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 $G \models \neg μ$? In fact, we verify a more general dichotomy conjecture from (Bodirsky et al., 2024) 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 $Δ_μ$ 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.

翻译：我们证明了有向图合取查询并集（即针对有向图签名$\{R\}$的存在正语句）的韧性问题的复杂度二分性。具体而言，对于每个有向图合取查询的并集$μ$，以下问题属于P类或NP完全问题：给定一个有向多重图$G$和一个自然数$u$，我们能否从$G$中移除$u$条边使得$G \models \neg μ$成立？事实上，我们验证了(Bodirsky等人，2024)中针对有向图特殊情形下所有韧性问题的更广义二分猜想，并证明对于此类查询并集$μ$，存在一个可数无限（"对偶"）赋值结构$Δ_μ$：该结构要么能原始正构造1-in-3-3-SAT问题，从而根据一般原理使得$μ$的韧性问题成为NP完全问题；要么具有伪循环典型分数多态性，此时$μ$的韧性问题属于P类。