Advertisement

Constructing Three Completely Independent Spanning Trees in Locally Twisted Cubes

  • Kung-Jui PaiEmail author
  • Ruay-Shiung Chang
  • Jou-Ming Chang
  • Ro-Yu Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11458)

Abstract

For the underlying graph G of a network, k spanning trees of G are called completely independent spanning trees (CISTs for short) if they are mutually inner-node-disjoint. It has been known that determining the existence of k CISTs in a graph is an NP-hard problem, even for \(k=2\). Accordingly, researches focused on the problem of constructing multiple CISTs in some famous networks. Pai and Chang [28] proposed a unified approach to recursively construct two CISTs with diameter \(2n-1\) in several n-dimensional hypercube-variant networks for \(n\geqslant 4\), including locally twisted cubes \(LTQ_n\). Later on, they provided a new construction for \(LTQ_n\) and showed that the diameter of two CISTs can be reduced to \(2n-2\) if \(n=4\) (and thus is optimal) and \(2n-3\) if \(n\geqslant 5\). In this paper, we intend to construct more CISTs of \(LTQ_n\). We develop a novel tree searching algorithm, called two-stages tree-searching algorithm, to construct three CISTs of \(LTQ_6\) and show that the three CISTs of the high-dimensional \(LTQ_n\) for \(n\geqslant 7\) can be constructed by recursion. The diameters of three CISTs for \(LTQ_n\) we constructed are 9, 12 and 14 when \(n=6\), and are \(2n-3\), \(2n-1\) and \(2n+1\) when \(n\geqslant 7\).

Keywords

Completely independent spanning trees Interconnection networks Locally twisted cubes Diameter 

Notes

Acknowledgments

This research was partially supported by MOST grants 107-2221-E-131-011 (K.-J. Pai), 107-2221-E-141-002 (R.-S. Chang) and 107-2221-E-141-001-MY3 (J.-M. Chang), from the Ministry of Science and Technology, Taiwan.

