Skip to main content
Springer Nature Link
Log in

Classification of Efficient Dominating Sets of Circulant Graphs of Degree 5

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

For a graph G, a subset D of V(G) is called an efficient dominating set of G if D is an independent set and every vertex in \(V(G) \setminus D\) is adjacent to exactly one vertex in the set D. In this paper, we give necessary and sufficient conditions for the existence of efficient dominating sets of circulant graphs of degree 5 and classify these sets.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Biggs, N.: Perfect codes in graphs. J. Combin. Theory Ser. B 15(3), 289–296 (1973)

    Article  MathSciNet  Google Scholar 

  2. Dejter, I.J., Serra, O.: Efficient dominating sets in Cayley graphs. Discrete Appl. Math. 129(2–3), 319–328 (2003)

    Article  MathSciNet  Google Scholar 

  3. Feng, R., Huang, H., Zhou, S.: Perfect codes in circulant graphs. Discrete Math. 340(7), 1522–1527 (2017)

    Article  MathSciNet  Google Scholar 

  4. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs Advanced Topics (Chapter 9). Marcel Dekker Inc, NY (1998)

    MATH  Google Scholar 

  5. Huang, H., Xia, B., Zhou, S.: Perfect codes in Cayley graphs. SIAM J. Discrete Math. 32(1), 548–559 (2018)

    Article  MathSciNet  Google Scholar 

  6. Kratochvíl, K.: Perfect codes of graphs. J. Combin. Theory Ser. B 40(2), 224–228 (1986)

    Article  MathSciNet  Google Scholar 

  7. Kwon, Y.S., Sohn, M.Y., Chen, X.: Existence of efficient total domination sets of circulant graphs of degree \(4\). Discrete Math. 343(3), 111742 (2020)

    Article  MathSciNet  Google Scholar 

  8. Reji Kumar, K., MacGillivray, G.: Efficient dominating sets in circulant graphs. Discrete Math. 313(6), 767–771 (2013)

    Article  MathSciNet  Google Scholar 

  9. Lee, J.: Independent perfect domination sets in Cayley graphs. J. Graph Theory 37(4), 213–219 (2001)

    Article  MathSciNet  Google Scholar 

  10. Lu, C.L., Tang, C.Y.: Weighted efficient domination problem on some perfect graphs. Discrete Appl. Math. 117(1–3), 163–182 (2002)

    Article  MathSciNet  Google Scholar 

  11. Ma, X., Walls, G.L., Wang, K., Zhou, S.: Subgroup perfect codes in Cayley graphs. SIAM J. Discrete Math. 34(1), 1909–1921 (2020)

    Article  MathSciNet  Google Scholar 

  12. Obradovi\(\acute{{\rm c}}\), N., Peters, J., Ru\(\breve{z}\)i\(\acute{c}\), G.: Efficient domination in circulant graphs with two chord lengths. Inf. Process. Lett. 102(6), 253–258 (2007)

  13. Yen, C.C., Lee, R.C.T.: The weighted perfect domination problem and its variants. Discrete Appl. Math. 66(2), 147–160 (1996)

    Article  MathSciNet  Google Scholar 

  14. Zhou, S.: Total perfect codes in Cayley grphs. Des. Codes Cryptogr. 81(3), 489–504 (2016)

    Article  MathSciNet  Google Scholar 

  15. Zhou, S.: Cyclotomic graphs and perfect codes. J. Pure Appl. Algebra 223(3), 931–947 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful for the referee’s constructive remarks and suggestions. This work was supported by the 2021 Yeungnam University Research Grant. The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2020R1I1A3A04036669).

Author information

Authors and Affiliations

  1. Mathematics, Yeungnam University, Kyongsan, 712-749, Korea

    Young Soo Kwon & Jaeun Lee

  2. Mathematics, Changwon National University, Changwon, 641-773, Korea

    Moo Young Sohn

Authors
  1. Young Soo Kwon
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jaeun Lee
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Moo Young Sohn
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Moo Young Sohn.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kwon, Y.S., Lee, J. & Sohn, M.Y. Classification of Efficient Dominating Sets of Circulant Graphs of Degree 5. Graphs and Combinatorics 38, 120 (2022). https://doi.org/10.1007/s00373-022-02527-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-022-02527-6

Keywords

Mathematics Subject Classification