Topological decomposition of a smooth manifold

Question

The decomposition of a smooth manifold for topological analysis requires that all vertices lie at the ends of edges, all edges bound the incident faces, and so on. Thus for example a decomposition of the sphere must be a graph. But I am unable to find a generic name for this kind of decomposition in n dimensions. Incidence complexes, CW complexes and so on seem to be more general constructs which apply to other kinds of object. On the other hand simplicial complexes are over-specific, as the elements need only be simple (i.e. topological balls) and not specifically simplices. Is there a specific term for such a decomposition of a manifold into simple elements, such that there are no rogue vertices, etc?