Towards the Independent Spanning Trees in the Line Graphs of Interconnection Networks

  • Baolei Cheng
  • Jianxi FanEmail author
  • Xiaoyan Li
  • Guijuan Wang
  • Jingya Zhou
  • Yuejuan Han
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11336)


Node/edge-Independent spanning trees (ISTs) have attracted a lot of attention in the past twenty years. Many results such as edge-disjoint Hamilton cycles, traceability, number of spanning trees, structural properties, topological indices, etc, have been obtained on line graphs, and researchers have applied the line graphs of some interconnection networks into data center networks, such as SWCube, BCDC, etc. However, node/edge conjecture is still open for n-node-connected interconnection network with \(n\ge \) 5. So far, results have been obtained on a lot of special interconnection networks, but few results are reported on the line graphs of them. In this paper, we consider the problem of constructing node-ISTs in a line graph G of an interconnection network \(G'\). We first give the construction of node-ISTs in \(G'\) based on the edge-ISTs in G. Then, an algorithm to construct node-ISTs in G based on the edge-ISTs in \(G'\) is presented. At the end, simulation experiments on the line graphs of hypercubes show that the maximal height of the constructed node-ISTs on the line graph of n-dimensional hypercube is \(n+1\) for \(n\ge 3\).


Independent spanning trees Internally disjoint paths Line graph Interconnection network 



This work is supported by National Natural Science Foundation of China (No. 61572337, No. 61502328, and No. 61602333), China Postdoctoral Science Foundation Funded Project (No. 2015M581858), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJA520009), the Jiangsu Planned Projects for Postdoctoral Research Funds (No. 1501089B and No. 1701173B), Opening Foundation of Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks (No. WSNLBKF201701), and Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_2005 and No. KYCX18_2510).


