This paper describes a novel and unified approach for Reeb graph construction and simplification as well as constriction approximation on 3D polygonal meshes. The key idea of our algorithm is that discrete contours - curves carried by the edges of the mesh and approximating the continuous contours of a mapping function - encode both topological and geometrical shape characteristics. Firstly, mesh feature points are computed. Then they are used as geodesic origins for the computation of an invariant mapping function that reveals the shape most significant features. Secondly, for each vertex in the mesh, its discrete contour is computed. As the set of discrete contours recovers the whole surface, each of them can be analyzed, both to detect topological changes or constrictions. Constriction approximations enable Reeb graphs refinement into more visually meaningful skeletons, that we refer as enhanced topological skeletons. Without pre-processing stages and without input parameters, our method provides nice-looking and affine- invariant skeletons, with satisfactory execution times. This makes enhanced topological skeletons good candidates for applications needing high level shape representations, such as mesh deformation (experimented in this paper), retrieval, compression, metamorphosis, etc.