Realization of a graph as the Reeb graph of a Morse function on a manifold

TL;DR

This paper demonstrates how any graph with a good orientation can be realized as the Reeb graph of a Morse function on a closed manifold, extending previous results and characterizing Reeb graphs of surfaces.

Contribution

It generalizes the realization of graphs as Reeb graphs for higher-dimensional manifolds and characterizes which graphs can appear as Reeb graphs of surfaces.

Findings

Any graph with a good orientation can be realized as a Reeb graph on an n-manifold.

Reeb graphs of simple Morse functions maximize the number of cycles.

Complete characterization of graphs that can be Reeb graphs of surfaces.

Abstract

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.