Hyper star structure connectivity of hierarchical folded cubic networks

With the increasing popularity and diversity of network environments, it is crucial to assess the fault tolerance and stability of the network. Structure connectivity and substructure connectivity are two novel indicators that can better measure the network’s fault tolerance compared to traditional connectivity. Additionally, analyzing a network’s minimum structure cuts and minimum substructure cuts is an interesting and important subject. For a graph G, let R and M be two connected subgraphs of G. An R-structure cut (resp. R-substructure cut) of G is a set of subgraphs of G, such that each subgraph in the set is isomorphic to R (resp. is isomorphic to a connected subgraph of R), whose deletion disconnects G. If the removal of any minimum R-structure cut (resp. R-substructure cut) divides G into exactly two components, one of which is isomorphic to M, then G is referred to as hyper R|_M -connected (resp. hyper sub- R|_M -connected). This paper first studies the K_1,r -structure connectivity and sub- K_1,r -structure connectivity of hierarchical folded cubic network HFQ_n . Specifically, we determine both of them are ⌈n+2/2⌉ for n≥ 7 and 2≤ r≤ n-1 . Then, we prove that HFQ_n is hyper K_1,r|_K_1 -connected and hyper sub- K_1,r|_K_1 -connected.