The reliability analysis based on the generalized connectivity in balanced hypercubes

Recently, network connectivity analysis in terms of reliability has received attention from the network research community. Although traditional connectivity can be used to assess the strength of the connection between two nodes, however, such measures are inadequate for evaluating the connectivity among a set of multiple nodes in a network. Given a set S of vertices in a graph G with vertical bar S vertical bar >= 2, we say that a tree in G is an S-tree if it connects all vertices of S. Two S-trees T and T' in G are internally disjoint if E(T)boolean AND E(T') = empty set and V(T) boolean AND V(T') = S. Let (kappa G)(S) denote the maximum number of S-trees in G such that every pair of them are internally disjoint. For an integer k >= 2, the generalized k-connectivity of graph G is defined as kappa(k) (G) = min{kappa(G)(S) vertical bar S subset of V(G) and vertical bar S vertical bar = k}. In this paper, we investigate the problem of finding the generalized 3-connectivity of the n-dimensional balanced hypercube BHn, which is a hypercube-variant network and is superior to hypercube due to having a smaller diameter. As a result, we prove that kappa(3)(BHn) = 2n - 1 for n >= 1. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

The paper explores recent advances in network connectivity analysis with a focus on a more accurate method for evaluating connectivity among network nodes. By studying the generalized k-connectivity problem, it is proven that the n-dimensional balanced hypercube has specific connectivity characteristics.