  Let M be a closed, irreducible, genus two 3-manifold, and F a maximal
collection of pairwise disjoint, closed, orientable, incompressible surfaces
embedded in M. Then each component manifold M_i of M-F has handle number at
most one, i.e. admits a Heegaard splitting obtained by attaching a single
1-handle to one or two components of boundary M_i. This result also holds for a
decomposition of M along a maximal collection of incompressible tori.
