Reeb Graphs of Piecewise Linear Functions

The Reeb graph is a popular tool in the field of computational topology for shape analysis. The Reeb graph is usually thought of as a transform from shapes, viewed as spaces endowed with functions, to graphs. It finds its roots in the classical Morse theory, where the Reeb graph transform is granted to produce a graph, but it finds its applications mostly in Computer Graphics. Therefore it is usually applied on objects that are not smooth but polyhedral. While the definition of the Reeb graph perfectly makes sense also in the polyhedral case, it is not straightforward to see that the output of the transform in this case is a graph. This paper is devoted to provide a formal guarantee of this fact.