  An out-branching $B^+_u$ (in-branching $B^-_u$) in a digraph $D$ is a
connected spanning subdigraph of $D$ in which every vertex except the vertex
$u$, called the root, has in-degree (out-degree) one. A {\bf good
$\mathbf{(u,v)}$-pair} in $D$ is a pair of branchings $B^+_u,B^-_v$ which have
no arc in common.
  Thomassen proved that is NP-complete to decide if a digraph has any good
pair. A digraph is {\bf semicomplete} if it has no pair of non adjacent
vertices. A {\bf semicomplete composition} is any digraph $D$ which is obtained
from a semicomplete digraph $S$ by substituting an arbitrary digraph $H_x$ for
each vertex $x$ of $S$.
  Recently the authors of this paper gave a complete classification of
semicomplete digraphs which have a good $(u,v)$-pair, where $u,v$ are
prescribed vertices of $D$. They also gave a polynomial algorithm which for a
given semicomplete digraph $D$ and vertices $u,v$ of $D$, either produces a
good $(u,v)$-pair in $D$ or a certificate that $D$ has such pair. In this paper
we show how to use the result for semicomplete digraphs to completely solve the
problem of deciding whether a given semicomplete composition $D$, has a good
$(u,v)$-pair for given vertices $u,v$ of $D$. Our solution implies that the
problem is polynomially solvable for all semicomplete compositions. In
particular our result implies that there is a polynomial algorithm for deciding
whether a given quasi-transitive digraph $D$ has a good $(u,v)$-pair for given
vertices $u,v$ of $D$. This confirms a conjecture of Bang-Jensen and Gutin from
1998.
