Generalized handlebody sets and non-Haken 3-manifolds

In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of nonseparating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance between the boundary sets of the handlebodies is zero if and only if the ambient manifold contains a nonseparating, two sided incompressible surface. We show that every vertex in the curve complex is within two edges of a point in the boundary set.