Generalized measures of fault tolerance for bubble sort networks☆
Introduction
In large parallel computing and communication systems, processors are connected by communication links according to some interconnection networks, where the processors and communication links correspond to the vertices and edges in the networks respectively. In the large-scale interconnection networks, it can not avoid faulty vertex and faulty edge occurred, thus estimating the fault tolerance for networks is of crucial importance. The connectivity κ(G) is the smallest size of vertices whose deletion disconnects the graph G. The edge-connectivity λ(G) is defined similarly, with vertices replaced by edges (see [20]). But, these two definitions are based on any subset of interconnection network can be faulty with equal possibility, which just occur in the worst case. Thus κ and λ underrate the fault tolerance of the large-scale networks. Motivated by this weakness, Harary [5] introduced the concept of conditional connectivity by imposing a number of additional requirements on the remaining networks. Thereafter, Latifi et al. [8] generalized this concept in some sense and proposed restricted k-connectivity by restricting each vertex has at least k fault-free neighbors. These generalized measurements can more accurately estimate the fault tolerance of an interconnection network in real applications.
Suppose G is a connected graph, and T is a subset in V(G). If is disconnected and we call T a k-vertex-cut of G. The k-connectivity κk(G) is defined as the cardinality of a minimum k-vertex-cut of G. The k-edge-cut and k-edge-connectivity λk(G) can be defined similarly, with T a set of edges. It is obvious that and .
For a graph G, to determine κk(G) and λk(G) for any integer k is not a easy task. For hypercubes Qn, κk and λk were determined for any k about two decades ago (see [14], [18], [19]). For the other networks except Qn, people concentrate upon on κk and λk for some small k’s at long durations [4]. Recently, using some new methods, κk and λk have been established for some well-known networks and for any k [11], [17]. For instance, κk and λk were given for exchanged hypercubes [12], star networks [13] and hierarchical cubic networks [10].
The bubble sort network Bn was first presented by Akers and Krishnamurthy [1], it has drawn considerable attention in recent years since it has high symmetry and simple hierarchical structure [1], [7]. Much properties for Bn have been investigated, such as Hamiltonian laceability [2], bipancyclicity [6], embedded connectivity [9], [22], subnetwork fault tolerance [16], conditional diagnosibility [25]. For n ≥ 3, Cheng and Lipták [3] determined ; afterwards, Yang et al. [21] shown that for n ≥ 4; Shi and Wu [15] recently determined that for n ≥ 6. Furthermore, it has been conjectured by Shi and Wu [15] that for n ≥ 3 and k ≤ n/2, . In this paper, we prove that for k ≤ n/2, which generalize the aforementioned results and also give an affirmative answer for the conjecture.
This paper proceeds as follows. Some structure properties of Bn and useful lemmas are given in Section 2, the main result and the proof are in Section 3. We conclude the paper in Section 4.
Section snippets
Preliminaries
Suppose integer n ≥ 2, let [n] denotes the set and P(n) denotes all the permutations on [n], that is, . For convenience, we do not distinguish the subset X of V(G) and the induced subgraph G[X]. We follow Xu [20] for graph terminology and notation not defined here. Definition 2.1 The n-dimensional bubble sort graph Bn has n ! vertices with each vertex represented by a permutation in P(n). Two vertices u and v are linked in Bn if and only if u and v differ in[1]
Main results
Theorem 3.1 If n ≥ 3 and 0 ≤ k ≤ n/2, then . Proof By Lemma 2.7, it suffices to prove for any k with k ≤ n/2, We use induction on n to prove (3.1). If by Lemma 2.3, we have and that is the conclusion holds for . If then Bn is C6, it is easy to see that . Similarly, we have λ1(B3) ≥ 2. Then the conclusion is correct for we suppose n ≥ 4 and k ≥ 1 below. Assume (3.1) holds for with
Concluding remarks
Two generalized measures of fault tolerance, κk and λk for the bubble sort graphs Bn have been investigated. We show that provided 0 ≤ k ≤ n/2, which also resolves a conjecture proposed by Shi and Wu [15]. When to determine the κk and λk for the bubble sort networks Bn is quite interesting, and is worth studying further. It should be mention that there does not exists a Qk in Bn provided that k > n/2, to study κk(Bn) and λk(Bn), we need to find a new technique.
Acknowledgments
The authors would like to express their gratitude to the anonymous referees for their kind comments and valuable suggestions on the original manuscript, which resulted in this version.
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This work was supported by NNSF of China (11571044, 91647204, 61673006, 11601041), YTFY(2015cqr23).