Abstract
A set of spanning trees in a graph G is called independent spanning trees (ISTs for short) if they are rooted at the same vertex, say r, and for each vertex \(v(\ne r)\) in G, the two paths from v to r in any two trees share no common vertex except for v and r. Constructing ISTs has applications on fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been used extensively to design interconnection networks, the study of constructing ISTs on Cayley graphs is very significative. It is well-known that star networks \(S_n\) and bubble-sort network \(B_n\) are two of the most attractive subclasses of Cayley graphs. Although it has been dealt with about two decades for the construction of ISTs on \(S_n\) (which has been pointed out that there is a flaw and has been corrected recently), so far the problem of constructing ISTs on \(B_n\) has not been dealt with. In this paper, we present an efficient algorithm to construct \(n-1\) ISTs of \(B_n\). It seems that our work is the latest breakthrough on the problem of ISTs for all subclasses of Cayley graphs except star networks.
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References
Akers, S.B., Krishnamurty, B.: A group theoretic model for symmetric interconnection networks. IEEE Trans. Comput. 28, 555–566 (1989)
Araki, T., Kikuchi, Y.: Hamiltonian laceability of bubble-sort graphs with edge faults. Inform. Sci. 177, 2679–2691 (2007)
Bao, F., Funyu, Y., Hamada, Y., Igarashi, Y.: Reliable broadcasting and secure distributing in channel networks. In: Proceedings of 3rd International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN 1997), Taipei, pp. 472–478 (1997)
Chang, Y.-H., Yang, J.-S., Hsieh, S.-Y., Chang, J.-M., Wang, Y.-L.: Construction independent spanning trees on locally twisted cubes in parallel. J. Comb. Optim. 33, 956–967 (2017)
Chang, J.-M., Yang, T.-J., Yang, J.-S.: A parallel algorithm for constructing independent spanning trees in twisted cubes. Discret. Appl. Math. 219, 74–82 (2017)
Cheriyan, J., Maheshwari, S.N.: Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs. J. Algorithms 9, 507–537 (1988)
Curran, S., Lee, O., Yu, X.: Finding four independent trees. SIAM J. Comput. 35, 1023–1058 (2006)
Hao, R.-X., Tian, Z.-X., Xu, J.-M.: Relationship between conditional diagnosability and 2-extra connectivity of symmetric graphs. Theor. Comput. Sci. 627, 36–53 (2012)
Itai, A., Rodeh, M.: The multi-tree approach to reliability in distributed networks. Inform. Comput. 79, 43–59 (1988)
Kikuchi, Y., Araki, T.: Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs. Inform. Process. Lett. 100, 52–59 (2006)
Kao, S.-S., Chang, J.-M., Pai, K.-J., Yang, J.-S., Tang, S.-M., Wu, R.-Y.: A parallel construction of vertex-disjoint spanning trees with optimal heights in star networks. In: Gao, X., Du, H., Han, M. (eds.) COCOA 2017. LNCS, vol. 10627, pp. 41–55. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71150-8_4
Kao, S.-S., Chang, J.-M., Pai, K.-J., Wu, R.-Y.: Open source for “Constructing independent spanning trees on bubble-sort networks". https://sites.google.com/ntub.edu.tw/ist-bs/. Accessed 8 Jan 2018
Lakshmivarahan, S., Jwo, J., Dhall, S.K.: Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey. Parallel Comput. 19, 361–407 (1993)
Kung, T.-L., Hung, C.-N.: Estimating the subsystem reliability of bubblesort networks. Theor. Comput. Sci. 670, 45–55 (2017)
Rescigno, A.A.: Vertex-disjoint spanning trees of the star network with applications to fault-tolerance and security. Inform. Sci. 137, 259–276 (2001)
Suzuki, Y., Kaneko, K.: An algorithm for disjoint paths in bubble-sort graphs. Syst. Comput. Jpn. 37, 27–32 (2006)
Suzuki, Y., Kaneko, K.: The container problem in bubble-sort graphs. IEICE Trans. Inf. Syst. E91–D, 1003–1009 (2008)
Wang, M., Guo, Y., Wang, S.: The 1-good-neighbour diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM* model. Int. J. Comput. Math. 94, 620–631 (2017)
Wang, M., Lin, Y., Wang, S.: The 2-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM* model. Theor. Comput. Sci. 628, 92–100 (2016)
Wang, S., Yang, Y.: Fault tolerance in bubble-sort graph networks. Theor. Comput. Sci. 421, 62–69 (2012)
Yang, Y., Wang, S., Li, J.: Subnetwork preclusion for bubble-sort graph networks. Inform. Process. Lett. 115, 817–821 (2015)
Yang, J.-S., Chan, H.-C., Chang, J.-M.: Broadcasting secure messages via optimal independent spanning trees in folded hypercubes. Discret. Appl. Math. 159, 1254–1263 (2011)
Zehavi, A., Itai, A.: Three tree-paths. J. Graph Theory 13, 175–188 (1989)
Zhou, S., Wang, J., Xu, X., Xu, J.-M.: Conditional fault diagnosis of bubble sort graphs under the PMC model. Intel. Comput. Evol. Comput. 180, 53–59 (2013). AISC
Acknowledgments
This research was partially supported by MOST grants MOST 104-2221-E-141-002-MY3 (Jou-Ming Chang), 105-2221-E-131-027 (Kung-Jui Pai) and 104-2221-E-262-005 (Ro-Yu Wu) from the Ministry of Science and Technology, Taiwan.
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Kao, SS., Chang, JM., Pai, KJ., Wu, RY. (2018). Constructing Independent Spanning Trees on Bubble-Sort Networks. In: Wang, L., Zhu, D. (eds) Computing and Combinatorics. COCOON 2018. Lecture Notes in Computer Science(), vol 10976. Springer, Cham. https://doi.org/10.1007/978-3-319-94776-1_1
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