Realization of a graph as the Reeb graph of a height function on an embedded surface

Abstract

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.