The tightly super 3-extra connectivity and 3-extra diagnosability of crossed cubes

Many multiprocessor systems have interconnection networks as underlying topologies and an interconnection network is usually represented by a graph where nodes represent processors and links represent communication links between processors. In 2016, Zhang et al. proposed the g-extra diagnosability of G, which restrains that every component of G − S has at least (g + 1) vertices. As an important variant of the hypercube, the n-dimensional crossed cube CQn has many good properties. In this paper, we prove that CQn is tightly (4n − 9) super 3-extra connected for n ≥ 7 and the 3-extra diagnosability of CQn is 4n − 6 under the PMC model (n ≥ 5) and MM∗ model (n ≥ 7).