A new structure for a vertex to be locally t-diagnosable in large multiprocessor systems

•To improve the local diagnosability of a vertex, we propose a new structure for a vertex to be locally t-diagnosable.•We also provide the corresponding diagnosis algorithm for the new structure.•We provide a sufficient and necessary condition for a vertex to be locally optimal diagnosed in faulty hypercubes.•We also give a sufficient and necessary condition for a vertex to be locally optimal diagnosed in faulty folded hypercubes.