Abstract: Given a triangle mesh representing a closed manifold surface of arbitrary genus, a method is proposed to automatically extract the Reeb graph of the manifold with respect to the height function. The method is based on a slicing strategy that traces contours while inserting them directly in the mesh as constraints. Critical areas, which identify isolated and non-isolated critical points of the surface, are recognized and coded in the extended Reeb graph (ERG). The remeshing strategy guarantees that topological features are correctly maintained in the graph, and the tiling of ERG nodes reproduces the original shape at a minimal, but topologically correct, geometric level.