The generalized 3-connectivity of a family regular networks

TL;DR

This paper introduces a new family of regular networks constructed from subgraphs and determines their generalized 3-connectivity, with applications to several well-known interconnection network topologies.

Contribution

The paper defines a new family of regular networks based on subgraphs and computes their generalized 3-connectivity, extending understanding of network robustness.

Findings

Determined the generalized 3-connectivity of the network family $H_n$.

Applied results to hierarchical star, cubic, and folded hypercube networks.

Provided insights into network resilience and fault tolerance.

Abstract

The generalized $k$-connectivity of a graph $G$, denoted by $\kappa_k(G)$, is the minimum number of internally edge disjoint $S$-trees for any $S\subseteq V(G)$ with $|S|=k$. The generalized $k$-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. In this paper, we firstly introduce a family of regular networks $H_n$ that can be obtained from several subgraphs $G_n^1, G_n^2, \cdots, G_n^{t_n}$ by adding a matching, where each subgraph $G_n^i$ is isomorphic to a particular graph $G_n$ ($1\le i\le t_n$). Then we determine the generalized 3-connectivity of $H_n$. As applications of the main result, the generalized 3-connectivity of some two-level interconnection networks, such as the hierarchical star graph $HS_n$, the hierarchical cubic network $HCN_n$ and the hierarchical folded hypercube $HFQ_n$, are determined directly.