**Conjecture**There exists an integer such that every -arc-strong digraph with specified vertices and contains an out-branching rooted at and an in-branching rooted at which are arc-disjoint.

Thomassen [T] showed that, given a digraph and two vertices and , deciding whether there are an out-branching rooted at and an in-branching rooted at which are arc-disjoint is NP-complete.

In contrast, one can decide in polynomial time whether there are arc-disjoint out-branchings with specified roots (some of which may be identical). This is a consequence of Edmonds’ well known branching theorem [E] states that a digraph has arc-disjoint out-branchings rooted at some fixed vertex if and only if there are arc-disjoint paths from to every other vertex of .

Bang-Jensen [B] proved this conjecture for tournaments.

A similar question can be asked about arc-disjoint strongly connected spanning subdigraphs. Several related problems are mentioned in the survey of Bang-Jensen and Kriesell [BK].

## Bibliography

[B] J. Bang-Jensen, Edge-disjoint in- and out-branching in tournaments and related path problems. J. Combin. Theory Ser. B 51 (1991), 1-23.

[BK] J. Bang-Jensen, M. Kriesell, Disjoint sub(di)graphs in digraphs, Electronic Notes in Discrete Mathematics 34 (2009), 179-183.

[E] J. Edmonds, Edge-disjoint branchings. In Combinatorial Algorithms, B. Rustin, ed., Acad. Press, New York (1973), 91-96.

*[T] C. Thomassen, Configurations in Graphs, Annals of The New York Acad. Sci. 555 (1989), 402-412.

* indicates original appearance(s) of problem.