在設計平行計算機VLSI系統快速發展的時代，有關在容錯情形下探討連結網路相關性質的研究，愈來愈受到人們的重視。在容錯情形下，有很多受到注意且有趣的性質值得深入探討，比如：圖形嵌入，或是偵錯能力等。在容錯情形下，將一個附屬圖形嵌入到一個主要圖形裡可以讓我們在連結網路中尋找一個最佳化或是最長的拓墣圖形。偵錯能力可以提供我們一些方法或數值來尋找一個圖形可能潛在錯誤的處理器。

在本篇論文裡，我們將研究成果分為兩個方面來展示：分別是圖形的容錯嵌入和偵錯能力。在容錯嵌入部分，首先，我們將證明在一個n維的莫氏立方體（Möbius cube）中，且至多只有n-1個錯誤的情形下，可嵌入一個無錯誤的漢米爾頓路徑（Hamiltonian path）。並且該圖具有容錯n-2個的泛迴圈性質。其次，在一個n維的超立方體（hypercube）中，且至多只有2n-5條錯邊和2n-4個錯誤的情形下，可以嵌入一個無錯誤的、其長度至少有2^n-2|F_v|的迴圈。第三，利用前面的結果，我們更說明一個n維的超立方體中，且至多只有2n-4個錯誤的情形下，我們可以嵌入一個無錯誤的、其長度從4到2^n-2|F_v|且是偶數的迴圈。最後，在偵錯能力部分，我們將證明一個n維的擴增立方體（augmented cube）在PMC模式下的條件偵錯度是8n-27。

With the rapid development of VLSI technology, a multiprocessor system may contain hundreds or even thousands of processors. Some of these processors may be faulty while the system is put in use. This motivates us the issues of fault-tolerant properties of multiprocessor systems. There are some popular and interesting fault-tolerant properties for multiprocessor systems such as graph embedding and system diagnosis.
In this dissertation, we demonstrate our work in two aspects. The first is fault-tolerant graph embedding, and the second is fault diagnosis. Let F_v be the set of faulty nodes in a system. For fault-tolerant graph embedding, we first show that an n-dimensional Möbius cube MQ_n with at most n-1 faulty elements contains a fault-free Hamiltonian path for n>=1, and MQ_n with at most n-2 elements is pancyclic for n>=2. Second, we show that an n-dimensional hypercube Q_n with at most 2n-5 non-free edges and at most 2n-4 faulty elements in which each node is incident to at least two free edges contains a fault-free cycle of length at least 2^n-2|F_v|. With the previous result, we further show that under the same fault condition, Q_n contains a fault-free cycle of every even length from 4 to 2^n-2|F_v|. For fault diagnosis, we evaluate that the conditional diagnosability of an n-dimensional augmented cube AQ_n under the PMC model is 8n-27 for n>=5.

[1] S. Abraham and K. Padmanabhan, "An analysis of the twisted cube topology," in Proceedings of the International conference on Parallel Processing, vol. 1, pp..116-120, 1989.
[2] S. Abraham and K. Padmanabhan, "The twisted cube topology for multiprocessors: A study in network asymmetry," Journal of Parallel and Distributed Computing, vol. 13, issue 1, pp. 104-110, 1991.
[3] B. Alspach, J. C. Bermond, and D. Sotteau, "Decomposition into cycles I:.Hamilton decompositions," in Proceedings of 1987 Cycles and Rays Colloquium, Montréal, pp. 9-18, 1987.
[4] T. Araki and Y. Shibata, "Diagnosability of networks represented by the Cartesian product," IEICE Transactions on Fundamentals, vol. E83-A, no. 3, pp. 465-470, 2000.
[5] J. R. Armstrong and F. G. Gray, "Fault diagnosis in a boolean n cube array of multiprocessors," IEEE Transactions on Computers, vol. 30, no. 8, pp. 587-590, 1981.
[6] C. Balbuena, "Extraconnectivity of s-geodetic digraphs and graphs," Discrete Mathematics, vol. 195, pp. 39-52, 1999.
