We prove a complexity dichotomy for the resilience problem for unions of conjunctive digraph queries (i.e., for existential positive sentences over the signature {R} of directed graphs). Specifically, for every union µ of conjunctive digraph queries, the following problem is in P or NP-complete: given a directed multigraph G and a natural number u, can we remove u edges from G so that G |= ¬µ? In fact, we verify a more general dichotomy conjecture from [6] for all resilience problems in the special case of directed graphs, and show that for such unions of queries µ there exists a countably infinite (‘dual’) valued structure ∆µ which either primitively positively constructs 1-in-3-3-SAT, and hence the resilience problem for µ is NP-complete by general principles, or has a pseudo cyclic canonical fractional polymorphism, and the resilience problem for µ is in P.