On conditional fault tolerance of hierarchical cubic networks

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 $HC{N}_n$, an attractive alternative network to the hypercube, and shows ${\kappa }^h(HC{N}_n)={\lambda }^h(HC{N}_n)=2^h(n+1-h)$ for any $h$ with $0\le h\le n-1$. The results imply that at least $2^h(n+1-h)$ vertices or edges have to be removed from $HC{N}_n$ to make it disconnected with no vertices of degree less than $h$, and generalize some known results.