Evaluating the reliability of complete Josephus cubes under extra link fault with the optimal solution of the edge isoperimetric problem

Abstract

The h-extra edge-connectivity of a connected graph G ${\lambda }_h\left(G\right)$, as a generalization of the classic Menger's Theorem, is the minimum number of edges that need to be removed to disconnect graph G and ensure that each component of the remaining graph has at least h vertices. This paper focuses on the reliability of the h-extra edge-connectivity of the complete Josephus cube $CJ{C}_n$ interconnection network, a variant of $Q_n$, as the underlying topology, denoted as ${\lambda }_h\left(CJ{C}_n\right)$. By analyzing the properties of the optimal solution of the edge isoperimetric problem on $CJ{C}_n$, the exact value and the ${\lambda }_h$-optimality of the ${\lambda }_h\left(CJ{C}_n\right)$ for $1\le h\le 2^{\lfloor n/2\rfloor +1}$ is characterized. For a sufficiently large positive integer n, about 44.44% of positive integers h in the well-defined interval $1\le h\le 2^{n-1}$, the corresponding ${\lambda }_h\left(CJ{C}_n\right)$ occurs a concentration phenomenon.