Edge-fault tolerance of hypercube-like networks☆
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 , 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 , an edge subset F of a connected graph G is called an h-super edge-cut, or h-edge-cut for short, if is disconnected and has the minimum degree . The h-super edge-connectivity of G, denoted by , is defined as the minimum cardinality over all h-edge-cuts of G. It is clear that , where is classical edge-connectivity of G. For , if exists, then .
For any graph G and a given integer h, determining is quite difficult since Latifi et al. [11] conjectured it is NP-hard, not proved so far. In fact, the existence of is an open problem so far when . Only few results have been known on for particular classes of graphs and small hʼs, such as, Xu [19] determined for .
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 [5], the twisted cube [1], [2], the locally twisted cube [21], the crossed cube [8], [10], the Möbius cube [6], the recursive circulant [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 , 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 for any and . 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 and be two disjoint graphs with the same order, σ a bijection from to . A 1-1 connection between and is defined as an edge-set . Let denote a graph . Clearly, is a perfect matching of G. Moreover, if σ is the identical permutation on , then , where × denotes the Cartesian product, and is a complete graph of order two.
Note that the operation may generate different graphs according to
Main results
In this section, our aim is to prove that for any and .
Lemma 3.1 for any and .
Proof Let . By Lemma 2.1 there is a sequence of graphs such that is one of the i-dimensional underlying graphs of for each i with . Let F be the set of edges between and . Then F is an edge-cut of . Since is n-regular and is h-regular, . We now show that F is an h-edge-cut of by proving
Conclusions
In this paper, we consider the generalized measures of edge-fault tolerance for the hypercube-like networks, called the h-super edge-connectivity . For the hypercube-like graph , we prove that for any n and . This result shows that at least edges of 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.
References (21)
- et al.
The twisted cube topology for multiprocessors: a study in network asymmetry
J. Parallel Distrib. Comput.
(1991) - et al.
Restricted connectivity for three families of interconnection networks
Appl. Math. Comput.
(2007) - et al.
Super-connectivity and super edge-connectivity for some interconnection networks
Appl. Math. Comput.
(2003) - et al.
Disjoint path covers in recursive circulants with faulty elements
Theor. Comput. Sci.
(2011) Connectivity of the crossed cube
Inf. Process. Lett.
(1997)- et al.
On the spanning connectivity and spanning laceability of hypercube-like networks
Theor. Comput. Sci.
(2007) - et al.
Recursive circulants and their embeddings among hypercubes
Theor. Comput. Sci.
(2000) - et al.
Panconnectivity and pancyclicity of hypercube-like interconnection networks with faulty elements
Theor. Comput. Sci.
(2007) - et al.
A note about some properties of BC graphs
Inf. Process. Lett.
(2008) - et al.
Edge fault tolerance of super edge connectivity for three families of interconnection networks
Inf. Sci.
(2012)