  Let M be a compact, orientable, irreducible, atoroidal 3-manifold with
boundary an incompressible torus. Techniques based on the characteristic
submanifold theory are used to bound the intersection number of two slopes
\alpha and \beta on the boundary of M. The method applies when \beta is the
boundary slope of an essential surface F that is not a semi-fiber (i.e. F is
not a fiber and does not split M into two twisted I-bundles), and the Dehn
filling M(\alpha) contains a suitable singular surface. One of the main results
is that if F is planar and if the fundamental group of M(\alpha) does not
contain a non-abelian free subgroup then the intersection number of \alpha and
\beta is at most 5.
