A finite graph is the Reeb graph of a Morse circle-valued function

Abstract

We show that any non-trivial finite connected graph (allowing loop edges and multiple edges) is isomorphic to the Reeb graph of a Morse circle-valued function on a closed n-manifold of a given dimension n ≥ 2; this manifold roughly resembles a thick version of the graph, we present its construction and study its properties. In the case of surfaces (n = 2), we prove a criterion for when a finite graph can be realized as the Reeb graph of such a function on a given orientable surface.