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.