  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. It is well-known that
there exists a polynomial algorithm for deciding whether a given digraph has
$k$ arc-disjoint out-branchings with prescribed roots ($k$ is part of the
input). In sharp contrast to this, it is already NP-complete to decide if a
digraph has one out-branching which is arc-disjoint from some in-branching. A
digraph is {\bf semicomplete} if it has no pair of non adjacent vertices. A
{\bf tournament} is a semicomplete digraph without directed cycles of length 2.
In this paper we give a complete classification of semicomplete digraphs which
have an out-branching $B^+_u$ which is arc-disjoint from some in-branching
$B^-_v$ where $u,v$ are prescribed vertices of $D$. Our characterization, which
is surprisingly simple, generalizes a complicated characterization for
tournaments from 1991 by the first author and our proof implies the existence
of a polynomial algorithm for checking whether a given semicomplete digraph has
such a pair of branchings for prescribed vertices $u,v$ and construct a
solution if one exists. This confirms a conjecture of Bang-Jensen for the case
of semicomplete digraphs.
