System level fault diagnosis theory is of great weight in the design and maintenance of multiprocessor systems. In this paper, we propose a novel kind of diagnosability, called non-inclusive g-extra diagnosability, that requires all faulty sets to be non-inclusive and every connected fault-free subgraph has at least g+1 vertices. Furthermore, we determine the non-inclusive g-extra diagnosability of a class of graphs under PMC model. As applications, the non-inclusive g-extra diagnosability of (n,k)-star graph Sn,k, data center networks DCell (Dk,n), BCDC (Bn) and BCube (BCn,k) under PMC model are determined successively.