Star structure fault tolerance of Bicube networks

Processor and communication link failures are inevitable in a large multiprocessor system, and so the fault tolerance of its underlying interconnection network has become a key scientific issue. Connectivity is an important parameter to characterize network fault tolerance, and there are many novel variants of classical connectivity to measure the fault tolerance of interconnection networks. However, these new strategies only consider a single faulty vertex. Structure connectivity and substructure connectivity make up for this deficiency, which underline the fault situation with certain specific structures. H-structure-connectivity $ \kappa (G;H) $ κ(G;H)(resp. H-substructure-connectivity $ \kappa ^s(G;H) $ κs(G;H)) of Gis the minimum cardinality of H-structure-cuts (resp. H-substructure-cuts). For the n-dimensional Bicube network $ BQ_n $ BQn, we establish the structure and substructure connectivity of Bicube networks, i.e. $ \kappa {(BQ_n;K_{1,1})}=\kappa ^s{(BQ_n;K_{1,1})}= n $ κ(BQn;K1,1)=κs(BQn;K1,1)=nfor odd $ n\geq 5 $ n≥5; $ \kappa {(BQ_n;K_{1,1})}=\kappa ^s{(BQ_n;K_{1,1})}= n-1 $ κ(BQn;K1,1)=κs(BQn;K1,1)=n−1for even $ n\geq 4 $ n≥4and $ \kappa {(BQ_n;K_{1,r})}=\kappa ^s{(BQ_n;K_{1,r})}=\lceil \frac {n}{2}\rceil $ κ(BQn;K1,r)=κs(BQn;K1,r)=⌈n2⌉for $ n\geq 6 $ n≥6and $ 2\leq r\leq n-1 $ 2≤r≤n−1.