Embedding Exchanged Hypercubes into Rings and Ladders

  • Weibei Fan
  • Jianxi Fan
  • Cheng-Kuan Lin
  • Zhijie Han
  • Peng Li
  • Ruchuan Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11335)


Graph embeddings are not only used to study the simulation capabilities of a parallel architecture but also to design its VLSI layout. The n-dimensional hypercube is one of the most popular topological structure for interconnection networks in parallel computing and communication systems. The exchanged hypercube \(EH_{s,t}\) (where \(s\ge 1\) and \(t\ge 1\)) is obtained by systematically deleting edges from a hypercube \(Q_{s+t+1}\), which retains several valuable and desirable properties of the hypercube such as a small diameter, bipancyclicity, and super connectivity. In this paper, we identify maximum induced subgraph of \(EH_{s,t}\) and study embeddings of \(EH_{s,t}\) into a ring and a ladder with minimum wirelength.


Interconnection networks \(EH_{s, t}\) Graph embedding Rings Ladders 



We would like to express our sincerest appreciation to Prof. Guoliang Chen for his constructive suggestions. This work is supported by National Key R&D Program of China (2018YFB1003201), Natural Science Foundation of China under grant (No. 61572337, No. 61602333, No. 61672296 and No. 61702351), China Postdoctoral Science Foundation (No. 172985), Scientific & Technological Support Project of Jiangsu Province (No. BE2016777, No. BE2016185), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Nos. 17KJB520036), Jiangsu Planned Projects for Postdoctoral Research Funds under Grant (No. 1701172B) and Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks Foundation (No. WSNLBKF201701).


  1. 1.
    Arockiaraj, M., Abraham, J., Quadras., J.: Linear layout of locally twisted cubes. Int. J. Comput. Math. 94(1), 56–65 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bezrukov, S.L., Das, S.K., Elsasser, R.: An edge-isoperimetric problem for powers of the Petersen graph. Ann. Combinatorics 4(2), 153–169 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bezrukov, S.L., Chavez, J.D., Harper, L.H., Röttger, M., Schroeder, U.P.: Embedding of hypercubes into grids. Mortar Fire Control System, pp. 693–701 (1998)Google Scholar
  4. 4.
    Boals, A.J., Gupta, A.K., Sherwani, N.A.: Incomplete hypercubes: algorithms and embeddings. J. Supercomputing 8(3), 263–294 (1994)CrossRefGoogle Scholar
  5. 5.
    Chen, Y., Shen, H.: Routing and wavelength assignment for hypercube in array-based WDM optical networks. J. Parallel Distrib. Comput. 70(1), 59–68 (2010)zbMATHCrossRefGoogle Scholar
  6. 6.
    Erbele, J., Chavez, J., Trapp, R.: The cyclic cutwidth of \(Q_{n}\). Technical report, California State UniversitySan Bernardino (CSUSB) (2003)Google Scholar
  7. 7.
    Fan, J., Jia, X., Lin, X.: Complete path embeddings in crossed cubes. Inf. Sci. 176(22), 3332–3346 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Fan, J., Jia, X., Lin, X.: Embedding of cycles in twisted cubes with edge-pancyclic. Algorithmica 51(3), 264–282 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Wang, X., Fan, J., Jia, X.: Embedding meshes into twisted-cubes. Inf. Sci. 181(14), 3085–3099 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness (1979)Google Scholar
  11. 11.
    Harper, L.H.: Global Methods for Combinatorial Isoperimetric Problems. Cambridge University Press, UK (2004)Google Scholar
  12. 12.
    Han, Y., Fan, J., Zhang, S.: Embedding meshes into locally twisted cubes. Inf. Sci. 180(19), 3794–3805 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Huang, K.E., Wu, J.: Area efficient layout of balanced hypercubes. Int. J. High Speed Electron. Syst. 6(04), 631–645 (1995)CrossRefGoogle Scholar
  14. 14.
    Hsu, L.-H., Lin, C.-K.: Graph Theory and Interconnection Networks. CRC, Boca Raton (2008)zbMATHGoogle Scholar
  15. 15.
    Liu, Y.-L., Wu, R.-C.: Implementing exchanged hypercube communication patterns on ring-connected WDM optical networks. IEICE Trans. Inf. Syst. 100(12), 2771–2780 (2017)CrossRefGoogle Scholar
  16. 16.
    Loh, P.K.K., Hsu, W.-J., Pan, Y.: The exchanged hypercube. IEEE Trans. Parallel Distrib. Syst. 16(9), 866–874 (2005)CrossRefGoogle Scholar
  17. 17.
    Katseff, H.: Incomplete hypercubes. IEEE Trans. Comput. 37(5), 604–608 (1988)CrossRefGoogle Scholar
  18. 18.
    Ma, M., Liu, B.: Cycles embedding in exchanged hypercubes. Inf. Process. Lett. 110(2), 71–76 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Manuel, P., Rajasingh, I., Rajan, B.: Exact wirelength of hypercubes on a grid. Discrete Appl. Math. 157(7), 1486–1495 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ma, M., Zhu, L.: The super connectivity of exchanged hypercubes. Inf. Process. Lett. 111(8), 360–364 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Miller, M., Rajan, R.S., Parthiban, N.: Minimum linear arrangement of incomplete hypercubes. Comput. J. 58(2), 331–337 (2015)CrossRefGoogle Scholar
  22. 22.
    Nakano, K.: Linear layout of generalized hypercubes. Int. J. Found. Comput. Sci. 14(01), 137–156 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Rostami, H., Habibi, J.: Minimum linear arrangement of Chord graphs. Appl. Math. Comput. 203(1), 358–367 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sýkora, O., Vrt’o, I.: On VLSI layouts of the star graph and related networks. Integr. VLSI J. 17(1), 83–93 (1994)zbMATHCrossRefGoogle Scholar
  25. 25.
    Wan, L., Liu., Y.: On the embedding genus distribution of ladders and crosses. Appl. Math. Lett. 22(5) 738–742 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Wang, D.: Hamiltonian embedding in crossed cubes with failed links. IEEE Trans. Parallel Distrib. Syst. 23(11), 2117–2124 (2012)CrossRefGoogle Scholar
  27. 27.
    Wang, S., Zhang, S.: Embedding hamiltonian paths in \(k\)-ary \(n\)-cubes with conditional edge faults. Theoret. Comput. Sci. 412(46), 6570–6584 (2011)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Yang, Y., Li, J., Wang, S.: Embedding various cycles with prescribed paths into \(k\)-ary \(n\)-cubes. Discrete Appl. Math. 220, 161–169 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Yang, X., David, J.E., Graham, M.: Maximum induced subgraph of a recursive circulant. Inf. Process. Lett. 95(1), 293–298 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Yeh, C. H., Varvarigos, E. A., Parhami, B.: Multilayer VLSI layout for interconnection networks. In: Proceedings of International Conference on IEEE Parallel Processing, pp. 33–40 (2000)Google Scholar
  31. 31.
    Yu, C., Yang, X.: Routing and wavelength assignment for 3-ary \(n\)-cube in array-based optical network. Inf. Process. Lett. 112(6), 252–256 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Weibei Fan
    • 1
  • Jianxi Fan
    • 1
  • Cheng-Kuan Lin
    • 1
  • Zhijie Han
    • 2
  • Peng Li
    • 2
  • Ruchuan Wang
    • 2
  1. 1.School of Computer Science and TechnologySoochow UniversitySuzhouChina
  2. 2.Jiangsu High Technology Research Key Laboratory for Wireless Sensor NetworksNanjingChina

Personalised recommendations