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In recent years, the most of studies on link fault diagnosis can only localize a faulty area with some uncertain links while link failures occurred. In order to overcome this issue, the core concept of k-link fault diagnostic capability is introduced in this research on all-optical mesh networks (AONs). According to this concept, the accuracy of fault diagnostic capability can be guaranteed so that the telecommunication and internet service operators can set a number of link faults k as the level of fault tolerance according to their needs on fault management. In addition, to meet the cost-effective objectives under each level of k, the probe numbers and the total probe length need to be optimal. Yet, these two factors are the negative correlation to each other, thus we propose a heuristic method, named an adaptive probe selection algorithm (PSA), to satisfy this trade-off issue. Finally, in order to better reflect the configuration facts and the establishment requirements on metro-edge fiber-optic networks, we adopt bi-directional links on the networks as example environments. It is because the one-way transmission characteristic of optical fibers is often overlooked. According to above, the simulations will demonstrate and compare the efficiency between different methods and networks, and the performance evaluation reveals that our method can function equally well even with large and complex networks.