Abstract
In this paper we first define (t, s)-completely independent spanning trees, which is a generalization of completely independent spanning trees. A set of t spanning trees of a graph is (t, s)-completely independent if, for any pair of vertices u and v, among the set of t paths from u to v in the t spanning trees, at least \(s\le t\) paths are internally disjoint. By (t, s)-completely independent spanning trees, one can ensure any pair of vertices can communicate each other even if at most \(s-1\) vertices break down. We prove that every maximal planar graph has a set of (3, 2)-completely independent spanning trees, every tri-connected planar graph has a set of (3, 2)-completely independent spanning trees, and every 3D grid graph has a set of (3, 2)-completely independent spanning trees. Also one can compute them in linear time.
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References
Di Battista, G., Tamassia, R., Vismara, L.: Output-sensitive reporting of disjoint paths. Algorithmica 23(4), 302–340 (1999)
Cheriyan, J., Maheshwari, S.N.: Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs. J. Algorithms 9(4), 507–537 (1988)
Curran, S., Lee, O., Xingxing, Yu.: Finding four independent trees. SIAM J. Comput. 35(5), 1023–1058 (2006)
Hasunuma, T.: Completely independent spanning trees in the underlying graph of a line digraph. Discrete Math. 234, 149–157 (2001)
Hasunuma, T.: Completely independent spanning trees in maximal planar graphs. In: Goos, G., Hartmanis, J., van Leeuwen, J., Kučera, L. (eds.) WG 2002. LNCS, vol. 2573, pp. 235–245. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36379-3_21
Hopcroft, J.E., Wong, K.: Linear time algorithm for isomorphism of planar graphs. In: Proceeding of the 6th Annual ACM Symposium on Theory of Computing, pp. 172–184 (1974)
John, E.: Hopcroft and Robert Endre Tarjan, efficient planarity testing. J. ACM 21(4), 549–568 (1974)
Huck, A.: Independent trees in graphs. Graphs Comb. 10(1), 29–45 (1994)
Huck, A.: Independent trees in planar graphs independent trees. Graphs Comb. 15(1), 29–77 (1999)
Itai, A., Rodeh, M.: The multi-tree approach to reliability in distributed networks. Inf. Comput. 79(1), 43–59 (1988)
Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)
Khuller, S., Schieber, B.: On independent spanning trees. Inf. Process. Letters 42(6), 321–323 (1992)
Miura, K., Takahashi, D., Nakano, S., Nishizeki, T.: A linear-time algorithm to find four independent spanning trees in four-connected planar graphs. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 310–323. Springer, Heidelberg (1998). https://doi.org/10.1007/10692760_25
Miura, K., Takahashi, D., Nakano, S.-I., Nishizeki, T.: A linear-time algorithm to find four independent spanning trees in four connected planar graphs. Int. J. Found. Comput. Sci. 10(2), 195–210 (1999)
Nagai, S., Nakano, S.: A linear-time algorithm to find independent spanning trees in maximal planar graphs. In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol. 1928, pp. 290–301. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-40064-8_27
Nagai, S., Nakano, S.: A linear-time algorithm for five-partitioning five-connected internally triangulated plane graphs. IEICE Trans. Fund. E84–A(9), 2330–2337 (2001)
Nakano, S.: Planar drawings of plane graphs. IEICE Trans. Inf. Syst. E83–D(3), 384–391 (2000)
Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of SODA, pp. 138–148 (1990)
Zehavi, A., Itai, A.: Three tree-paths. J. Graph Theory 13(2), 175–188 (1989)
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Nakano, Si. (2024). (t, s)-Completely Independent Spanning Trees. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_26
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DOI: https://doi.org/10.1007/978-981-97-0566-5_26
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