The $g$-extra $H$-structure connectivity and $g$-extra $H$-substructure connectivity of hypercubes

At present, the reliability of interconnection networks of multiprocessing systems has become a hot topic of research concern for parallel computer systems. Conditional connectivity is an important parameter to measure the reliability of an interconnected network. In reality, the failure of one node will inevitably have a negative impact on the surrounding nodes. Often it is the specific structures that fail in an interconnected network. Therefore, we propose two novel kinds of connectivity, called $g$-extra $H$-structure connectivity and $g$-extra $H$-substructure connectivity, to go for a more accurate measure of the reliability of the network. Hypercube network is the most dominant interconnection network topology used by computer systems today, for example, the famous parallel computing systems Cray $T3D$, Cray $T3E$, $IBM$ Blue Gene, etc. are built with it as the interconnection network topology. In this paper, we obtain the results of the $g$-extra $H$-structure connectivity and the $g$-extra $H$-substructure connectivity of the hypercubes when the specific structure is $P_k$ and $g=1$.