Handle Additions On Simple 3-Manifolds

Let M be a simple 3-manifold.Let F be a component of ?M with genus at least two.? and ? are separating slopes on F.We attach 2-handles to M along?? and ? respectively and denote the resulting manifolds by M(?) and M(?).Suppose that M(?) and M(?) are non-simple.In this paper,we use the labeled graph to study the relations between?? and ? and obtained the following results:If M(?) and M(?) are ?-reducible,then the minimal geometric intersection numberof?? and ? is at most 8.If M(?) contains an essential annulus and M(?) is ?-reducible,then the minimalgeometric intersection number of?? and ? is at most 8.If M(?) contains an essential annulus and M(?) is reducible,then the minimal geometric intersection number of?? and ? is at most 8.If M(?) contains an essential torus and M(?) is ?-reducible,then the minimal geometricintersection number of?? and ? is at most 8.If M(?) contains an essential torus and M(?) is reducible,then the minimal geometric intersection number of?? and ? is at most 8.If M(?) contains an essential annulus but no essential tori and M(?) is reducible but?-irreducible,then the minimal geometric intersection number of?? and ? is at most 2.