Generalized Measures of Fault Tolerance in Exchanged Hypercubes

TL;DR

This paper introduces generalized fault tolerance measures for exchanged hypercubes, providing exact formulas for the minimum vertex and edge removals needed to disconnect the network while maintaining certain degree conditions.

Contribution

It derives explicit formulas for generalized fault tolerance measures in exchanged hypercubes, extending previous results and enhancing understanding of their robustness.

Findings

Fault tolerance measures are exactly 2^h(s+1-h) for exchanged hypercubes.

The results generalize known fault tolerance bounds.

Provides insights into the resilience of exchanged hypercubes against vertex and edge failures.

Abstract

The exchanged hypercube $EH(s,t)$, proposed by Loh {\it et al.} [The exchanged hypercube, IEEE Transactions on Parallel and Distributed Systems 16 (9) (2005) 866-874], is obtained by removing edges from a hypercube $Q_{s+t+1}$. This paper considers a kind of generalized measures $\kappa^{(h)}$ and $\lambda^{(h)}$ of fault tolerance in $EH(s,t)$ with $1\leqslant s\leqslant t$ and determines $\kappa^{(h)}(EH(s,t))=\lambda^{(h)}(EH(s,t))= 2^h(s+1-h)$ for any $h$ with $0\leqslant h\leqslant s$. The results show that at least $2^h(s+1-h)$ vertices (resp. $2^h(s+1-h)$ edges) of $EH(s,t)$ have to be removed to get a disconnected graph that contains no vertices of degree less than $h$, and generalizes some known results.