Edge-Fault Tolerance of Hypercube-like Networks

TL;DR

This paper introduces a generalized fault tolerance measure for hypercube-like networks, proving a precise formula that enhances understanding of their resilience to edge failures.

Contribution

It establishes a new fault tolerance measure $lambda_s^{(h)}$ for hypercube-like networks and proves its exact value, improving theoretical understanding of network robustness.

Findings

$lambda_s^{(h)}(G_n)= 2^h(n-h)$ for all relevant $h$

At least $2^h(n-h)$ edges must be removed to disconnect the network without low-degree vertices

The result improves the theoretical fault-tolerance bounds of hypercube-like networks.

Abstract

This paper considers a kind of generalized measure $\lambda_s^{(h)}$ of fault tolerance in a hypercube-like graph $G_n$ which contain several well-known interconnection networks such as hypercubes, varietal hypercubes, twisted cubes, crossed cubes and M\"obius cubes, and proves $\lambda_s^{(h)}(G_n)= 2^h(n-h)$ for any $h$ with $0\leqslant h\leqslant n-1$ by the induction on $n$ and a new technique. This result shows that at least $2^h(n-h)$ edges of $G_n$ have to be removed to get a disconnected graph that contains no vertices of degree less than $h$. Compared with previous results, this result enhances fault-tolerant ability of the above-mentioned networks theoretically.