  We analyze how a family of essential annuli in a compact 3-manifold will
induce, from a strongly irreducible generalized Heegaard splitting of the
ambient manifold, generalized Heegaard splittings of the complementary
components. There are specific applications to the subadditivity of tunnel
number of knots, improving somewhat bounds of Kowng.
  For example, in the absence of 2-bridge summands, the tunnel number of the
sum of n knots is no less than 2/5 the sum of the tunnel numbers.
