A Novel Links Fault Tolerant Analysis: $g$-Good $r$-Component Edge-Connectivity of Interconnection Networks With Applications to Hypercubes

Abstract

The underlying topology of the interconnection network of parallel and distributed systems is usually modelled by a simple connected graph $G$. In order to quantitatively analyze the reliability and fault tolerance of these networks more accurately, this study introduces a novel topology parameter. The $g$-good $(r+1)$-component edge-connectivity $\lambda _{g,r+1}(G)$ of $G$, if any, is the smallest cardinality of faulty link set, whose malfunction yields a disconnected graph with at least $r+1$ connected components, and with the neighboring edges of any vertex being at least $g$. When designing and maintaining parallel and distributed systems, the hypercube network $Q_{n}$ is one of the most attractive interconnection network models. This article offers a unified method to derive an upper bound for $g$-good $(r+1)$-component edge-connectivity $\lambda _{g,r+1}(Q_{n})$ of $Q_{n}$. When $n\geq 4$, this upper bound is proved to be tight for $1\leq 2^{g}\cdot r\leq 2^{\lfloor \frac{n}{2}\rfloor }$ or $r=2^{k_{0}}$, $0\leq k_{0}< \lfloor \frac{n}{2}\rfloor$, $0\leq g\leq n-2k_{0}-1$. The conclusions for the $g$-good-neighbor edge-connectivity of $Q_{n}$ from Xu and the $(r+1)$-component edge-connectivity of $Q_{n}$ from Zhao et al. are contained as corollaries of our main results for $r=1$, $0\leq g\leq n-1$ and $1\leq r\leq 2^{\lfloor \frac{n}{2}\rfloor }$, $g=0$, respectively.