Skip to main content
SpringerLink
Log in

On Some Structural Properties of Star and Pancake Graphs

  • Chapter
Information Theory, Combinatorics, and Search Theory

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7777))

Abstract

In this paper we give a report on some results for the Star and Pancake graphs obtained after the conference “Search Methodologies” which was held in October, 2010. The graphs are defined as Cayley graphs on the symmetric group with the generating sets of all prefix–transpositions and prefix–reversals, correspondingly. They are also known as the Star and Pancake networks in computer science. In this paper we give the full characterization of perfect codes for these graphs. We also investigate a cycle structure of the Pancake graph and present an explicit description of small cycles as well as their number in the graph.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akers, S.B., Krishnamurthy, B.: A group–theoretic model for symmetric interconnection networks. IEEE Trans. Comput. 38(4), 555–566 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arumugam, S., Kala, R.: Domination parameters of Star graph. Ars Combinatoria 44(1), 93–96 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Asai, S., Kounoike, Y., Shinano, Y., Kaneko, K.: Computing the Diameter of 17-Pancake Graph Using a PC Cluster. In: Nagel, W.E., Walter, W.V., Lehner, W. (eds.) Euro-Par 2006. LNCS, vol. 4128, pp. 1114–1124. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  4. Cibulka, J.: On average and highest number of flips in pancake sorting. Theoretical Computer Science 412, 822–834 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dejter, I.J., Serra, O.: Efficient dominating sets in Cayley graphs. Discrete Appl. Math. 129, 319–328 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dweighter, H.: E 2569 in: Elementary problems and solutions. Amer. Math. Monthly 82(1), 1010 (1975)

    MathSciNet  Google Scholar 

  7. Gates, W.H., Papadimitriou, C.H.: Bounds for sorting by prefix–reversal. Discrete Math. 27, 47–57 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hyedari, M.H., Sudborough, I.H.: On the diameter of the pancake network. Journal of Algorithms 25(1), 67–94 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jwo, J.S., Lakshmivarahan, S., Dhall, S.K.: Embedding of cycles and grids in star graphs. J. Circuits, Syst., Comput. 1(1), 43–74 (1991)

    Article  Google Scholar 

  10. Kanevsky, A., Feng, C.: On the embedding of cycles in pancake graphs. Parallel Computing 21, 923–936 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Konstantinova, E.: Perfect codes in the pancake networks, http://www.math.uniri.hr/NATO-ASI/abstracts/Konstantinova_abstract.pdf

  12. Konstantinova, E.V., Medvedev, A.N.: Cycles of length seven in the Pancake graph. Diskretn. Anal. Issled. Oper. 17(5), 46–55 (2010) (in Russian)

    MathSciNet  MATH  Google Scholar 

  13. Konstantinova, E.V., Medvedev, A.N.: Cycles of length nine in the Pancake graph. Diskretn. Anal. Issled. Oper. 8(6), 33–60 (2011) (in Russian)

    Google Scholar 

  14. Konstantinova, E.V., Medvedev, A.N.: Small cycles in the Pancake graph. Submitted to Ars Mathematica Contemporanea

    Google Scholar 

  15. Konstantinova, E.V., Savin, M.Y.: Efficient dominating sets of the Star and Pancake graphs. Submitted to Diskretn. Anal. Issled. Oper. (in Russian)

    Google Scholar 

  16. Lakshmivarahan, S., Jwo, J.S., Dhall, S.K.: Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey. Parallel Comput. 19(4), 361–407 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Qiu, K.: Optimal broadcasting algorithms for multiple messages on the star and pancake graphs using minimum dominating sets. Congress Numerantium 181, 33–39 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Sheu, J.J., Tan, J.J.M., Chu, K.T.: Cycle embedding in pancake interconnection networks. In: Proc. 23rd Workshop on Combinatorial Mathematics and Computation Theory, Taiwan, pp. 85–92 (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk-90, 630090, Russia

    Elena Konstantinova

Authors
  1. Elena Konstantinova
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Bielefeld, Universitätsstr. 25, 33615, Bielefeld, Germany

    Harout Aydinian  & Christian Deppe  & 

  2. Department of Computer Science, University of Salerno, via Ponte don Melillo, 84084, Fisciano, Italy

    Ferdinando Cicalese

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Konstantinova, E. (2013). On Some Structural Properties of Star and Pancake Graphs. In: Aydinian, H., Cicalese, F., Deppe, C. (eds) Information Theory, Combinatorics, and Search Theory. Lecture Notes in Computer Science, vol 7777. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36899-8_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-36899-8_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36898-1

  • Online ISBN: 978-3-642-36899-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation