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Line Digraph Iterations and Connectivity Analysis of de Bruijn and Kautz Graphs
May 1993 (vol. 42 no. 5)
pp. 612-616

A graph has spread (m, k, l) if for any m+1 distinct nodes x, y/sub 1/, . . ., y/sub m/ and m positive integers r/sub 1/, . . ., r/sub m/, such that Sigma /sub i/r/sub i/=k, there exist k node-disjoint paths of length at most 1 from x to the y/sub i/, where r/sub i/ of them end at y/sub i/. This concept contains, and is related to many important concepts used in communications and graph theory. The authors prove an optimal general theorem about the spreads of digraphs generated by line digraph iterations. Useful graphs, like the de Bruijn and Kautz digraphs, can be thus generated. The theorem is applied to the de Bruijn and Kautz digraphs to derive optimal bounds on their spreads, which implies previous results and resolves open questions on their connectivity, diameter, k-diameter, vulnerability, and some other measures related to length-bound disjoint paths.

[1] J.-C. Bermond, N. Homobono, and C. Peyrat, "Large fault-tolerant interconnection networks,"Graphs and Combinatorics, vol. 5, no. 2, pp. 107-123, 1989.
[2] L.W. Beineke, "On derived graphs and digraphs,"Beiträge zur Graphentheorie, pp. 17-23, 1968.
[3] F.T. Boesch, "Synthesis of reliable networks--A survey,"IEEE Trans. Reliability, vol. R-35, no. 3, pp. 240-246, Aug. 1986.
[4] B. Bollobás,Extremal Graphs Theory. New York: Academic, 1978.
[5] B. Bollobás,Graph Theory: an Introductory Course. Berlin, Germany: Springer-Verlag, 1979.
[6] J. Bond and C. Peyrat, "Diameter vulnerability of some large interconnection networks,"Congressus Numerantium, vol. 66, pp. 267-282, 1988.
[7] N. G. de Bruijn, "A combinatorial problem," inProc. Akademe Van Wetenschappen, 49, part 2, 1946, pp. 758-764.
[8] F. Buckley and F. Harary,Distance in Graphs. Reading, MA: Addison-Wesley, 1990.
[9] G. Chartrand and M. J. Stewart, "The connectivity of line-graphs,"Mathematische Annalen, vol. 182, pp. 170-174, 1969.
[10] M. Dietzfelbinger, S. Madhavapeddy, and I. H. Sudborough, "Three disjoint path paradigms in star networks," inProc. 3rd IEEE Symp. Parallel Distributed Processing, 1991, pp. 400-406.
[11] B. Elspas, "Topological constraints on interconnection-limited logic," inProc. 5th Annu. Symp. Switching Circuit Theory and Logical Design, 1964, pp. 133-137.
[12] A.-H. Esfahanian and S. L. Hakimi, "Fault-tolerant routing in de Bruijn communications networks,"IEEE Trans. Comput., vol. C-34, pp. 777-788, Sept. 1985.
[13] D.-Z. Du, D. F. Hsu, and Y.-D. Lyuu, "Line digraph iterations and the spread concept--with application to graph theory, fault tolerance, and routing," inProc. Graph-Theoretical Concepts in Computer Science, 1991.Lecture Notes in Computer Science, 570, 1992, pp. 169-179.
[14] M. A. Fiol, J. L. A. Yebra, and I. Alegre, "Line digraph iterations and the (d,k) digraph problem,"IEEE Trans. Comput., vol. C-33, no. 5, pp. 400-403, May 1984.
[15] F. Harry and R. Z. Norman, "Some properties of line digraphs,"Rendiconti del Circolo Matematico de Palermo, vol. 9, pp. 161-168, 1960.
[16] D. F. Hsu and Y.-D. Lyuu, "A graph-theoretical study of transmission delay and faul tolerance," inProc. Fourth ISMM Int. Conf. Parallel Distributed Comput. Syst., 1991, pp. 20-24.
[17] M. Imase, T. Soneoka, and K. Okada, "Connectivity of regular directed graphs with small diameters,"IEEE Trans. Comput., vol. C-34, no. 3, pp. 267-273, Mar. 1985.
[18] M. Imase, T. Soneoka, and K. Okada, "Fault-tolerant processor interconnection networks,"Syst. Comput. Japan, vol. 17, no. 8, pp. 21-30, Aug. 1986. Translated fromDenshi Tsushin Gakkai Ronbunshi, vol. 68-D, no. 8, pp. 1449-1456, Aug. 1985.
[19] W. H. Kautz, "Bounds on directed (d,k) graphs,"Theory of Cellular Logic Networks and Machines, AFCKL-68-0668 Final Rep., 1968, pp. 20-28.
[20] F. T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, and Hypercubes. Palo Alto, CA: Morgan Kaufmann, 1992.
[21] Y.-D. Lyuu, "Fast fault-tolerant parallel communication for de Bruijn networks using information dispersal," inProc. 3rd IEEE Symp. Parallel Distributed Processing, 1991, pp. 466-473.Networks, to be published.
[22] F. J. Meyer and D. K. Pradhan, "Flip-trees: fault-tolerant graphs with wide containers,"IEEE Trans. Comput., vol. C-37, no. 4, pp. 472-478, Apr. 1988.
[23] D. K. Pradhan and S. M. Reddy, "A fault-tolerant communication architecture for distributed systems,"IEEE Trans. Comput., vol. C-31, no. 9, pp. 863-870, Sept. 1982.
[24] M. O. Rabin, "Efficient dispersal of information for security, load balancing, and fault tolerance,"J. ACM, vol. 36, no. 2, Apr. 1989.
[25] S. M. Reddy, J. G. Kuhl, S. H. Hosseini, and H. Lee, "On digraphs with minimum diameter and maximum connectivity," inProc. 20th Annu. Allerton Conf. Commun., Contr., Comput., 1982, pp. 1018-1026.
[26] Y. Saad and M. H. Schultz, "Topological properties of hypercubes,"IEEE Trans. Comput., vol. C-37, no. 7, pp. 867-872, July 1988.
[27] M. R. Samatham and D. K. Pradhan, "The de Bruijn multiprocessor network: A versatile parallel processing and sorting network for VLSI,"IEEE Trans. Comput., vol. C-38, no. 4, pp. 567-581, Apr. 1989.
[28] A. S. Tanenbaum,Computer Networks, Englewood Cliffs, NJ: Prentice-Hall, 1981.

Index Terms:
de Bruijn digraphs; digraph iterations; connectivity analysis; Kautz graphs; node-disjoint paths; graph theory; optimal general theorem; optimal bounds; length-bound disjoint paths; directed graphs; iterative methods.
Citation:
Ding-Zhu Du, Y.-D. Lyuu, D.F. Hsu, "Line Digraph Iterations and Connectivity Analysis of de Bruijn and Kautz Graphs," IEEE Transactions on Computers, vol. 42, no. 5, pp. 612-616, May 1993, doi:10.1109/12.223681
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