# Line digraph iterations and the spread concept—with application to graph theory, fault tolerance, and routing

## Abstract

This paper is concerned with the spread concept, line digraph iterations, and their relationship. A graph has spread (*m, k, l*) if for any *m* + 1 distinct nodes *x,y*_{1},..., *y*_{ m } and *m* positive integers *r*_{1},..., *r*_{ m } such that ⌆_{i}r_{i}= *k*, there exist *k* node-disjoint paths of length at most *l* from *x* to the *y*_{ i }, where *r*_{ i } of them end at *y*_{ i }. This concept contains, and is related to, many important concepts used in communications and graph theory. The line digraph of a digraph *G(V, E)* is the digraph *L(G)* where nodes represent the edges of *G* and there is an edge (*x,y*) in *L(G)* if and only if *x* represents the edge (*u, v*) in *G* and *y* represents the edge (*v,w*) in *G* for some *u, v,w ε V(G)*. Many useful graphs, like the de Bruijn and Kautz digraphs, can be generated by line digraph iterations. We prove an *optimal* general theorem about the spreads of digraphs generated by line digraph iterations. Then we apply it to the de Bruijn and Kautz digraphs to derive optimal bounds on their spreads, which improve, re-prove, or resolve previous results and open questions on the connectivity, diameter, *k*-diameter, diameter vulnerability, and some other issues related to length-bounded disjoint paths, of these two graphs.

## Key words

graph connectivity spread line digraph iteration fault tolerance diameter vulnerability de Bruijn graph Kautz graph container,*k*-diameter

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]J.-C. Bermond, N. Homobono, and C. Peyrat,
*Large Fault-Tolerant Interconnection Networks*, Graphs and Combinatorics, 5, No. 2 (1989), pp. 107–123.Google Scholar - [2]L.W. Beineke,
*On Derived Graphs and Digraphs*, Beiträge zur Graphentheorie, Teubner, Leipzig, 1968, pp. 17–23.Google Scholar - [3]B. Bollobás,
*Extremal Graphs Theory*, Academic Press, New York, 1978.Google Scholar - [4]F. Buckley and F. Harary,
*Distance in Graphs*, Addison-Wesley, Reading, Massachusetts, 1990.Google Scholar - [5]G. Chartrand and M.J. Stewart,
*The Connectivity of Line-Graphs*, Mathematische Annalen 182 (1969), pp. 170–174.Google Scholar - [6]M.A. Fiol, J.L.A. Yebra, and I. Alegre,
*Line Digraph Iterations and the (d,k) Digraph Problem*, IEEE Trans. on Computers, C-33, No. 5 (May 1984), pp. 400–403.Google Scholar - [7]F. Harary and R.Z. Norman,
*Some Properties of Line Digraphs*, Rendiconti del Circolo Matematico di Palermo, 9 (1960), pp. 161–168.Google Scholar - [8]D.F. Hsu and Y.-D. Lyuu,
*A Graph-Theoretical Study of Transmission Delay and Fault Tolerance*, to appear in*Proc. Fourth ISMM International Conference on Parallel and Distributed Computing and Systems*, 1991.Google Scholar - [9]M. Imase, T. Soneoka, and K. Okada,
*Connectivity of Regular Directed Graphs with Small Diameters*, IEEE Trans. on Computers, C-34, No. 3 (March 1985), pp. 267–273.Google Scholar - [10]M. Imase, T. Soneoka, and K. Okada,
*Fault-Tolerant Processor Interconnection Networks*, Systems and Computers in Japan, 17, No. 8 (August 1986), pp. 21–30. Translated from Denshi Tsushin Gakkai Ronbunshi, 68-D, No. 8 (August 1985), pp. 1449–1456.Google Scholar - [11]W.H. Kautz,
*Bounds on Directed (d, k) Graphs*, Theory of Cellular Logic Networks and Machines, AFCKL-68-0668 Final Report, 1968, pp. 20–28.Google Scholar - [12]F.J. Meyer and D.K. Pradhan,
*Flip-Trees: Fault-Tolerant Graphs with Wide Containers*, IEEE Trans. on Computers, C-37, No. 4 (April 1988), pp. 472–478.Google Scholar - [13]D.K. Pradhan and S.M. Reddy,
*A Fault-Tolerant Communication Architecture for Distributed Systems*, IEEE Trans. on Computers, C-31, No. 9 (September 1982), pp. 863–870.Google Scholar - [14]M.O. Rabin,
*Efficient Dispersal of Information for Security, Load Balancing, and Fault Tolerance*, J. ACM, 36, No. 2 (April 1989), pp. 335–348.Google Scholar - [15]S.M. Reddy, J.G. Kuhl, S.H. Hosseini, and H. Lee,
*On Digraphs with Minimum Diameter and Maximum Connectivity*, Proc. 20th Annual Allerton Conference on Communication, Control, and Computing, 1982, pp. 1018–1026.Google Scholar - [16]Y. Saad and M.H. Schultz,
*Topological Properties of Hypercubes*, IEEE Trans. on Computers, C-37, No. 7 (July 1988), pp. 867–872.Google Scholar - [17]M.R. Samatham and D.K. Pradhan,
*The De Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI*, IEEE Trans. on Computers, C-38, No. 4 (April 1989), pp. 567–581.Google Scholar - [18]A.S. Tanenbaum,
*Computer Networks*, Englewood Cliffs, Prentice-Hall, New Jersey, 1981.Google Scholar