[7] C. Balbuena, A. Carmona, J. Fàbrega, and M. A. Fiol, "Extraconnectivity of graphs with large minimum degree and girth," Discrete Mathematics, vol. 167-.168, pp. 85-100, 1997.
[8] C. Balbuena, M. Cera, A. Diánez, P. García-Vázquez, and X. Marcote, "Suffcient conditions for λ-optimality of graphs with small conditional diameter," Information Processing Letters, vol. 95, issue 4, pp. 429-434, 2005.
[9] C. Balbuena, M. Cera, A. Diánez, P. García-Vázquez, and X. Marcote, "On the edge-connectivity and restricted edge-connectivity of a product of graphs," Discrete Applied Mathematics, vol. 155, issue 18, pp. 2444-2455, 2007.
[10] C. Balbuena, M. Cera, A. Diánez, P. García-Vázquez, and X. Marcote, "On the restricted connectivity and superconnectivity in graphs with given girth," Discrete.Mathematics, vol. 307, issue 6, pp. 659-667, 2007.
[11] C. Balbuena, M. Cera, A. Diánez, P. García-Vázquez, and X. Marcote, "Diameter-girth suffcient conditions for optimal extraconnectivity in graphs," Discrete Mathematics, vol. 308, issue 16, pp. 3526-3536, 2008.
[12] C. Balbuena, D. González-Moreno, and X. Marcote, "On the 3-restricted edge connectivity of permutation graphs," Discrete Applied Mathematics, vol. 157, issue 7, pp. 1586-1591, 2009.
[13] L. Bhuyan and D. P. Agrawal, "Generalized hypercubes and hyperbus structure for a computer network," IEEE Transactions on Computers, vol. 33, pp. 323-333, 1984.
[14] J. Bruck, R. Cypher, and D. Soroker, "Embedding cube-connected-cycles graphs into faulty hypercubes," IEEE Transactions on Computers, vol. 43, no. 10, pp. 1210-1220, 1994.
[15] M. Chan, "The distinguishing number of the augmented cube and hypercube powers," Discrete Mathematics, vol. 308, no. 11, pp. 2330-2336, 2008.
[16] M. Y. Chan and S. J. Lee, "On the existence of Hamiltonian circuits in faulty hypercubes," SIAM Journal on Discrete Mathematics, vol. 4, issue 4, pp. 511-527, 1991.
[17] M. Y. Chan and S. J. Lee, "Distributed fault-tolerant embeddings of rings in hypercubes," Journal of Parallel and Distributed Computing, vol. 11, issue 1, pp. 63-71, 1991.
[18] M. Y. Chan and S. J. Lee, "Fault-tolerant embeddings of complete binary trees in hypercubes," IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 3, pp. 540-547, 1993.
[19] G. Y. Chang, G. J. Chang, and G. H. Chen, "Diagnosabilities of regular networks," IEEE Transactions on Parallel and Distributed Systems, vol. 16, no. 4, pp. 314-323, 2005.
[20] F. B. Chedid, "On the generalized twisted cube," Information Processing Letters, vol. 55, no. 1, pp. 49-52, 1995.
[21] M. S. Chen and K. G. Shin, "Processor allocation in an N-cube multiprocessor using gray codes," IEEE Transactions on Computers, vol. 36, issue 12, pp. 1396-1407, 1987.
[22] Y. C. Chen and J. J. M. Tan, "Restricted connectivity for three families of interconnection networks," Applied Mathematics and Computation, vol. 188, issue 2, pp. 1848-1855, 2007.
[23] S. A. Choudum and V. Sunitha, "Augmented Cubes," Networks, vol. 40, issue. 2, pp. 71-84, 2002.
[24] P. Cull, and S. M. Larson, "The Möbius cubes," in Proceedings of the sixth Distrib. Memory Computing Conference, vol. 44, pp. 699-702, 1991.
[25] P. Cull and S.M. Larson, "The Möbius cubes," IEEE Transactions on Computers, vol. 44, no. 5, pp. 647-659, 1995.
