The 4-component connectivity of alternating group networks

Abstract

The -component connectivity (or -connectivity for short) of a graph G, denoted by κ (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least components or a graph with fewer than vertices. This generalization is a natural extension of the classical connectivity defined in term of minimum vertexcut. As an application, the -connectivity can be used to assess the vulnerability of a graph corresponding to the underlying topology of an interconnection network, and thus is an important issue for reliability and fault tolerance of the network. So far, only a little knowledge of results have been known on -connectivity for particular classes of graphs and small 's. In a previous work, we studied the -connectivity on n-dimensional alternating group networks AN n and obtained the result κ 3 (AN n ) = 2n − 3 for n 4. In this sequel, we continue the work and show that κ 4 (AN n ) = 3n − 6 for n 4.

Measuring the Vulnerability of Alternating Group Graphs and Split-Star Networks in Terms of Component Connectivity

“…Recently, Guo [26] and Guo et al [27] determined the {3, 4}-connectivity of twisted cubes and locally twisted cubes, respectively. Also, Chang et al [3], [4] determined the {3, 4}-connectivity of alternating group networks AN n . Note that the two classes of AG n and AN n are definitely different.…”

The m-Component Connectivity of Leaf-Sort Graphs

“…Zhang et al [18] determined the (m + 1)-component connectivity of augmented cubes. In addition, the m-component connectivity (m = 3, m = 4) has been studied for many graphs: alternating group networks AN n [19,20], split-star networks [20], hierarchical star networks [21], balanced hypercube [22], bubble-sort star graphs, and burnt pancake graphs [23]. However, most of these results are related to m = 3 and m = 4 .…”

A Validation of the Phenomenon of Linearly Many Faults on Burnt Pancake Graphs with Its Applications

“…Henceforth, we adopt appropriate terms called component connectivity and component edge connectivity, suggested by Hsu et al [17] and Zhao et al [18], respectively. For the recent results of interconnection networks, please refer to [19][20][21][22][23] for extra (edge) connectivity, [24][25][26][27][28][29] for component (edge) connectivity, and [12,[30][31][32] for relationship between these two kinds of (edge) connectivity. In addition, for research on connectivity related to diverse graph indices (such as the Wiener index, the Zagreb index, the Randic index, etc.)…”