Elsevier

Discrete Applied Mathematics

Volume 292, 31 March 2021, Pages 19-32
Discrete Applied Mathematics

The reliability analysis based on the generalized connectivity in balanced hypercubes

https://doi.org/10.1016/j.dam.2020.12.011Get rights and content

Abstract

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 |S|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)E(T)= and V(T)V(T)=S. Let κG(S) denote the maximum number of S-trees in G such that every pair of them are internally disjoint. For an integer k2, the generalized k-connectivity of graph G is defined as κk(G)=min {κG(S)SV(G)and|S|=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 κ3(BHn)=2n1 for n1.

Introduction

It is well known that interconnection networks constitute the communication backbone for the multiprocessor system. An interconnection network can be modeled as an undirected connected graph G=(V,E), where V represents the set of processors and E represents the set of communication links between processors. To evaluate the reliability and fault tolerance of a network in communication, many related concepts and graph parameters have been put forward. As usual, one can use traditional connectivity to assess the strength of the connection between two nodes in a network. Past and recent analyses revealed that the classic connectivity metrics are insufficient to determine how well-connected or well-designed a network is against the node or link failures [14], [25]. In particular, it is important to evaluate the connectivity among a set of multiple nodes in a distributed network when we are in an application for multi-party computation or communication [1], [3]. Hence, the generalized connectivity as one generalization of the traditional connectivity via some tree structures connecting more than two nodes is of great significance.

The connectivity of a connected graph G, denoted by κ(G), is the minimum number of vertices whose deletion results in a disconnected graph. A graph G is k-connected if κ(G)k. A well-known result by Whitney [38] (as a corollary of Menger’s Theorem [34]) provided an equivalent definition of connectivity as follows: A graph G is k-connected if and only if there exist k internally disjoint paths between any two vertices in G. Given a set SV(G) with |S|2, a tree in G is said to be an S-tree if it connects all vertices of S. For an integer k2, the generalized k-connectivity of a graph G is defined as κk(G)=min {κG(S)SV(G)and|S|=k}, where κG(S) denotes the maximum number of S-trees in G such that every pair of them, say T and T, are internally disjoint (i.e., E(T)E(T)= and V(T)V(T)=S). Clearly, the generalized 2-connectivity is exactly the traditional connectivity, i.e., κ(G)=κ2(G).

The generalized k-connectivity was firstly introduced by Hager [11], which has been received much attention in recent years. Li et al. [22] proved that κk(G)κk1(G) is not always true in general case, while κ3(G)κ(G) for every connected graph G. Currently, the exact values of κk(G) are known only for small classes of graphs, such as complete graphs [4], complete bipartite graphs [17], [35], complete equipartition 3-partite graphs [18]. In addition, it has pointed out in [16], [22] that the investigation of κk(G) for general k is very difficult (see also [5] for complexity results). Thus, many pieces of research focused on the study of κ3(G) or κ4(G). For example, results of κ3(G) were obtained for product graphs [21], [23], [24], star graphs and bubble-sort graphs [27], alternating group graphs and (n,k)-star graphs [47], (n,k)-bubble-sort graphs [52], recursive circulants [28], data center networks DCell [12], line graph and the total graph of the complete bipartite graph [15], Cayley graphs generated by trees and cycles [26], Cayley graphs generated by complete graphs and wheel graphs [49], and m-regular m-connected graphs with recursive structure [51]. Also, results of κ4(G) are known for hypercubes [31], exchanged hypercubes [48], and dual cubes [50]. For more works related to κk(G), one may also refer to [19], [20], [36], [37] and a monograph [25].

The n-dimensional balanced hypercube BHn, proposed by Wu and Huang [39], is a hypercube-variant network with high symmetry and superior properties. BHn is a bipartite graph with vertex-symmetry [39] and edge-symmetry [53]. So far, there are many research results of balanced hypercubes. For example, the explorations of (conditional) matching preclusion number [32], completely independent spanning trees [44], diverse connectivities and edge connectivities [9], [30], [43], [53], diagnosabilities in distinct models [10], [41], [42], [46]. In particular, research on path and cycle (fault-tolerant) embeddings have many applications [6], [7], [8], [13], [29], [33], [40], [45]. To the best of our knowledge, there is no research result to deal with the generalized connectivity of BHn.

In this paper, we will consider the generalized 3-connectivity of BHn. Due to the recursive structure, BHn consists of four copies of BHn1, denoted by BHn1i for i{0,1,2,3} (see Definition 2 in the next section). However, the relationships between pairs of copies of BHn1 are distinct. For example, there exist edges between BHn10 and BHn11, while there are no edges between BHn10 and BHn12, which makes the discussion in constructing S-trees more difficult. Moreover, since BHn is a bipartite graph, each copy BHn1i contains two kinds of vertices (called white vertices and black vertices, respectively) whose neighbors outside the copy are completely different. Thus, the situations in the proof of κ3(BHn) become more complex. As a consequence, we determine the following result.

Theorem 1

κ3(BHn)=2n1 for n1.

Section snippets

Preliminaries

Let G=(V,E) be a simple undirected graph. The subgraph of G induced by a subset UV(G) is denoted by G[U], and the subgraph G[V(G)U] is denoted by GU. Let dG(v) stand for the number of edges incident with a vertex v in G. A graph G is k-regular if dG(v)=k for all vV(G). We denote by δ(G) the minimum degree of the vertices of G. For any two vertices x,yV(G), an (x,y)-path is a path starting at x and ending at y. Two (x,y)-paths P and Q are internally disjoint if E(P)E(Q)= and V(P)V(Q)={x,

Proof of Theorem 1

As BHn is 2n-regular, by Lemma 8, we have κ3(BHn)2n1. We now prove that κ3(BHn)2n1 by induction on n. For n=1, the proof is trivial. For n=2, by Lemma 4, κ(BH2)=4=41+0. By Lemma 9, κ3(BH2)3=2n1, it implies that the result holds for n=2. Assume that n3 and the result of Theorem 1 holds for BHn1.

Let S={u,v,w} be an arbitrary 3-subset of V(BHn). The sets of the out-neighbors of u, v and w are denoted by {u,u}, {v,v} and {w,w}, respectively. By Lemma 3(1), without loss of generality,

Concluding remarks

Connectivity is one of the most important concepts in graph theory, and it has been extensively used for measuring the reliability of networks in theoretical computer science. To meet the need for a more refined measure of the reliability in networks, the concept of generalized connectivity is proposed. In this paper, we determine the generalized 3-connectivity of the n-dimensional balanced hypercube BHn. Even though BHn has the recursive structure with four copies of BHn1, the relationship

Acknowledgments

This work was supported by the National Natural Science Foundation of China under the grants 11971054 and 11731002 (R.-X. Hao) and by the Ministry of Science and Technology of Taiwan under the grant 107-2221-E-141-001-MY3 (J.-M. Chang). The authors express their sincere thanks to the anonymous referees and editors for their valuable suggestions.

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