Adjacent vertices fault tolerance hamiltonian laceability of star graphs 楊俊彥、洪春男
E-mail: [email protected]
ABSTRACT
Let Sn be an n-dimensional Star graph. In this paper, we show that Sn ?F is Hamiltonian laceable where F is the set of f ?(n?4) pairs of adjacent faulty vertices, Sn?F is Hamiltonian where F is the set of f ?(n?3) pairs of adjacent faulty vertices. We also show that Sn ?F is hyper-Hamiltonian laceable where F is the set of f ?(n ?4) pairs of adjacent faulty vertices. Applying these results, we also construct the fault-free cycle with length n! ?2f + 2 in Sn ?F’ where F’ is the faulty vertices set with at least a black vertex and a white vertex for |F’| = f ?n?2 and the fault-free path with length n! ?2f+ 1 for any two different color vertices in Sn?F’where F’is the faulty vertices set with at least a black vertex and a white vertex for |F'| = f ?n?3 and n!?2f for any two same color vertices in Sn ?F
′where F′is the faulty vertices set for |F'| = f ?n?3
Keywords : Sn ; Hamiltonian laceable ; hyper-Hamiltonian laceable. ; Star graph ; Hamiltonian Table of Contents
封面內頁 簽名頁 授權書...iii 中文摘要...iv 英文 摘要...v 誌謝...vi 目
錄...vii 圖目錄...viii 表目
錄...ix Chapter 1 Introduction and Definitions...1 Chapter 2 The Adjacency Hamiltonian Laceablility of Star Graphs...4 Chapter 3 Fault-free Cycle and Path in Star Graphs with Faulty Vertices...24 Section 3.1. Fault-free Cycle...24 Section 3.2. Fault-free
Path...25 Chapter 4 Conclusion...31 Bibliography...32
REFERENCES
[1] S. B. Akers and B. Krishnamurthy, “A group-theoretic model for symmetric interconnection networks,” IEEE Transaction on Computers, 38, pp. 555- 566, 1989.
[2] S.B. Akers, D. Harel, B. Krishnamurthy, “The star graph: an attractive alternative to the n-cube”, Proc. Internat. Conf. Parallel Processing, pp. 216-223, 1986.
[3] S.G. Akl, “Parallel Computation: Models and Methods, Prentice-Hall”, NJ, 1997.
[4] N. Bagherzadeh, M. Dowd, N. Nassif, “Embedding an arbitrary tree into the star graph”, IEEE Trans. Comput. pp. 475-481, 1996.
[5] J.C. Bermond (Ed.), “Interconnection networks”, Discrete Appl. Math. 37+38 (1992) (special issue).
[6] R.V. Boppana, S. Chalasani, C.S. Raghavendra, “Resource deadlock and per- formance of whormhole multicast routing algorithms”, IEEE Trans. Parallel Distributed Systems pp. 535-549, 1998.
[7] J.H. Chang, C.S. Shin, K.Y. Chwa, “Ring embedding in faulty star graphs”, IEICE Trans. Fund. E82-A. pp. 1953-1964, 1999.
[8] Y. H. Chang, C. N Hung, “Adjacent Vertices Fault Tolerance Ham-iltonian Laceability of Hypercube Graphs”, W.C.M.C.T., pp. 301-309, 2005.
[9] M. Y. Chen, S.-J. Lee, “Distributed fault-tolerant embedding of rings in hypercubes”, J. Parallel Distrib. Comput. pp. 63-71, 1991.
[10] J. S. Fu, “Fault-tolerant cycle embedding in the hypercube”, Paral-lel Com- puting, pp. 821-832, 2003.
[11] C. N. Hung, Y. H. Chang, and C. M. Sun, “Longest paths and cycles in faulty hypercubes,” Proceedings of the IASTED International Conference on Parallel and Distributed Computing and Networks, pp. 101-110, 2006.
[12] S. Y. Hsieh, “Embedding longest fault-free paths onto star graphs with more vertex faults,” Theoretical Computer Science, 337, pp.
370-378, 2005.
[13] S.Y. Hsieh, G.H. Chen, C.W. Ho, “Longest fault-free paths in star graphs with vertex faults”, Theoret. Comput. pp. 215-227, 2001.
[14] S.Y. Hsieh, G.H. Chen, C.W. Ho, “Longest fault-free paths in star graphs with edge faults”, IEEE Trans. Comput. pp. 960-971, 2001.
[15] S.Y. Hsieh, G.H. Chen, C.W. Ho, “Fault-free Hamiltonian cycles in faulty arrangement graphs”, IEEE Transactions on Parallel and Distributed Sys- tems 10 pp. 223-237, 1993.
[16] D.F. Hsu, “Interconnection Networks and Algorithms”, Networks 23 (4) (1993) (special issue).
[17] S. Latifi, N. Bagherzadeh, “Hamiltonicity of the clustered-star graph with embedding applications”, Proc. I.C.P.D.P.T.. pp. 734-744, 1996.
[18] S. Latifi, S.Q. Zheng, N. Bagherzadeh, “Optimal ring embedding in hy- percubes with faulty links”, Proceedings of the IEEE Sympo-sium on Fault- Tolerant Computing pp. 178-184, 1992.
[19] T. K. Li, Jimmy J.M. Tan, and L. H. Hsu, “Hyper hamiltonian laceability on edge fault star graph,” Information Sciences, Vol. 165, pp.
59-71, 2004.
[20] C. K. Lin, H. M. Huanga, and L. H. Hsub,“The super connectivityof the pancake graphs and the super laceability of the star graphs,”
Theoretical Computer Science, 339, pp. 257-271, 2005.
[21] J. S. Jwo, S. Lakshmivarahan, S.K. Dhall, “Embedding of cycles and grids in star graphs”, J. Circuits, Systems, and Comput. pp. 43-74, 1991.
[22] Z. Miller, D. Pritikin, and I.H. Sudborough, “Near embeddings of hyper- cubes into Cayley graphs on the symmetric group,” IEEE Transaction on Computers, 43, pp. 13-22, 1994.
[23] J. H. Park and H. C. Kim, “Longest paths and cycles in faulty star graphs,” Journal of Parallel and Distributed Computing, 64, pp.
1286-1296, 2004.
[24] K. Qiu, S.G. Akl, H. Meijer, On some properties and algorithms for the star and pancake interconnection networks, J. Parallel Distributed Comput. pp. 16-25, 1994.
[25] S. Ranka, J.C.Wang, N. Yeh, “Embedding meshes on the star graph”, J. Parallel Distributed Comput. pp. 131-135, 1993.
[26] Abhijit Sengupta, “On ring embedding in hypercubes with faulty nodes and links”, Information Processing Letters, pp. 207-214, 1998.
[27] Yu-Chee Tseng, S.H. Chang, J.P. Sheu, Fault-tolerant ring embedding in star graphs with both link and node failures, IEEE Trans. Parallel Dis- tributed Systems. pp. 1185-1195, 1997.
[28] Y. C. Tseng, Embedding a ring in a hypercube with both faulty links and faulty nodes, Information Processing Letters, pp. 217-222, 1996.
[29] D.J. Wang, Embedding Hamiltonian cycles into folded hyper-cubes with link faults, Journal of Parallel and Distributed Com-puting 61 pp.
545-564, 2001.
[30] P.J. Yang, S.B. Tien, C.S. Raghavendra, Embedding of rings and meshes onto faulty hypercubes using free dimensions, IEEE Transactions on Com- puters 43 pp. 608-613, 1994.
