Disjoint A-paths in digraphs

For a set A of vertices of a digraph G, let E G - ( A ) and E G + ( A ) denote the set of edges whose terminal, respectively, initial vertex is in A. An A-path is a path in G connecting two distinct vertices of A and having no internal vertex in A. proved that for a set A of vertices of a finite digraph G, there exist k disjoint A-paths if and only if | S ∩ T | + ∑ ⌊ 1 2 | C ∩ ( A ∪ S ∪ T ) | ⌋ ⩾ k for all S , T ⊆ V ( G ) with S ∩ A = T ∩ A , where the respective sum is taken over the weak components C of G - ( E G - ( S ) ∪ E G + ( T ) ) . eneralizes a graph result of Gallai to digraphs.