk Michalak. Realization of a graph as the Reeb graph of a Morse function on a manifold. Topol. Methods Nonlinear Anal., 52(2):749762, 2018.

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    Łukasz Patryk Michalak. Combinatorial modifications of Reeb graphs and the realization problem. Discrete Comput. Geom., 65(4):10381060, 2021.

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    Osamu Saeki. Reeb spaces of smooth functions on manifolds. Int. Math. Res. Not., February 2021.

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    Vladimir Vasilievich Sharko. About Kronrod-Reeb graph of a function on a manifold. Meth-ods Funct. Anal. Topol., 12(4):389396, 2006.

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    Bin Yu. Lyapunov graphs of nonsingular Smale fiows on S1 × S2. Trans. Amer. Math. Soc., 365(2):767783, 2013.

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2021  
Web of Science  
Total Cites
WoS
589
Journal Impact Factor 0,739
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5 Year
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Journal Citation Indicator 0,57
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Scimago
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Scimago Quartile Score Mathematics (miscellaneous) (Q3)
Scopus  
Scopus
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Scopus
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2020  
Total Cites 536
WoS
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without
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Citable 32
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Articles
Total 0
Reviews
Scimago 24
H-index
Scimago 0,307
Journal Rank
Scimago Mathematics (miscellaneous) Q3
Quartile Score  
Scopus 139/130=1,1
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Scopus General Mathematics 204/378 (Q3)
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Scopus 1,069
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Days from  85
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2019  
Total Cites
WoS
463
Impact Factor 0,468
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without
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0,468
5 Year
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0,413
Immediacy
Index
0,135
Citable
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37
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37
Total
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0
Cited
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Citing
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Article Influence
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0,196
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H-index
23
Scimago
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0,234
Scopus
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Scopus
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Studia Scientiarum Mathematicarum Hungarica
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