Arc-disjoint out- and in-branchings in compositions of digraphs

Abstract

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 good$\mathrm{(u,v)}$-pair in $D$ is a pair of branchings $B_u^{+},B_v^{-}$ which have no arc in common. Thomassen proved that it is NP-complete to decide if a digraph has any good pair. A digraph is semicomplete if it has no pair of non-adjacent vertices. A 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. 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 no such pair. In this paper we show how to use the result for semicomplete digraphs to completely solve the problem of characterizing semicomplete compositions which have a good $(u,v)$-pair for given vertices $u,v$. Our solution implies that the problem of deciding the existence of a good $(u,v)$-pair and finding such a pair when it exists is polynomially solvable for all semicomplete compositions. We also completely solve the problem of deciding the existence of a good $(u,v)$-pair and finding one when it exists for digraphs that are compositions of transitive digraphs. Combining these two results we obtain 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 proves a conjecture of Bang-Jensen and Gutin from 1998.