On conditional fault tolerance of hierarchical cubic networks

TL;DR

This paper analyzes the conditional fault tolerance of hierarchical cubic networks, establishing exact values for their $h$-super connectivity and edge-connectivity, which quantify their resilience under vertex or edge failures.

Contribution

It provides exact formulas for the $h$-super connectivity and edge-connectivity of $HCN_n$, generalizing previous results and enhancing understanding of their fault tolerance.

Findings

$ ext{kappa}^h(HCN_n)= ext{lambda}^h(HCN_n)=2^h(n+1-h)$ for $0 \\leq h \\leq n-1$

At least $2^h(n+1-h)$ vertices or edges must be removed to disconnect $HCN_n$ with minimum degree at least $h$

Generalizes known results on hierarchical cubic networks' fault tolerance.

Abstract

This paper considers the conditional fault tolerance, $h$-super connectivity $\kappa^{h}$ and $h$-super edge-connectivity $\lambda^{h}$ of the hierarchical cubic network $HCN_n$, an attractive alternative network to the hypercube, and shows $\kappa^h(HCN_n)=\lambda^h(HCN_n)=2^h(n+1-h)$ for any $h$ with $0\leq h\leq n-1$. The results imply that at least $2^h(n+1-h)$ vertices or edges have to be removed from $HCN_n$ to make it disconnected with no vertices of degree less than $h$, and generalize some known results.