@article{Gelbukh_2023, title={Realization of a graph as the Reeb graph of a height function on an embedded surface}, volume={61}, url={https://apcz.umk.pl/TMNA/article/view/42231}, DOI={10.12775/TMNA.2021.058}, abstractNote={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.}, number={2}, journal={Topological Methods in Nonlinear Analysis}, author={Gelbukh, Irina}, year={2023}, month={Jan.}, pages={591–610} }