Dehn filling with non-degenerate boundary slope rows

We prove that Dehn filling a small link exterior with a non-degenerate boundary slope row produces a 3-manifold which is either Haken and irreducible or one of very restricted typies of reducible manifolds (Theorem 2), generalizing a result of Culler, Gordon, Luecke and Shalen in the case of a knot exterior (Theorem 1). The result provides some interesting applications on exceptional Dehn fillings (Corollaries 3 and 4) and on telling if a link is small (Corollaries 5 and 6).