Realization of a graph as the Reeb graph of a Morse function on a manifold
By L. Michalak. Topological Methods in Nonlinear Analysis, 2018
66 citations recorded by the source corpus.
We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $\Gamma$ admitting the so-called good orientation there exist an $n$-manifold $M$ and a Morse function $f\colon M\rightarrow \mathbb{R} $ such that its Reeb graph $\mathcal{R}(f)$ is isomorphic to $\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces.
Subfields: Artificial Intelligence, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/7ddfa6f4f9400c259e87fe50c91aa884daf47267
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Citations in the corpus
- Reeb Spaces of Smooth Functions on Manifolds (2020, background): “by Sharko [29], Mart́ınez-Alfaro, Meza-Sarmiento and Oliveira [18, 19, 20], Masumoto and Saeki [21], Gelbukh [2, 3, 4, 5], Kaluba, Marzantowicz and Silva [12], Michalak [22, 23], Michalak and Marzantowicz [17], Kitazawa [13, 14, 15], etc.”