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.