Decomposing locally semicomplete digraphs into strong spanning subdigraphs

We prove that the arc set of every 2-arc-strong locally semicomplete digraph D=(V,A) which is not the second power of an even cycle can be partitioned into two sets A1,A2 such that both of the spanning subdigraphs D1=(V,A1) and D2=(V,A2) are strongly connected. Moreover, we show that such a partition (if it exists) can be obtained in polynomial time. This generalizes a result from Bang-Jensen and Yeo (2004) [5], on semicomplete digraphs and implies that every 2-arc-strong locally semicomplete digraph D=(V,A) has a pair of arc-disjoint branchings such that is an in-branching rooted at u and is an out-branching rooted at v where u,v∈V can be chosen arbitrarily. This generalizes results from Bang-Jensen (1991) [2], for tournaments and Bang-Jensen and Yeo (2004) [5] for semicomplete digraphs.