A formula on Heegaard genus of self-amalgamated 3-manifolds

Let M be an orientable compact irreducible and ∂-irreducible 3-manifold, and suppose ∂M consists of two boundary components F1 and F2 with g(F1)=g(F2)>1. Let Mf be the closed orientable 3-manifold obtained by identifying F1 and F2 via a homeomorphism f:F1→F2. With the assumption that M is small or g(M,F1)=g(M)+g(F1), we show that if f is sufficiently complicated, then g(Mf)=g(M,∂M)+1.