[26] P. Dundar, "Augmented cubes and its connectivity numbers," Neural Network World, vol. 15, no. 1, pp. 1-8, 2005.
[27] K. Efe, "A variation on the hypercube with lower diameter," IEEE Transactions on Computers, vol. 40, no. 11, pp. 1312-1316, 1991.
[28] A. H. Esfahanian, "Generalized measures of fault tolerance with application to n-cube networks," IEEE Transactions on Computers, vol. 38, no. 11, pp. 1586-1591,.1989.
[29] A. H. Esfahanian and S. L. Hakimi, "On computing a conditional edge-connectivity of a graph," Information Processing Letters, vol. 27, issue 4, pp. 195-199, 1988.
[30] J. Fàbrega and M. A. Fiol, "Extraconnectivity of graphs with large girth," Discrete Mathematics, vol. 127, pp. 163-170, 1994.
[31] J. Fàbrega and M. A. Fiol, "On the extraconnectivity of graphs ," Discrete Math-.ematics, vol. 155, pp. 49{57, 1996.
[32] J. X. Fan, "Diagnosability of Crossed Cubes under the two strategies," Chinese Journal of Computers, vol. 21, no. 5, pp. 456-462, 1998.
[33] J. X. Fan, "Diagnosability of the Mobius cubes," IEEE Transactions on Parallel and Distributed Systems, vol. 9, no. 9, pp. 923-928, 1998.
[34] J. X. Fan, "Hamiltonian-connectivity and cycle-embedding of the Möbius cubes," Information Processing Letters, vol. 82, pp. 113-117, 2002.
[35] J. X. Fan and L. Q. He, "BC interconnection networks and their properties," Chinese Journal of Computers, vol. 26, no. 1, pp. 1-7, 2003.
[36] J. X. Fan, X. L. Lin, and X. H. Jia, "Optimal path embedding in crossed cubes," IEEE Transactions on Parallel and Distributed Systems, vol. 16, no. 12, pp. 1190-1200, 2005.
[37] J. S. Fu and G. H. Chen, "Hamiltonicity of the hierarchical cubic network," Theory of Computing Systems, vol. 35, no. 1, pp. 59-79, 2002.
[38] J. S. Fu, "Fault-tolerant cycle embedding in the hypercube," Parallel Computing, vol. 29, issue 6, pp. 821-832, 2003.
[39] H. Fugiwara and K. Kinoshita, "On the computational complexity of system diagnosis," IEEE Transactions on Computers, vol. 27, no. 10, pp. 881-885, 1978.
[40] A. Ghafoor, "A Class of Fault-Tolerant Multiprocessor Networks," IEEE Transactions on Reliability, vol. 38, no. 1, pp. 5-15, 1989.
[41] A. Ghafoor, "Partitioning of Even Networks for Improved Diagnosability," IEEE Transactions on Reliability, vol. 39, no. 3, pp. 281-286, 1990.
[42] S. L. Hakimi and A. T. Amin, "Characterization of connection assignment of diagnosable systems," IEEE Transactions on Computers, vol. 23, pp. 86-88, 1974.
[43] F. Harary, "Conditional connectivity," Networks, vol. 13, issue 3, pp. 347-357, 1983.
[44] F. Harary, J. P. Hayes, and H. J.Wu "A survey of the theory of hypercube graphs," Computers and Mathematics with Applications, vol. 15, issue 4, pp. 277-289, 1988.
[45] P.A.J. Hilbers, M.R.J. Koopman, and J.L.A. van de Snepscheut, "The twisted cube," Parallel Architectures and Languages Europe, pp. 152-159, 1987.
[46] S. Y. Hsieh, "Fault-tolerant cycle embedding in the hypercube with more both faulty vertices and faulty edges," Parallel Computing, vol. 32, issue 1, pp. 84-91, 2006.
[47] S. Y. Hsieh and C. H. Chen, "Pancyclicity on Möbius cubes with maximal edge faults," Parallel Computing, vol. 30, pp. 407-421, 2004.
