Realization of a graph as the Reeb graph of a height function on an embedded surface
DOI:
https://doi.org/10.12775/TMNA.2021.058Słowa kluczowe
{Reeb graph, height function, Morse-Bott function, orientable surface, embeddingAbstrakt
We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $\mathbb{R}^3$ such that the Reeb graph of the associated height function has the structure of $G$. In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011. We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functions and in the class of round Morse-Bott functions. In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott; we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.Bibliografia
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Prawa autorskie (c) 2023 Irina Gelbukh
Utwór dostępny jest na licencji Creative Commons Uznanie autorstwa – Bez utworów zależnych 4.0 Międzynarodowe.
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