Panconnectivity and edge-fault-tolerant pancyclicity of augmented cubes

Ma, Meijie; Liu, Guizhen; Xu, Jie

doi:10.1016/j.parco.2006.11.008

Cited by 75 publications

(30 citation statements)

References 18 publications

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“…First, it is known that AQ n is pancyclic for n 2 [2] and panconnected for n 1 [10]. There are several other generalized results.…”

Section: Conclusion and Problemsmentioning

confidence: 99%

“…There are several other generalized results. For example, AQ n is (2n − 3)-edge-fault-tolerant pancyclic for n 2 [10], (2n − 3)-fault-tolerant pancyclic for n 4 [18], (2n − 3)-fault-tolerant hamiltonian, and (2n − 4)-fault-tolerant hamiltonian connected for n 4 [7]. The first question is, is AQ n (2n − 4)-fault-tolerant panconnected for some large d 2 or n 4?…”

Section: Conclusion and Problemsmentioning

confidence: 99%

“…Hsu et al [7] showed that AQ n is (2n − 3)-fault-tolerant hamiltonian and (2n − 4)-fault-tolerant hamiltonian connected for n 4. Ma et al [10] showed that AQ n is panconnected for n 1 and (2n − 3)-edge-fault-tolerant pancyclic for n 2. Wang et al [18] showed that AQ n is (2n − 3)-fault-tolerant pancyclic for n 4.…”

Section: Introductionmentioning

confidence: 99%

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Fault-tolerant panconnectivity of augmented cubes

Wang

2009

Front. Math. China

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The augmented cube AQ n is a variation of the hypercube Q n . This paper considers the panconnectivity of AQ n (n 3) with at most 2n− 5 faulty vertices and/or edges and shows that, for any two fault-free vertices u and v with distance d in AQ n , there exist fault-free uv-paths of every length from d + 2 to 2 n − f − 1, where f is the number of faulty vertices in AQ n . The proof is based on an inductive construction.

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“…First, it is known that AQ n is pancyclic for n 2 [2] and panconnected for n 1 [10]. There are several other generalized results.…”

Section: Conclusion and Problemsmentioning

confidence: 99%

Section: Conclusion and Problemsmentioning

confidence: 99%

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Fault-tolerant panconnectivity of augmented cubes

Wang

2009

Front. Math. China

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“…It is well known that the hypercube is one of the most popular interconnection networks for parallel computer/communication system. As an enhancement on the hypercube Q n , the augmented cube AQ n , proposed by Choudum and Sunitha [2], not only retains some of the favorable properties of Q n but also possesses some embedding properties that Q n does not (see, for example, [6,9]). In this paper, we prove that κ (AQ n ) = 4n−8 for n 6 and λ (AQ n ) = 4n − 4 for n 5.…”

Section: Networkmentioning

confidence: 99%

The super connectivity of augmented cubes

Liu

2008

Information Processing Letters

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The augmented cube AQ n , proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a (2n − 1)-regular (2n − 1)-connected graph (n = 3). This paper determines that the super connectivity of AQ n is 4n − 8 for n 6 and the super edge-connectivity is 4n − 4 for n 5. That is, for n 6 (respectively, n 5), at least 4n − 8 vertices (respectively, 4n − 4 edges) of AQ n are removed to get a disconnected graph that contains no isolated vertices. When the augmented cube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.

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“…The notions in the preceding paragraph have been investigated in the context of a number of interconnection networks: for example, in crossed cubes [12], [31], Mö bius cubes [14], augmented cubes [20], alternating group graphs [9], star graphs [29], bubble-sort graphs [17], and in hypercubes and hypercube-like networks [13], [19], [22], [25], [26], [28], [30]. With regard to k-ary n-cubes, these notions have been considered in [15] and [27].…”

Section: Introductionmentioning

confidence: 99%

Bipanconnectivity and Bipancyclicity in k-ary n-cubes

Stewart

Xiang

2009

IEEE Trans. Parallel Distrib. Syst.

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Publisher's copyright statement:This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder. c 2009 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. is bipanconnected and edge-bipancyclic, when k ! 3 and n ! 2, and we also show that when k is odd, Q k n is m-panconnected, for m ¼ nðkÀ1Þþ2kÀ6 2, and ðk À 1Þ-pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening, then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q k n , even in the presence of a faulty processor.

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Panconnectivity and edge-fault-tolerant pancyclicity of augmented cubes

Cited by 75 publications

References 18 publications

Fault-tolerant panconnectivity of augmented cubes

Fault-tolerant panconnectivity of augmented cubes

The super connectivity of augmented cubes

Bipanconnectivity and Bipancyclicity in k-ary n-cubes

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