The Heegaard genera of surface sums

Let M be a compact orientable 3-manifold, and let F be a separating (resp. non-separating) incompressible surface in M which cuts M into two 3-manifolds M1 and M2 (resp. a manifold M1). Then M is called the surface sum (resp. self surface sum) of M1 and M2 (resp. M1) along F, denoted by M=M1∪FM2 (resp. M=M1∪F). In this paper, we will study how g(M) is related to χ(F), g(M1) and g(M2) when both M1 and M2 have high distance Heegaard splittings.