Connectivity and diagnosability of the complete Josephus cube networks under h-extra fault-tolerant model

The h-extra connectivity and the h-extra diagnosability are key parameters for evaluating the reliability and fault-tolerance of the interconnection networks of the multiprocessor systems, and play an important role in designing and maintaining interconnection networks. Recently, various self-diagnostic models have emerged to assess the fault-tolerance in interconnection networks. These interconnection networks are typically expressed by a connected graph $G(V,E)$. For a non-complete graph G and $h\ge 0$, the h-extra cut signifies a vertex subset R of G, whose removal results in $G-R$ disconnected, with each remaining component containing at least $h+1$ vertices. And the h-extra connectivity of G is defined as the minimum cardinality of all h-extra cuts of G. The h-extra diagnosability for a graph G denotes the maximum number of detectable faulty vertices when focusing on these h-extra faulty sets only. The complete Josephus cube $CJ{C}_n$, a variant of $Q_n$, exhibits superior properties compared to hypercube $Q_n$, and also boasts higher connectivity. In this study, with the help of the exact value of the h-extra connectivity of $CJ{C}_n$, the explicit expression of h-extra diagnosability of $CJ{C}_n$ under both the PMC model for $n\ge 5$ and $1\le h\le \lfloor \frac{n-3}{2}\rfloor$ and the MM* model for $n\ge 5$ and $2\le h\le \lfloor \frac{n-3}{2}\rfloor$ are identified to share the same value $(h+1)n-\left(\begin{array}{c}h-1\\ 2\end{array}\right)+1$.