[48] S. Y. Hsieh and Y. S. Chen, "Strongly diagnosable product networks under the comparison diagnosis model," IEEE Transactions on Computers, vol. 57, no. 6,.pp. 721-732, 2008.
[49] S. Y. Hsieh and Y. S. Chen, "Strongly diagnosis systems under the comparison diagnosis model," IEEE Transactions on Computers, vol. 57, no. 12, pp. 1720-1725, 2008.
[50] S. Y. Hsieh, G. H. Chen, and C. W. Ho, "Fault-free Hamiltonian cycles in faulty arrangement graphs," IEEE Transactions on Parallel Distributed Systems, vol. 10,.no. 32, pp. 223-237, 1999.
[51] S. Y. Hsieh and T. H. Shen "Edge-bipancyclicity of a hypercube with faulty vertices and edges," Discrete Applied Mathematics, vol. 156, issue 10, pp. 1802-1808, 2008.
[52] S. Y. Hsieh and J. Y. Shiu, "Cycle embedding of augmented cubes," Applied Mathematics and Computation, vol. 191, no. 2, pp. 314-319, 2007.
[53] G. H. Hsu, C. F. Chiang, L. M. Shih, L. H. Hsu, and J.J.M. Tan, "Conditional diagnosability of hypercubes under the comparison diagnosis model," Journal of Systems Architecture, vol. 55, no. 2, pp. 140-146, 2009.
[54] G. H. Hsu and J. J. M. Tan, "Conditional diagnosability of the BC Networks under the comparison diagnosis model," International Computer Symposium, vol..1, pp. 269-274, 2008.
[55] W. T. Huang, W. K. Chen, and C. H. Chen, "Pancyclicity of Möbius cubes", in Proceedings of the 9th International Conference on Parallel and Distributed Systems (ICPADS'02), pp. 591-596, 2002.
[56] W. Huang, Y. Chuang, J. J. M. Tan, L. Hsu, "Fault-free Hamiltonian cycles in faulty Möbius cubes," Computación y Systemas, vol. 4, no. 2, pp. 106-114, 2000.
[57] A. Kanevsky and C. Feng, "On the embedding of cycles in pancake graphs," Parallel Computing, vol. 21, pp. 923-936, 1995.
[58] A. Kavianpour, "Sequential diagnosability of star graphs," Computers and Electrical Engineering, vol. 22, no. 1, pp. 37-44, 1996.
[59] A. Kavianpour and K. H. Kim, "Diagnosability of hypercubes under the pessimistic one-step diagnosis strategy," IEEE Transactions on Computers, vol. 40,.no. 2, pp. 232-237, 1991.
[60] A. Kavianpour and K. H. Kim, "A Comparative Evaluation of Four Basic System-Level Diagnosis Strategies for Hypercubes," IEEE Transactions on Reliability, vol..41, no. 1, pp. 26-37, 1992.
[61] H. C. Keh and J. C. Lin, "On fault-tolerant embedding of Hamiltonian cycles, linear arrays and rings in a flexible hypercube," Parallel Computing, vol. 26, pp. 769-781, 2000.
[62] P. L. Lai, J. J.M. Tan, C. P. Chang, and L. H. Hsu, "Conditional diagnosability measures for large multiprocessor systems," IEEE Transactions on Computers,.vol. 54, no. 2, pp. 165"175, 2005.
[63] S. Latifi and N. Bagherzadeh, "Hamiltonicity of the clustered-star graph with embedding applications," in Proceedings of the International Conference on Parallel.and Distributed Processing Techniques and Applications, 1996, pp. 734-744.
[64] S. Latifi, M. Hegde, and M. Naraghi-Pour, "Conditional connectivity measures for large multiprocessor systems," IEEE Transactions on Computers, vol. 43, no. 2, pp. 218-222, 1994.
[65] S. Latifi, S. Q. Zheng, and N. Bagherzadeh, "Optimal ring embedding in hypercubes with faulty links," in: Proceedings of the Twenty-Second International.Symposium on Fault-Tolerant Computing, pp. 178-184, 1992.