  1. 1.
    Bonomom, F., Durán, G., Safe, M.D., Wagler, A.K.: Clique-perfectness of complements of line graphs. Discret. Appl. Math. 186(1), 19–44 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bao, F., Funyu, Y., Hamada, Y., Igarashi, Y.: Reliable broadcasting and secure distributing in channel networks. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E81–A, 796–806 (1998)Google Scholar
  3. 3.
    Bao, F., Igarashi, Y., Öhring, S.R.: Reliable broadcasting in product networks. Discret. Appl. Math. 83(1–3), 3–20 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chen, Y.-S., Chiang, C.-Y., Chen, C.-Y.: Multi-node broadcasting in all-ported 3-D wormhole-routed torus using an aggregation-then-distribution strategy. J. Syst. Arch. 50(9), 575–589 (2004)CrossRefGoogle Scholar
  5. 5.
    Cheng, B., Fan, J., Jia, X., Zhang, S.: Independent spanning trees in crossed cubes. Inf. Sci. 233(1), 276–289 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cheng, B., Fan, J., Jia, X., Wang, J.: Dimension-adjacent trees and parallel construction of independent spanning trees on crossed cubes. J. Parallel Distrib. Comput. 73, 641–652 (2013)zbMATHCrossRefGoogle Scholar
  7. 7.
    Cheng, B., Fan, J., Lyu, Q., Zhou, J., Liu, Z.: Constructing independent spanning trees with height \(n\) on the \(n\)-dimensional crossed cube. Futur. Gener. Comput. Syst. 87, 404–415 (2018)Google Scholar
  8. 8.
    Cheng, B., Fan, J., Jia, X., Jia, J.: Parallel construction of independent spanning trees and an application in diagnosis on Möbius cubes. J. Supercomput. 65(3), 1279–1301 (2013)CrossRefGoogle Scholar
  9. 9.
    Cheriyan, J., Maheshwari, S.N.: Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs. J. Algorithms 9(4), 507–537 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Curran, S., Lee, O., Yu, X.: Finding four independent trees. SIAM J. Comput. 35(5), 1023–1058 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dong, F., Yan, W.: Expression for the number of spanning trees of line graphs of arbitrary connected graphs. J. Graph Theory 85(1), 74–93 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gopalan, A., Ramasubramanian, S.: A counterexample for the proof of implication conjecture on independent spanning trees. Inf. Process. Lett. 113(14–16), 522–526 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Gopalan, A., Ramasubramanian, S.: On constructing three edge independent spanning trees. SIAM J. Comput. (2011, submitted)Google Scholar
  14. 14.
    Hasunuma, T.: Structural properties of subdivided-line graphs. J. Discret. Algorithms 31, 69–86 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Harvey, D.J., Wood, D.R.: Treewidth of the line graph of a complete graph. J. Graph Theory 79(1), 48–54 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hoyer, A., Thomas, R.: Four edge-independent spanning tree. SIAM J. Discret. Math. 32(1), 233–248 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Huck, A.: Independent trees in planar graphs. Graphs Comb. 15(1), 29–77 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hussain, Z., AlBdaiwi, B., Cerny, A.: Node-independent spanning trees in Gaussian networks. J. Parallel Distrib. Comput. 109, 324–332 (2017)CrossRefGoogle Scholar
  19. 19.
    Li, D., Wu, J.: On data center network architectures for interconnecting dual-port servers. IEEE Trans. Comput. 64(11), 3210–3222 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Itai, A., Rodeh, M.: The multi-tree approach to reliability in distributed networks. Inf. Comput. 79(1), 43–59 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Khuller, S., Schieber, B.: On independent spanning trees. Inf. Process. Lett. 42(6), 321–323 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Kim, J.-S., Lee, H.-O., Cheng, E., Lipták, L.: Independent spanning trees on even networks. Inf. Sci. 181(13), 2892–2905 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Kim, J.-S., Lee, H.-O., Cheng, E., Lipták, L.: Optimal independent spanning trees on odd graphs. J. Supercomput. 56(2), 212–225 (2011)CrossRefGoogle Scholar
  24. 24.
    Li, H., He, W., Yang, W., Bai, Y.: A note on edge-disjoint Hamilton cycles in line graphs. Graphs Comb. 32, 741–744 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Liu, Y.-J., Chou, W.Y., Lan, J.K., Chen, C.: Constructing independent spanning trees for locally twisted cubes. Theor. Comput. Sci. 412(22), 2237–2252 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Obokata, K., Iwasaki, Y., Bao, F., Igarashi, Y.: Independent spanning trees of product graphs and their construction. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E79–A(11), 1894–1903 (1996)zbMATHGoogle Scholar
  27. 27.
    Su, G., Xu, L.: Topological indices of the line graph of subdivision graphs and their Schur-bounds. Appl. Math. Comput. 253, 395–401 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Tian, T., Xiong, L.: Traceability on 2-connected line graphs. Appl. Math. Comput. 321, 1339–1351 (2018)MathSciNetGoogle Scholar
  29. 29.
    Tseng, Y.-C., Wang, S.-Y., Ho, C.-W.: Efficient broadcasting in wormhole-routed multicomputers: a network-partitioning approach. IEEE Trans. Parallel Distrib. Syst. 10(1), 44–61 (1999)CrossRefGoogle Scholar
  30. 30.
    Tang, S.-M., Wang, Y.-L., Leu, Y.-H.: Optimal independent spanning trees on hypercubes. J. Inf. Sci. Eng. 20(1), 143–155 (2004)MathSciNetGoogle Scholar
  31. 31.
    Wang, X., Fan, J., Lin, C.-K., Zhou, J., Liu, Z.: BCDC: a high-performance, server-centric data center network. J. Comput. Sci. Technol. 33(2), 400–416 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yang, J.-S., Tang, S.-M., Chang, J.-M., Wang, Y.-L.: Parallel construction of optimal independent spanning trees on hypercubes. Parallel Comput. 33(1), 73–79 (2007)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zehavi, A., Itai, A.: Three tree-paths. J. Graph Theory 13(2), 175–188 (1989)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Baolei Cheng
    • 1
    • 2
    • 3
  • Jianxi Fan
    • 1
    • 2
    Email author
  • Xiaoyan Li
    • 1
  • Guijuan Wang
    • 1
  • Jingya Zhou
    • 1
  • Yuejuan Han
    • 1
  1. 1.School of Computer Science and TechnologySoochow UniversitySuzhouChina
  2. 2.Jiangsu High Technology Research Key Laboratory for Wireless Sensor NetworksNanjingChina
  3. 3.Provincial Key Laboratory for Computer Information Processing TechnologySoochow UniversitySuzhouChina

Personalised recommendations