References

  1. 1.
    Araki, T.: Dirac’s condition for completely independent spanning trees. J. Graph Theor. 77, 171–179 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chang, H.-Y., Wang, H.-L., Yang, J.-S., Chang, J.-M.: A note on the degree condition of completely independent spanning trees. IEICE Trans. Fundam. E98-A, 2191–2193 (2015)CrossRefGoogle Scholar
  3. 3.
    Chang, J.-M., Pai, K.-J., Yang, J.-S., Chan, H.-C.: Embedding two disjoint multi-dimensional meshes into locally twisted cubes. J. Internet Tech. 16, 541–546 (2015)Google Scholar
  4. 4.
    Chang, J.-M., Chang, H.-Y., Wang, H.-L., Pai, K.-J., Yang, J.-S.: Completely independent spanning trees on 4-regular chordal rings. IEICE Trans. Fundam. E100-A, 1932–1935 (2017)CrossRefGoogle Scholar
  5. 5.
    Chang, N.-W., Hsieh, S.-Y.: \(\{2,3\}\)-extraconnectivities of hypercube-like networks. J. Comput. Syst. Sci. 79, 669–688 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cheng, B., Wang, D., Fan, J.: Constructing completely independent spanning trees in crossed cubes. Discrete Appl. Math. 219, 100–109 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Darties, B., Gastineau, N., Togni, O.: Completely independent spanning trees in some regular graphs. Discrete Appl. Math. 217, 163–174 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fan, G., Hong, Y., Liu, Q.: Ore’s condition for completely independent spanning trees. Discrete Appl. Math. 177, 95–100 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo, L., Su, G., Lin, W., Chen, J.: Fault tolerance of locally twisted cubes. Appl. Math. Comput. 334, 401–406 (2018)MathSciNetGoogle Scholar
  11. 11.
    Han, Y., Fan, J., Zhang, S., Yang, J., Qian, P.: Embedding meshes into locally twisted cubes. Inform. Sci. 180, 3794–3805 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hasunuma, T.: Completely independent spanning trees in the underlying graph of a line digraph. Discrete Math. 234, 149–157 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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_21CrossRefzbMATHGoogle Scholar
  14. 14.
    Hasunuma, T.: Minimum degree conditions and optimal graphs for completely independent spanning trees. In: Lipták, Z., Smyth, W.F. (eds.) IWOCA 2015. LNCS, vol. 9538, pp. 260–273. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29516-9_22CrossRefGoogle Scholar
  15. 15.
    Hasunuma, T., Morisaka, C.: Completely independent spanning trees in torus networks. Networks 60, 59–69 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hsieh, S.-Y., Huang, H.-W., Lee, C.-W.: \(\{2,3\}\)-restricted connectivity of locally twisted cubes. Theor. Comput. Sci. 615, 78–90 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hsieh, S.-Y., Tu, C.-J.: Constructing edge-disjoint spanning trees in locally twisted cubes. Theor. Comput. Sci. 410, 926–932 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hsieh, S.-Y., Wu, C.-Y.: Edge-fault-tolerant Hamiltonicity of locally twisted cubes under conditional edge faults. J. Combin. Optim. 19, 16–30 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hong, X., Liu, Q.: Degree condition for completely independent spanning trees. Inform. Process. Lett. 116, 644–648 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lai, C.-J., Chen, J.-C., Tsai, C.-H.: A systematic approach for embedding of Hamiltonian cycles through a prescribed edge in locally twisted cubes. Inform. Sci. 289, 1–7 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, T.-K., Lai, C.-J., Tsai, C.-H.: A novel algorithm to embed a multi-dimensional torus into a locally twisted cube. Theor. Comput. Sci. 412, 2418–2424 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lin, J.-C., Yang, J.-S., Hsu, C.-C., Chang, J.-M.: Independent spanning trees vs. edge-disjoint spanning trees in locally twisted cubes. Inform. Process. Lett. 110, 414–419 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liu, Y.-J., Lan, J.K., Chou, W.Y., Chen, C.: Constructing independent spanning trees for locally twisted cubes. Theor. Comput. Sci. 412, 2237–2252 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu, Z., Fan, J., Zhou, J., Cheng, B., Jia, Z.: Fault-tolerant embedding of complete binary trees in locally twisted cubes. J. Parallel Distrib. Comput. 101, 69–78 (2017)CrossRefGoogle Scholar
  25. 25.
    Ma, M.-J., Xu, J.-M.: Panconnectivity of locally twisted cubes. Appl. Math. Lett. 19, 673–677 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Matsushita, M., Otachi, Y., Araki, T.: Completely independent spanning trees in (partial) \(k\)-trees. Discuss. Math. Graph Theor. 5, 427–437 (2015)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Moinet, A., Darties, B., Gastineau, N., Baril, J.-L., Togni, O.: Completely independent spanning trees for enhancing the robustness in ad-hoc Networks. In: Proceedings of 10th IEEE International Workshop on Selected Topics in Mobile and Wireless Computing (STWiWob 2017), Rome, Italy, 9–11 October, pp. 63–70 (2017)Google Scholar
  28. 28.
    Pai, K.-J., Chang, J.-M.: Constructing two completely independent spanning trees in hypercube-variant networks. Theor. Comput. Sci. 652, 28–37 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pai, K.-J., Chang, J.-M.: Improving the diameters of completely independent spanning trees in locally twisted cubes. Inform. Process. Lett. 141, 22–24 (2019)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Pai, K.-J., Tang, S.-M., Chang, J.-M., Yang, J.-S.: Completely independent spanning trees on complete graphs, complete bipartite graphs and complete tripartite graphs. In: Chang, R.S., Jain, L., Peng, S.L. (eds.) Advances in Intelligent Systems and Applications - Volume 1. Smart Innovation, Systems and Technologies, vol. 20, pp. 107–113. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-35452-6_13CrossRefGoogle Scholar
  31. 31.
    Pai, K.-J., Yang, J.-S., Yao, S.-C., Tang, S.-M., Chang, J.-M.: Completely independent spanning trees on some interconnection networks. IEICE Trans. Inform. Syst. E97-D, 2514–2517 (2014)CrossRefGoogle Scholar
  32. 32.
    Péterfalvi, F.: Two counterexamples on completely independent spanning trees. Discrete Math. 312, 808–810 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ren, Y., Wang, S.: The \(g\)-good-neighbor diagnosability of locally twisted cubes. Theor. Comput. Sci. 697, 91–97 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Wei, Y.-L., Xu, M.: The \(g\)-good-neighbor conditional diagnosability of locally twisted cubes. J. Oper. Res. Soc. China 6, 333–347 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Xu, X., Huang, Y., Zhang, P., Zhang, S.: Fault-tolerant vertex-pancyclicity of locally twisted cubes. J. Parallel Distrib. Comput. 88, 57–62 (2016)CrossRefGoogle Scholar
  36. 36.
    Xu, X., Zhai, W., Xu, J.-M., Deng, A., Yang, Y.: Fault-tolerant edge-pancyclicity of locally twisted cubes. Inform. Sci. 181, 2268–2277 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Yang, H., Yang, X.: A fast diagnosis algorithm for locally twisted cube multiprocessor systems under the MM* model. Comput. Math. Appl. 53, 918–926 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Yang, X., Evans, D.J., Megson, G.M.: The locally twisted cubes. Int. J. Comput. Math. 82, 401–413 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Kung-Jui Pai
    • 1
    Email author
  • Ruay-Shiung Chang
    • 2
  • Jou-Ming Chang
    • 2
  • Ro-Yu Wu
    • 3
  1. 1.Department of Industrial Engineering and ManagementMing Chi University of TechnologyNew Taipei CityTaiwan
  2. 2.Institute of Information and Decision SciencesNational Taipei University of BusinessTaipeiTaiwan
  3. 3.Department of Industrial ManagementLunghwa University of Science and TechnologyTaoyuanTaiwan

Personalised recommendations