Elsevier

Information Processing Letters

Volume 113, Issues 19–21, September–October 2013, Pages 760-763
Information Processing Letters

Edge-fault tolerance of hypercube-like networks

https://doi.org/10.1016/j.ipl.2013.07.010Get rights and content

Highlights

  • We study generalized measures of fault tolerance of hypercube-like networks.

  • We completely determine h-edge-connectivity of such networks.

  • This result contains some known results.

Abstract

This paper considers a kind of generalized measure λs(h) of fault tolerance in a hypercube-like graph Gn which contains several well-known interconnection networks such as hypercubes, varietal hypercubes, twisted cubes, crossed cubes, Möbius cubes and the recursive circulant G(2n,4), and proves λs(h)(Gn)=2h(nh) for any h with 0hn1 by the induction on n and a new technique. This result shows that at least 2h(nh) edges of Gn have to be removed to get a disconnected graph that contains no vertices of degree less than h. Compared with previous results, this result enhances fault-tolerant ability of the above-mentioned networks theoretically.

Introduction

It is well known that interconnection networks play an important role in parallel computing/communication systems. An interconnection network can be modeled by a graph G=(V,E), where V is the set of processors and E is the set of communication links in the network. For graph terminology and notation not defined here we follow [20].

The edge-connectivity of a graph G is an important measurement for fault tolerance of the network, and the larger the edge-connectivity is, the more reliable the network is. However, computing this parameter, one implicitly assumes that all links incident with the same processor may fail simultaneously. Consequently, this measurement is inaccurate for large-scale processing systems in which some subsets of system components cannot fail at the same time in real applications. To overcome such a shortcoming, Esfahanian [7] proposed the concept of restricted connectivity, in which the links incident with the same processor cannot fail at the same time. Latifi et al. [11] generalized it to the restricted h-connectivity, in which at least h links incident with the same processor cannot fail. This parameter can measure fault tolerance of an interconnection network more accurately than the classical connectivity. The concepts stated here are slightly different from theirs.

For a given integer h(0), an edge subset F of a connected graph G is called an h-super edge-cut, or h-edge-cut for short, if GF is disconnected and has the minimum degree δ(GF)h. The h-super edge-connectivity of G, denoted by λs(h)(G), is defined as the minimum cardinality over all h-edge-cuts of G. It is clear that λs(0)(G)=λ(G), where λ(G) is classical edge-connectivity of G. For h1, if λs(h)(G) exists, then λs(h1)(G)λs(h)(G).

For any graph G and a given integer h, determining λs(h)(G) is quite difficult since Latifi et al. [11] conjectured it is NP-hard, not proved so far. In fact, the existence of λs(h)(G) is an open problem so far when h1. Only few results have been known on λs(h)(G) for particular classes of graphs and small hʼs, such as, Xu [19] determined λs(h)(Qn)=2h(nh) for hn1.

It is widely known that the hypercube has been one of the most popular interconnection networks for parallel computer/communication system. However, the hypercube has the large diameter correspondingly. To minimize diameter, various networks are proposed by twisting some pairs of links in hypercubes, such as the varietal hypercube VQn [5], the twisted cube TQn [1], [2], the locally twisted cube LTQn [21], the crossed cube CQn [8], [10], the Möbius cube MQn [6], the recursive circulant G(2n,4) [13] and so on. Because of the lack of the unified perspective on these variants, results of one topology are hard to be extended to others. To make a unified study of these variants, Vaidya et al. [16] introduced the class of hypercube-like graphs HLn, which contains all the above-mentioned networks. Thus, the hypercube-like graphs have received much attention in recent years [3], [4], [12], [14], [15], [17], [18].

In this paper, we determine λs(h)(Gn)=2h(nh) for any GnHLn and 0hn1. Our result contains many know conclusions and enhances the fault-tolerant ability of the hypercube-like networks theoretically.

The proof of this result is in Section 3 by the induction on n and a new technique. Section 2 recalls the definition and Section 4 gives a conclusion on our work.

Section snippets

Hypercube-like graphs

Let G0=(V0,E0) and G1=(V1,E1) be two disjoint graphs with the same order, σ a bijection from V0 to V1. A 1-1 connection between G0 and G1 is defined as an edge-set Mσ={xσ(x)|xV0,σ(x)V1}. Let G0σG1 denote a graph G=(V0V1,E0E1Mσ). Clearly, Mσ is a perfect matching of G. Moreover, if σ is the identical permutation on V(G0), then G0σG0=G0×K2, where × denotes the Cartesian product, and K2 is a complete graph of order two.

Note that the operation σ may generate different graphs according to

Main results

In this section, our aim is to prove that λs(h)(Gn)=2h(nh) for any GnHLn and hIn1.

Lemma 3.1

λs(h)(Gn)2h(nh) for any GnHLn and hIn1.

Proof

Let GnHLn. By Lemma 2.1 there is a sequence of graphs {Gh,Gh+1,,Gn1,Gn} such that Gi is one of the i-dimensional underlying graphs of Gi+1 for each i with hin1. Let F be the set of edges between Gh and GnGh. Then F is an edge-cut of Gn. Since Gn is n-regular and Gh is h-regular, |F|=|Gh|(nh)=2h(nh).

We now show that F is an h-edge-cut of Gn by proving δ(GnF)

Conclusions

In this paper, we consider the generalized measures of edge-fault tolerance for the hypercube-like networks, called the h-super edge-connectivity λs(h). For the hypercube-like graph GnHLn, we prove that λs(h)(Gn)=2h(nh) for any n and hIn1. This result shows that at least 2h(nh) edges of Gn have to be removed to get a disconnected graph that contains no vertices of degree less than h. Thus, when the hypercube-like networks are used to model the topological structure of a large-scale

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their kind suggestions and comments on the original manuscript, which resulted in this version.

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The work was supported by NNSF of China (No. 11071233, 61272008).

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