This article proposes a new approach to segment a discrete 3-D object into a structure of characteristic topological primitives with attached qualitative features. This structure can be seen itself as a qualitative description of the object, because : - it is intrinsic to the 3-D object, which means it is stable to rigid transformations (rotations and translations); - it is locally defined, and therefore stable to partial occlusions and local modifications of the object structure; - it is robust to noise and small deformations, as confirmed by our experimental results. Our approach concentrates on topological properties of discrete surfaces. These surfaces may correspond to the external surface of the objects extracted by a 3-D edge detector, or to the skeleton surface obtained by a new thinning algorithm. Our labeling algorithm is based on very local computations, allowing massively parallel computations and real-time computations. An indirect result of these topological properties is a new characterization of simple points. We present a realistic experiment to characterize and locate spatially a complex 3-D medical object using the proposed segmentation of its skeleton.