Arc-disjoint out-branchings and in-branchings in semicomplete digraphs

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 semicomplete if it has no pair of nonadjacent vertices. A tournament is a semicomplete digraph without directed cycles of length 2. In this paper we give a complete classification of semicomplete digraphs that 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.