Generalized Measures of Edge Fault Tolerance in (n,k)-star Graphs

TL;DR

This paper introduces a generalized fault tolerance measure for (n,k)-star graphs, providing exact values that demonstrate their robustness in large-scale parallel systems.

Contribution

It determines the exact generalized edge fault tolerance measure for (n,k)-star graphs, extending understanding of their robustness.

Findings

The measure $ ext{lambda}_s^{(h)}$ equals $ ext{min}ig\{(n-h-1)(h+1), (n-k+1)(k-1)ig\}$.

At least $ ext{min}ig\{(n-k+1)(k-1), (n-h-1)(h+1)ig\}$ edges must be removed to disconnect the graph without low-degree vertices.

The results confirm the robustness of (n,k)-star graphs in modeling large-scale parallel systems.

Abstract

This paper considers a kind of generalized measure $\lambda_s^{(h)}$ of fault tolerance in the $(n,k)$-star graph $S_{n,k}$ for $2\leqslant k \leqslant n-1$ and $0\leqslant h \leqslant n-k$, and determines $\lambda_s^{(h)}(S_{n,k})=\min\{(n-h-1)(h+1), (n-k+1)(k-1)\}$, which implies that at least $\min\{(n-k+1)(k-1),(n-h-1)(h+1)\}$ edges of $S_{n,k}$ have to remove to get a disconnected graph that contains no vertices of degree less than $h$. This result shows that the $(n,k)$-star graph is robust when it is used to model the topological structure of a large-scale parallel processing system.