ð-reducible Dehn surgery and annular Dehn surgery

Let M be a compact, orientable, irreducible, ∂-irreducible, anannular 3-manifold with one component T of ∂M a torus. Suppose that r1 and r2 are two slopes on T. In this paper, we shall prove that if M(r1) is ∂-reducible while M(r2) contains an essential annulus, then Δ(r1, r2) ⩽ 3.