  We show that if two 3-manifolds with toroidal boundary are glued via a
`sufficiently complicated' map then every Heegaard splitting of the resulting
3-manifold is weakly reducible. Additionally, if Z is a manifold obtained by
gluing X and Y, two connected small manifolds with incompressible boundary,
along a closed surface F. Then the genus g(Z) of Z is greater than or equal to
1/2(g(X)+g(Y)-2g(F)). Both results follow from a new technique to simplify the
intersection between an incompressible surface and a strongly irreducible
Heegaard splitting.
