  In this paper, we give an algorithm to build all compact orientable atoroidal
Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so
that in both cases, there are embedded closed orientable separating
incompressible surfaces which are not tori. Next, such incompressible surfaces
are related to Heegaard splittings. For simplicity, we focus on the case of
separating incompressible surfaces, since non-separating ones have been
extensively studied. After putting the surfaces into Morse position relative to
the height function associated to the Heegaard splittings, a thin position
method is applied so that levels are thin or thick, depending on the side of
the surface. The complete description of the surface in terms of these
thin/thick levels gives a hierarchy. Also this thin/thick description can be
related to properties of the curve complex for the Heegaard surface.
