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Interdependent directed networks model real-life systems, like trade flows and social interactions, where asymmetric edges drive one-way cascades and mutual dependencies amplify vulnerabilities. Dismantling these networks to minimize the largest mutually strongly connected component (MSCC) is an NP-hard problem. We propose Dismantling Directed Interdependent Networks (DDIN), a novel combination of Reinforcement Learning (RL) and Graph Neural Networks (GNN) framework, to address this problem. Our contributions include (i) a directed GraphSAGE encoder separating in/out aggregations for asymmetry, (ii) multi-relational attention fusing layer semantics, and (iii) sum-tree prioritized n-step Deep Q-Network (DQN) for efficient policy search in sparse states. Evaluated on three directed multiplexes (FAO Trade, Homo Genetic, Sanremo 2016), DDIN achieves 17-22% lower AUDC values compared to heuristics like High Degree Attack (HDA) and Directed Collective Influence (DCI).