Boundary reducible handle additions on simple 3-manifolds

Let $M$ be a simple manifold, and $F$ be a component of $\partial M$ of genus two. For a slope $\gamma$ on $F$, we denote by $M(\gamma)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $\gamma$ on $F$. In this paper, we shall prove that there is at most one separating slope $\gamma$ on $F$ so that $M(\gamma)$ is $\partial$-reducible.