Longest fault-free paths in hypercubes with vertex faults

Fu, Jung-Sheng

doi:10.1016/j.ins.2005.01.011

Cited by 65 publications

(38 citation statements)

References 14 publications

(21 reference statements)

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“…; k À 1g representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang et al [22] and by Fu [11]. Furthermore, we extend known results, obtained by Kim and Park [15], for the case when n ¼ 2.…”

Section: Embedding Long Paths In K-ary N-cubes With Faulty Nodes Andsupporting

confidence: 81%

“…They provide conditions when a twodimensional torus with at most two faulty nodes is Hamiltonian, Hamiltonian-connected, and bi-Hamiltonian connected. In [11], Fu proves that an n-dimensional hypercube with f n À 2 faulty nodes is such that there is a path of length at least 2 n À 2f À between any two distinct healthy nodes, where ¼ 1 if the two nodes have different parities and ¼ 2 otherwise. In [13], Hsieh and Chang show that Fu's result holds even when f 2n À 5 but only so long as every healthy node is adjacent to at least two healthy nodes (the socalled conditional fault assumption).…”

Section: Introductionmentioning

confidence: 99%

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Embedding Long Paths in k-Ary n-Cubes with Faulty Nodes and Links

Stewart

Xiang

2008

IEEE Trans. Parallel Distrib. Syst.

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Publisher's copyright statement: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. Abstract-Let k ! 4 be even and let n ! 2. Consider a faulty k-ary n-cube Q k n in which the number of node faults f v and the number of link faults f e are such that f v þ f e 2n À 2. We prove that given any two healthy nodes s and e of Q k n , there is a path from s to e of length at least k n À 2f v À 1 (respectively, k n À 2f v À 2) if the nodes s and e have different (respectively, the same) parities (the parity of a node in Q k n is the sum modulo 2 of the elements in the n-tuple over f0; 1; . . . ; k À 1g representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang et al. [22] and by Fu [11]. Furthermore, we extend known results, obtained by Kim and Park [15], for the case when n ¼ 2.

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Section: Embedding Long Paths In K-ary N-cubes With Faulty Nodes Andsupporting

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Embedding Long Paths in k-Ary n-Cubes with Faulty Nodes and Links

Stewart

Xiang

2008

IEEE Trans. Parallel Distrib. Syst.

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“…It remains to show that there exists an edge xy satisfying (8). Recall that α(F i:R ) is the number of vertices z in Q i:R with |F i:R (z)| ≥ 4.…”

Section: Lemma 51 If There Exists a Dimension I ∈ [N] Such That Atmentioning

confidence: 99%

Long cycles in hypercubes with optimal number of faulty vertices

Fink

Gregor

2011

J Comb Optim

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Let f (n) be the maximum integer such that for every set F of at most f (n) vertices of the hypercube Q n , there exists a cycle of length at least 2 n − 2|F | in Q n − F . Castañeda and Gotchev conjectured that f (n) = n 2 − 2. We prove this conjecture. We also prove that for every set F of at most (n 2 + n − 4)/4 vertices of Q n , there exists a path of length at least 2 n − 2|F | − 2 in Q n − F between any two vertices such that each of them has at most 3 neighbors in F . We introduce a new technique of potentials which could be of independent interest.

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“…A path or cycle in a graph is called Hamiltonian if it contains all the vertices of the graph. This problem has attracted much attention in the literature, such as works on faulty hypercubes [9,10,17]. The disjoint path cover problem is closely related to the Hamiltonian problem in that a Hamiltonian path joining a pair of vertices can be viewed as any type of 1-DPC joining them, and a Hamiltonian path joining a pair of vertices that passes through k − 1 prescribed edges can be obtained directly from some paired k-DPC of the graph [29].…”

Section: Introductionmentioning

confidence: 99%

Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

Jo¹,

Park²,

Chwa³

2013

Information Sciences

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A paired many-to-many k-disjoint path cover (paired k-DPC for short) of a graph is a set of k vertex-disjoint paths joining k distinct source-sink pairs that altogether cover every vertex of the graph. We consider the problem of constructing paired 2-DPC's in an m-dimensional bipartite HL-graph, X m , and its application in finding the longest possible paths. It is proved that every X m , m ≥ 4, has a fault-free paired 2-DPC if there are at most m − 3 faulty edges and the set of sources and sinks is balanced in the sense that it contains the same number of vertices from each part of the bipartition. Furthermore, every X m , m ≥ 4, has a paired 2-DPC in which the two paths have the same length if each source-sink pair is balanced. Using 2-DPC properties, we show that every X m , m ≥ 3, with either at most m − 2 faulty edges or one faulty vertex and at most m − 3 faulty edges is strongly Hamiltonian-laceable.

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Longest fault-free paths in hypercubes with vertex faults

Cited by 65 publications

References 14 publications

Embedding Long Paths in k-Ary n-Cubes with Faulty Nodes and Links

Embedding Long Paths in k-Ary n-Cubes with Faulty Nodes and Links

Long cycles in hypercubes with optimal number of faulty vertices

Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

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