Embedded connectivity of recursive networks

TL;DR

This paper investigates the embedded connectivity and edge-connectivity of recursive networks like hypercubes, star graphs, and bubble-sort networks, providing exact values for these measures to understand their fault tolerance.

Contribution

It determines the h-embedded and edge-connectivity for hypercubes and star graphs, and the edge-connectivity for bubble-sort networks, advancing the understanding of network robustness.

Findings

Calculated $ ext{zeta}_h$ and $ ext{eta}_h$ for hypercubes and star graphs.

Established $ ext{eta}_3$ for bubble-sort networks.

Enhanced understanding of fault tolerance in recursive networks.

Abstract

Let $G_n$ be an $n$-dimensional recursive network. The $h$-embedded connectivity $\zeta_h(G_n)$ (resp. edge-connectivity $\eta_h(G_n)$) of $G_n$ is the minimum number of vertices (resp. edges) whose removal results in disconnected and each vertex is contained in an $h$-dimensional subnetwork $G_h$. This paper determines $\zeta_h$ and $\eta_h$ for the hypercube $Q_n$ and the star graph $S_n$, and $\eta_3$ for the bubble-sort network $B_n$.