Embedding longest fault-free paths in arrangement graphs with faulty vertices

Lo, Ray-Shang; Chen, Gen-Huey

doi:10.1002/1097-0037(200103)37:2<84::aid-net3>3.0.co;2-a

Cited by 10 publications

(6 citation statements)

References 17 publications

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“…Certain special mapping functions enable networks to be embedded within other networks [2,7,[10][11][12][13]15,17,18,24,30]. All algorithms developed in a network can be also used in the network that encloses it.…”

Section: Introductionmentioning

confidence: 99%

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Proof that pyramid networks are 1-Hamiltonian-connected with high probability

Duh

Chen²,

2007

Information Sciences

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Section: Introductionmentioning

confidence: 99%

“…Topological properties of the faulty networks were investigated in the literature [2,5,10,12,13,15,17,27,30]. These studies related to the faulty networks include computing diameter, fault diameter, and wide diameter, providing routing, multicasting, broadcasting and embedding rules.…”

Section: Introductionmentioning

confidence: 99%

Proof that pyramid networks are 1-Hamiltonian-connected with high probability

Duh

Chen²,

2007

Information Sciences

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“…We remark that there is no connection between our arrangement graphs and another family of graphs that bears the same name and which play a role in the field of network design [6,21].…”

Section: Introductionmentioning

confidence: 99%

Hamiltonicity and colorings of arrangement graphs

Felsner

Hurtado

Noy

et al. 2006

Discrete Applied Mathematics

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We study connectivity, Hamilton path and Hamilton cycle decomposition, 4-edge and 3-vertex coloring for geometric graphs arising from pseudoline (affine or projective) and pseudocircle (spherical) arrangements. While arrangements as geometric objects are well studied in discrete and computational geometry, their graph theoretical properties seem to have received little attention so far. In this paper we show that they provide well-structured examples of families of planar and projective-planar graphs with very interesting properties. Most prominently, spherical arrangements admit decompositions into two Hamilton cycles; this is a new addition to the relatively few families of 4-regular graphs that are known to have Hamiltonian decompositions. Other classes of arrangements have interesting properties as well: 4-connectivity, 3-vertex coloring or Hamilton paths and cycles. We show a number of negative results as well: there are projective arrangements which cannot be 3-vertex colored. A number of conjectures and open questions accompany our results.

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“…They are often used as a connection structure for local area network and as a control and data flow structure for distributed computations in arbitrary networks because of its nice properties such as low connectivity, simplicity, extensibility, and its feasible implementation. Hence, embedding rings (called a separating cycle) into various networks has been discussed in [5] [6][12] [13] [16].…”

Section: Introduction and Notationmentioning

confidence: 99%

“…Moreover, Lo and Chen generalized Hsieh's work by embedding a Hamiltonian path between two arbitrary distinct vertices of the same arrangement graph Ò [5] [12]. Again, Lo and Chen obtained more powerful results over Hsieh's work [5] [13]. Hsieh et al proved that the longest fault-free paths which can be embedded in an Ò-dimensional star graph with edge faults [6].…”

Section: Introduction and Notationmentioning

confidence: 99%

On the fault-tolerant pancyclicity of crossed cubes

Huang

Chen

Ninth International Conference on Parallel and Distributed Systems, 2002. Proceedings.

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A graph is pancyclic if it contains all cycles from lengths 4 to Î´ µ . An Ò-dimensional crossed cube, an important variation of hypercube denoted as É Ò , has been proved to be pancyclic because it contains all cycles whose lengths range from 4 to Î´ É Ò µ . Since vertex and edge faults may occur when a network is used, it is practical and meaningful to evaluate the performance of a faulty network. Moreover, the vertex fault-tolerant Hamiltonicity and the edge fault-tolerant Hamiltonicity measure the performances of the Hamiltonian properties in the faulty networks. From this fault-tolerant concept, we propose using the fault-tolerant pancyclicity of networks to measure the performance of faulty networks. In this paper, we consider a faulty crossed Ò-cube with vertex and/or edge faults here. Let the faulty set be a subset of Î´ É Ò µ ´ É Ò µ. We prove that any cycle of length Ð (4 Ð Î´ É Ò µ Ú ) can be embedded into a faulty crossed Ò-cube É Ò with dilation 1, where Ú · is less than Ò ¾, Ú is the number of faulty vertices of , is the number of faulty edges of , and Ò is greater than 2. The results can readily be used in the optimum embedding of a ring of the specified length in a faulty crossed cube.

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Embedding longest fault-free paths in arrangement graphs with faulty vertices

Cited by 10 publications

References 17 publications

Proof that pyramid networks are 1-Hamiltonian-connected with high probability

Proof that pyramid networks are 1-Hamiltonian-connected with high probability

Hamiltonicity and colorings of arrangement graphs

On the fault-tolerant pancyclicity of crossed cubes

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