  Given a 3-manifold M containing an incompressible surface Q, we obtain an
inequality relating the Heegaard genus of M and the Heegaard genera of the
components of M - Q. Here the sum of the genera of the components of M - Q is
bounded above by a linear expression in terms of the genus of M, the Euler
characteristic of Q and the number of parallelism classes of essential annuli
for which representatives can be simultaneously imbedded in the components of M
- Q.