[66] F. T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes, Morgan Kaufmann, San Mateo, CA, 1992.
[67] C. K. Lin, J. J. M. Tan, L. H. Hsu, E. Cheng, and L. Lipták, Conditional diagnosability of Cayley Graphs generated by transposition trees," Journal of Interconnection Networks, vol. 9, nos. 1-2, pp. 83-97, 2008.
[68] S. W. Lin and S. Y. Wang, "Super p-restricted edge connectivity of line graphs," Information Sciences, vol. 179, issue 18, pp. 3122-3126, 2009.
[69] R. S. Lo and G. H. Chen, "Embedding Hamiltonian paths in faulty arrangement graphs with the backtracking method," IEEE Transactions on Parallel and Distributed Systems, vol. 12, no. 2, pp. 209-222, 2001.
[70] R. S. Lo and G. H. Chen, "Embedding longest fault-free paths in arrangement graphs with faulty vertices," Networks, vol. 37, no. 2, pp. 84-93, 2001.
[71] M. Lü, G. L. Chen, and X. R. Xu, "On super edge-connectivity of product graphs," Applied Mathematics and Computation, vol. 207, issue 2, pp. 300-306, 2009.
[72] M. Ma, G. Liu, and J. M. Xu, "Panconnectivity and edge-fault-tolerant pancyclicity of augmented cubes," Parallel Computing, vol. 33, no. 1, pp. 36-42, 2007.
[73] M. Ma, G. Liu, and J. M. Xu, "The super connectivity of augmented cubes," Information Processing Letters, vol. 106, no. 2, pp. 59-63, 2008.
[74] J. Maeng and M. Malek, "A comparison connection assignment for self-diagnosis of multiprocessors systems," in Proceedings of the 11th International Symposium.on Fault-Tolerant Computing, pp. 173-175, 1981.
[75] M. Malek, "A comparison connection assignment for diagnosis of multiprocessors systems," in Proceedings of the 7th International Symposium on Computer Architecture, pp. 31-36, 1980.
[76] J. X. Meng and Y. H. Ji, "On a kind of restricted edge connectivity of graphs," Discrete Applied Mathematics, vol. 117, pp. 183-193, 2002.
[77] J. H. Park and K. Y. Chwa, "Recursive circulant: a new topology for multicomputer networks," in Proceedings of International Symposium on Parallel Architectures, Algorithms and Networks, pp. 73-80, 1994.
[78] J. H. Park and K. Y. Chwa, "Recursive circulants and their embeddings among hypercubes," Theoretical Computer Science vol. 244, issue 1-2, pp. 35-62, 2000.
[79] C. D. Park and K. Y. Chwa, "Hamiltonian properities on the class of hypercube-like networks," Information Processing Letters, vol. 91, issue 1, pp. 11-17, 2004.
[80] J. H. Park, H. C. Kim, and H. S. Lim, "Many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements," IEEE Transactions on Parallel and Distributed Systems, vol. 17, no. 3, pp. 227-240, 2006.
[81] F. P. Preparata, G. Metze, and R.T. Chien, "On the connection assignment problem of diagnosis systems," IEEE Transactions on Electronic Computers, vol. 16, no. 12, pp. 848-854, 1967.
[82] R. A. Rowley and B. Bose, "Fault-tolerant ring embedding in deBruijn networks," IEEE Transacions on Computers, vol. 42, no. 12, pp. 1480-1486, 1993.
[83] R. A. Rowley and B. Bose, "Distributed ring embedding in faulty De Bruijn networks," IEEE Transaction on Computers, vol. 46, no. 2, pp. 187-190, 1997.
[84] Y. Saad and M. H. Schultz, "Topological properties of hypercubes," IEEE Transactions on Computers, vol. 37, no. 7, pp. 867-872, 1988.
[85] H. Sarbazi-Azad, M. Ould-Khaoua, and L. M. Mackenzie, "Algorithmic construction of Hamiltonian in pyramids," Information Processing Letters, vol. 80, no. 2, pp. 75-79, 2001.
