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Optimally Reliable Graphs for Both Vertex and Edge Failures

Published online by Cambridge University Press:  12 September 2008

D. H. Smith
D. H. Smith
Affiliation:
Department of Mathematics and Computing, University of Glamorgan, Pontypridd, Mid-Glamorgan, CF37 1DL, Wales, U.K.
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Abstract

We consider networks in which both the nodes and the links may fail. We represent the network by an undirected graph G. Vertices of the graph fail with probability p and edges of the graph fail with probability q, where all failures are assumed independent. We shall be concerned with minimising the probability P(G) that G is disconnected for graphs with given numbers of vertices and edges. We show how to construct these optimal graphs in many cases when p and q are ‘small’.

Information

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 1 , March 1993 , pp. 93 - 100
Copyright
Copyright © Cambridge University Press 1993

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References

[1]Bauer, D., Boesch, F., Suffel, C. and Tindell, R. (1981) Connectivity extremal problems and the design of reliable probabilistic networks. In: Chartrand, G., Alavi, Y., Goldsmith, D., Lesniak-Forster, L. and Lick, D. (Eds.) The Theory and Application of Graphs, Wiley 45–54.Google Scholar
[2]Bauer, D., Boesch, F., Suffel, C. and Tindell, R. (1985) Combinatorial optimization problems in the analysis and design of probabilistic networks. Networks 15 257–;271.CrossRefGoogle Scholar
[3]Bauer, D., Boesch, F. T., Suffel, C. and Van Slyke, R. (1987) On the validity of a reduction of reliable network design to a graph extremal problem. IEEE Trans. on Circuits and Systems CAS34 (12) 1579–1581.CrossRefGoogle Scholar
[4]Boesch, F. T. (1986) Synthesis of reliable networks – a survey. IEEE Trans. Reliability R35 (3) 240–246.CrossRefGoogle Scholar
[5]Boesch, F. T. (1986) On unreliability polynomials and graph connectivity in reliable network synthesis. J. Graph Theory 10 (3) 339–352.CrossRefGoogle Scholar
[6]Boesch, F. T. and Felzer, A. P. (1971) On the invulnerability of the regular complete k-partite graphs. SIAM J. Appl. Math. 20 176–182.CrossRefGoogle Scholar
[7]Boesch, F. T. and Frisch, I. T. (1968) On the smallest disconnecting set in a graph. IEEE Trans. Circuit Theory 15 (3) 286–288.CrossRefGoogle Scholar
[8]Boesch, F. T., Li, X. and Suffel, C. (1991) On the existence of uniformly optimally reliable networks. Networks 21 181–194.CrossRefGoogle Scholar
[9]Frank, H. (1969) Some new results in the design of survivable networks. Proc. 12th Ann. Midwest Circuit Theory Symp.,University of Texas, 13.1–13.8.Google Scholar
[10]Frank, H. (1969) Maximally reliable node weighted graphs. Proc. 3rd Ann. Conf. Information Sciences and Systems,Princeton University, 1–6.Google Scholar
[11]Hakimi, S. L. and Amin, A. T. (1973) On the design of reliable networks. Networks 3 241–260.CrossRefGoogle Scholar
[12]Smith, D. H. (1984) Graphs with the smallest number of minimum cut sets. Networks 14 47–61.CrossRefGoogle Scholar
[13]Smith, D. H. and Doty, L. L. (1990) On the construction of optimally reliable graphs. Networks 20 723–729.CrossRefGoogle Scholar