Comparing Heegaard splittings -the bounded case

In a recent paper we used Cerf theory to compare strongly irreducible Heegaard splittings of the same closed irreducible orientable 3-manifold. This captures all irreducible splittings of non-Haken 3-manifolds. One application is a solution to the stabilization problem for such splittings: If are the genera of two splittings, then there is a common stabilization of genus . Here we show how to obtain similar results even when the 3-manifold has boundary.