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

An out-branching Bu+ (in-branching Bu−) 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(u,v)-pair in D is a pair of branchings Bu+,Bv− 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 Hx 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.