The vulnerability of the diameter of enhanced hypercubes

TL;DR

This paper investigates the wide diameters and fault diameters of enhanced hypercubes, demonstrating their robustness by deriving exact formulas for these parameters based on network connectivity and disjoint paths.

Contribution

The paper provides the first exact formulas for the wide and fault diameters of enhanced hypercubes, extending previous bounds and confirming their high robustness.

Findings

Exact formulas for $D_\omega(Q_{n,k})$ and $d_\omega(Q_{n,k})$ are derived.

Enhanced hypercubes exhibit high fault tolerance and robustness.

Results confirm enhanced hypercubes as reliable interconnection networks.

Abstract

For an interconnection network $G$, the {\it $\omega$-wide diameter} $d_\omega(G)$ is the least $\ell$ such that any two vertices are joined by $\omega$ internally-disjoint paths of length at most $\ell$, and the {\it $(\omega-1)$-fault diameter} $D_{\omega}(G)$ is the maximum diameter of a subgraph obtained by deleting fewer than $\omega$ vertices of $G$. The enhanced hypercube $Q_{n,k}$ is a variant of the well-known hypercube. Yang, Chang, Pai, and Chan gave an upper bound for $d_{n+1}(Q_{n,k})$ and $D_{n+1}(Q_{n,k})$ and posed the problem of finding the wide diameters and fault diameters of $Q_{n,k}$. By constructing internally disjoint paths between any two vertices in the enhanced hypercube, for $n\ge3$ and $2\le k\le n$ we prove $$ D_\omega(Q_{n,k})=d_\omega(Q_{n,k})=\begin{cases} d(Q_{n,k}) & \textrm{for $1 \leq \omega < n-\lfloor\frac{k}{2}\rfloor$;}\\ d(Q_{n,k})+1 & \textrm{for $n-\lfloor\frac{k}{2}\rfloor \leq \omega \leq n+1$.} \end{cases} $$ where $d(Q_{n,k})$ is the diameter of $Q_{n,k}$. These results mean that interconnection networks modelled by enhanced hypercubes are extremely robust.