Edge isoperimetric method for link fault tolerance of the complete Josephus cube under five models: A unified approach

The significance of large parallel processing systems in semiconductor technology underscores the need to delve into both qualitative and quantitative parameters for assessing fault tolerance and reliability. Conditional edge-connectivity dynamically evaluates the traits of isolated components in the face of diverse link faulty models, offering a more accurate assessment of the fault tolerance and reliability of extensive parallel processing systems. Admitting network sizes in powers of two, the complete Josephus cube $CJ{C}_n$ is a link-augmented Josephus cube. This configuration is well-suited for implementation in expansive hierarchical interconnection networks. By integrating elements from fractal geometry and iterative processes, we introduce a unified method that leverages the edge isoperimetric problem in combinatorics to scrutinize the $P$-conditional edge-connectivity of $CJ{C}_n$. This examination encompasses the investigation of $2^{c-1}$-extra edge-connectivity, modified $c$-embedded edge-connectivity, $c$-super edge-connectivity and $c$-average degree edge-connectivity. For $3\le c\le n-1$ and $n\ge 4$, these measures exhibit uniform values, specifically $(n-c+2)2^{c-1}$, signifying the minimum cardinalities of faulty links that lead to a $(c-1)$-dimensional complete Josephus cube from $CJ{C}_n$. Furthermore, we establish the exact value of the cyclic edge-connectivity of $CJ{C}_n$ for $n\ge 3$.