[86] A. Sen, A. Sengupta, and S. Bandyopadhyay, "On some topological properties of hypercube, incomplete hypercube and supercube," in Proceedings of the International Parallel Processing Symposium, Newport Beach, pp. 636-642, 1993.
[87] A. Sengupta, "On ring embedding in hypercubes with faulty nodes and links," Information Processing Letters, vol. 68, issue 4, pp. 207-214, 1998.
[88] A. Sengupta and A. Dahbura, "On self-diagnosable multiprocessor system: diagnosis by the comparison approach," IEEE Transactions on Computers, vol. 41,.no. 11, pp. 1386-1396, 1992.
[89] N. K. Singhvi and K. Ghose, "The Mcube: a symmetrical cube based network with twisted links," in Proceedings of the 9th IEEE International Parallel Processing Symposium IPPS 1995, pp. 11-16, 1995.
[90] G. Tel, Topics in distributed algorithms, Cambridge Int'l Series on Parallel Computation, Cambridge University Press, 1991.
[91] C. H. Tsai, "Linear array and ring embeddings in conditional faulty hypercubes," Theoretical Computer Science, vol. 314, issue 3, pp. 431-443, 2004.
[92] C. H. Tsai, "Fault-tolerant cycles embedded in hypercubes with mixed link and node failures," Applied Mathematics Letters, vol. 21, issue 8, pp. 855-860, 2008.
[93] C. H. Tsai, J. Tan, T. Liang, L. H. Hsu, "Fault-tolerant hamiltonian laceability of hypercube," Information Processing Letters, vol. 83, issue 6, pp. 301-306, 2002.
[94] Y. C. Tseng, "Embedding a ring in a hypercube with both faulty links and faulty nodes," Information Processing Letters, vol 59, issue 4, pp. 217-222, 1996.
[95] Y. C. Tseng, S. H. Chang, and J. P. Sheu, "Fault-tolerant ring embedding in star graphs with both link and node failures," IEEE Transactions on Parallel and Distributed Systems, vol. 8, no. 12, pp. 1185-1195, 1997.
[96] A. S. Vaidya, P.S. N. Rao, and S. R. Shankar, "A class of hypercube-like networks," in Proceedings of the Fifth IEEE Symposium on Parallel and Distributed Processing, pp. 800-803, 1993.
[97] L. Volkmann, "Restricted arc-connectivity of digraphs," Information Processing Letters, vol. 103, issue 6, pp. 234-239, 2007.
[98] M. D. Wagh and J. Mo, "Hamiltonian cycles in trivalent Cayley graphs," Information Processing Letters, vol. 60, no. 4, pp. 177-181, 1996.
[99] M. Wan and Z. Zhang, "A kind of conditional vertex connectivity of star graphs," Applied Mathematics Letters, vol. 22, issue 2, pp. 264-267, 2009.
[100] D.Wang, "Diagnosability of enhanced hypercubes," IEEE Transactions on Computers, vol. 43, no. 9, pp. 1054-1061, 1994.
[101] D. Wang, "Diagnosability of hypercubes and enhanced hypercubes under the comparison diagnosis model," IEEE Transactions on Computers, vol. 48, pp. 1369-1374, 1999.
[102] D. J. Wang, "Embedding Hamiltonian cycles into folded hypercubes with faulty links," Journal of Parallel and Distributed Computing, vol. 61, no. 4, pp. 545-564, 2001.
[103] Y. Q. Wang, "Super restricted edge-connectivity of vertex-transitive graphs," Discrete Mathematics, vol. 289, pp. 199-205, 2004.
[104] M. Wang and Q. Li, "Conditional edge connectivity properties, reliability comparisons and transitivity of graphs," Discrete Mathematics, vol. 258, pp. 205-214, 2002.
[105] S. Y. Wang, S. W. Lin, and C. F. Li, "Suffiient conditions for super k-restricted edge connectivity in graphs of diameter 2," Discrete Mathematics, vol. 309, issue.4, pp. 908-919, 2009.
