  Let $M$ be a surface sum of 3-manifolds $M_1$ and $M_2$ along a bounded
connected surface $F$ and $\partial_i$ be the component of $\partial M_i$
containing $F$. If $M_i$ has a high distance Heegaard splitting, then any
minimal Heegaard splitting of $M$ is the amalgamation of those of $M^1, M^2$
and $M^*$, where $M^i=M_i\setminus\partial_i\times I$, and
$M^{*}=\partial_1\times I\cup_{F} \partial_2\times I$. Furthermore, once both
$\partial_i\setminus F$ are connected, then $g(M) = Min\bigl\{g(M_1)+g(M_2),
\alpha\bigr\}$, where $\alpha = g(M_1) + g(M_2) + 1/2(2\chi(F) + 2 -
\chi(\partial_1) - \chi(\partial_2)) - Max\bigl\{g(\partial_1),
g(\partial_2)\bigl\}$; in particular $g(M)=g(M_1)+g(M_2)$ if and only if
$\chi(F)\geq 1/2Max\bigl\{\chi(\partial_1), \chi(\partial_2)\bigr\}.$
  The proofs rely on Scharlemann-Tomova's theorem.
