Abstract
This paper proves that for every positive integers n,k and any positive constant ε, we can explicitly construct a graph G with n+O(k 1 + ε) vertices and a constant degree such that even after removing any k vertices from G, the remaining graph still contains an n-vertex dipath. This paper also proves that for every positive integers n,k and any positive constant ε, we can explicitly construct a graph H with n+O(k 2 + ε) vertices and a constant degree such that even after removing any k vertices from H, the remaining graph still contains an n-vertex 2-dimensional square mesh.
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References
Alon, N., Chung, F.: Explicit construction of linear sized tolerant networks. Discrete Math. 72, 15–19 (1988)
Bruck, J., Cypher, R., Ho, C.: Fault-tolerant meshes and hypercubes with minimal numbers of spares. IEEE Trans. on Comput., 1089–1104 (1993)
Bruck, J., Cypher, R., Ho, C.: Tolerating faults in a mesh with a row of spare nodes. Theoretical Computer Science 128(1-2), 241–252 (1994)
Bruck, J., Cypher, R., Ho, C.: Fault-tolerant meshes with small degree. SIAM. Journal on Computing 26, 1764–1784 (1997)
Dutt, S., Hayes, J.: Designing fault-tolerant systems using automorphisms. Journal of Parallel and Distributed Computing 12, 249–268 (1991)
Erdös, P., Graham, R., Szemerédi, E.: On sparse graphs with dense long paths. Comp. and Math. with Appl. 1, 145–161 (1975)
Farrag, A.: New algorithm for constructing fault-tolerant solutions of the circulant graph configuration. Parallel Computing 22, 1239–1254 (1996)
Harary, F., Hayes, J.: Node fault tolerance in graphs. Networks 27, 19–23 (1996)
Hayes, J.: A graph model for fault-tolerant computing systems. IEEE Trans. on Comput. C-25, 875–883 (1976)
Paoli, M., Wong, W., Wong, C.: Minimum k-hamiltonian graphs, II. J. Graph Theory 10, 79–95 (1986)
Tamaki, H.: Construction of the mesh and the torus tolerating a large number of faults. In: Proc. 6th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA 1994), pp. 268–277 (1994)
Ueno, S., Bagchi, A., Hakimi, S., Schmeichel, E.: On minimum fault-tolerant networks. SIAM J. on Discrete Mathematics 6(4), 565–574 (1993)
Wong, W., Wong, C.: Minimum k-hamiltonian graphs. J. Graph Theory 8, 155–165 (1984)
Yamada, T., Ueno, S.: Optimal fault-tolerant linear arrays. In: Proc. 15th ACM SPAA, pp. 60–64 (2003)
Zhang, L.: Fault tolerant networks with small degree. In: Proceedings of Twelfth annual ACM Symposium on Parallel Algorithms and Architectures, pp. 64–69 (2000)
Zhang, L.: Fault-tolerant meshes with small degree. IEEE Transactions on Computers 51(5), 553–560 (2002)
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© 2004 Springer-Verlag Berlin Heidelberg
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Yamada, T. (2004). Fault-Tolerant Meshes with Constant Degree. In: Chwa, KY., Munro, J.I.J. (eds) Computing and Combinatorics. COCOON 2004. Lecture Notes in Computer Science, vol 3106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27798-9_43
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DOI: https://doi.org/10.1007/978-3-540-27798-9_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22856-1
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