[106] W. W. Wang, M. J. Ma, and J. M. Xu, "Fault-tolerant pancyclicity of augmented cubes," Information Processing Letters, vol. 103, no. 2, pp. 52-56, 2007.
[107] S. Y. Wang, J. Yuan, and A. X. Liu, "k-restricted edge connectivity for some interconnection networks," Applied Mathematics and Computation, vol. 201, issues.1-2, pp. 587-596, 2008.
[108] D. B. West, Introduction to Graph Theory, Prentice Hall Publishers, 2001.
[109] J. M. Xu, Topological structure and analysis of interconnection networks, Kluwer Academic Publishers, 2001.
[110] J. M. Xu, M. Lü, M. J. Ma, and A. Hellwig, "Super connectivity of line graphs," Information Processing Letters, vol. 94, issue 4, pp. 191-195, 2005.
[111] J. M. Xu, J. W. Wang, and W. W. Wang, "On super and restricted connectivity of some interconnection networks," Ars Combinatoria, vol. 94, 2010.
[112] M. Xu, K. Thulasiraman, and X. D. Hu, "Conditional Diagnosability of Matching Composition Networks Under the PMC Model," IEEE Transactions on Circuits and Systems II, vol. 56, no. 11, pp. 875-879, 2009.
[113] M. Xu and J. M. Xu, "The forwarding indices of augmented cubes," Information Processing Letters, vol. 101, no. 5, pp. 158-159, 2007.
[114] J. M. Xu, Q. Zhu, and M. Xu, "Fault-tolerant analysis of a class of networks," Information Processing Letters, vol. 103, issue 6, pp. 222-226, 2007.
[115] X. F. Yang, D. J. Evans, and G. M. Megson, "The locally twisted cubes," International Journal of Computer Mathematics, vol. 82, no. 4, pp. 401-413, 2005.
[116] W. H. Yang and J. X. Meng, "Extraconnectivity of hypercubes," Applied Mathematics Letters, vol. 22, issue 6, pp. 887-891, 2009.
[117] P. J. Yang, S. B. Tien, and C. S. Raghavendra, "Embedding of rings and meshes onto faulty hypercubes using free dimensions," IEEE Transactions on Computers, vol. 43, issue 5, pp. 608-613, 1994.
[118] Z. Zhang, "Sufficient conditions for restricted-edge-connectivity to be optimal," Discrete Mathematics, vol. 307, issue 22, pp. 2891-2899, 2007.
[119] Z. Zhang, "Extra edge connectivity and isoperimetric edge connectivity," Discrete Mathematics, vol. 308, issue 20, pp. 4560-4569, 2008.
[120] S. Zhou, "The Conditional Diagnosability of Möbius cubes under the Comparison Model," Proceedings of the 2009 IEEE International Conference on Information and Automation, pp. 96-100, 2009.
[121] J. Zhao, F. J.Meyer, N. Park, and F. Lombardi, "Sequential Diagnosis of Processor Array Systems," IEEE Transactions on Reliability, vol. 53, pp. 487-498, 2004.
[122] S. Zhou, "The Conditional Diagnosability of Locally Twisted Cubes," Proceedings of 2009 4th International Conference on Computer Science and Education, pp. 221-226, 2009.
[123] S. Zhou, "The Conditional Diagnosability of Twisted cubes under the Comparison Model," 2009 IEEE International Symposium on Parallel and Distributed Processing with Applications, pp. 696-701, 2009.
[124] Q. Zhu, "On conditional diagnosability and reliability of the BC networks," The Journal of Supercomputing, vol. 45, no. 2, pp. 173-184, 2008.
[125] Q. Zhu, J. M. Xu, X. M. Hou, and M. Xu, "On reliability of the folded hypercubes," Information Sciences, vol. 177, issue 8, pp. 1782-1788, 2007.
[126] Q. Zhu, S.Y. Liu, and M. Xu, "On conditional diagnosability of the folded hypercubes," Information Sciences, vol. 178, no. 4, pp. 1069-1077, 2008.