Arc-disjoint spanning sub(di)graphs in digraphs

We prove that a number of natural problems concerning the existence of arc-disjoint directed and “undirected” (spanning) subdigraphs in a digraph are NP-complete. Among these are the following of which the first settles an open problem due to Thomassé (see e.g. Bang-Jensen and Gutin (2009) [1, Problem 9.9.7] and Bang-Jensen and Kriesell (2009) [5,4]) and the second settles an open problem posed in Bang-Jensen and Kriesell (2009) [5]. •Given a directed graph D and a vertex s of D; does D contain an out-branching Bs+ rooted at s such that the digraph remains connected (in the underlying sense) after removing all arcs of Bs+?•Given a strongly connected directed graph D; does D contain a spanning strong subdigraph D′ such that the digraph remains connected (in the underlying sense) after removing all arcs of D′?