Arc-disjoint paths in decomposable digraphs

We prove that the weak k-linkage problem is polynomial for every fixed k for totally Φ-decomposable digraphs, under appropriate hypothesis on Φ. We then apply this and recent results by Fradkin and Seymour (on the weak k-linkage problem for digraphs of bounded independence number or bounded cut-width) to get polynomial algorithms for some classes of digraphs like quasi-transitive digraphs, extended semicomplete digraphs, locally semicomplete digraphs (all of which contain the class of semicomplete digraphs as a subclass) and directed cographs.