In the present paper, we consider a class of compact orientable 3-manifolds
with one boundary component, and suppose that the manifolds are ∂-reducible and
admit complete surface systems. One of our main results says that for a compact
orientable, irreducible and ∂-reducible 3-manifold M with one boundary component
F of genus n > 0 which admits a complete surface system S′
, if D is a collection
of pairwise disjoint compression disks for ∂M, then there exists a complete surface
system S for M, which is equivalent to S′
, such that D is disjoint from S. We also
obtain some properties of such 3-manifolds which can be embedded in S3.