Arc-disjoint spanning sub(di)graphs in digraphs

4 We prove that a number of natural problems concerning the existence of arc-disjoint directed 5 and “undirected” (spanning) subdigraphs in a digraph are NP-complete. Among these are the 6 following of which the first settles an open problem due to Thomassé (see e.g. [1, Problem 9.9.7] 7 and [3, 5]) and the second settles an open problem posed in [3]. 8 • Given a directed graph D and a vertex s of D; does D contain an out-branching B s rooted 9 at s such that the digraph remains connected (in the underlying sense) after removing all 10 arcs of B s ? 11 • Given a strongly connected directed graph D; does D contain a spanning strong subdigraph 12 D ′ such that the digraph remains connected (in the underlying sense) after removing all arcs 13 of D? 14