SpringerLink
Log in

Isolation of the Diamond Graph

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

A graph is H-free if it contains no H as a subgraph. The diamond graph is the graph obtained from \(K_4\) by deleting one edge. We prove that if G is a connected graph with order \(n\ge 10\), then there exists a subset \(S\subseteq V(G)\) with \(|S|\le n/5\) such that the subgraph induced by \(V(G)\setminus N[S]\) is diamond-free, where N[S] is the closed neighborhood of S. Furthermore, the bound is sharp.

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
EUR 32.99 /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 includes VAT (Thailand)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Bondy, J.A., Murty, U.S.R.: Graph theory. In: GTM, vol 244. Springer, New York, (2008)

  2. Borg, P.: Isolation of cycles. Gr. Combin. 36(3), 631–637 (2020)

    Article  MathSciNet  Google Scholar 

  3. Borg, P., Fenech, K., Kaemawichanurat, P.: Isolation of \(k\)-cliques. Discrete Math. 343, 111879 (2020)

    Article  MathSciNet  Google Scholar 

  4. Borg, P., Kaemawichanurat, P.: Partial domination of maximal outerplanar graphs. Discrete Appl. Math. 283, 306–314 (2020)

    Article  MathSciNet  Google Scholar 

  5. Caro, Y., Hansberg, A.: Partial domination - the isolation number of a graph. Filomat 31(12), 3925–3944 (2017)

    Article  MathSciNet  Google Scholar 

  6. Favaron, O., Kaemawichanurat, P.: Inequalities between the \(K_k\)-isolation number and the independent \(K_k\)-isolation number of a graph. Discrete Appl. Math. 289, 93–97 (2021)

    Article  MathSciNet  Google Scholar 

  7. Ore, O.: Theory of Graphs, in: American Mathematical Society Colloquium Publications, vol. 38, American Mathematical Society, Providence, RI, (1962)

  8. Tokunaga, S., Jiarasuksakun, T., Kaemawichanurat, P.: Isolation number of maximal outerplanar graphs. Discrete Appl. Math. 267, 215–218 (2019)

    Article  MathSciNet  Google Scholar 

  9. West, D.B.: Introduction to graph theory. Prentice Hall, Inc.,Upper Saddle River, NJ, (1996)

  10. Yu, H., Wu, B.: Admissible property of graphs in terms of radius. Gr. Combin. 38, 6 (2022). https://doi.org/10.1007/s00373-021-02431-5

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang, G., Wu, B.: \(K_{1,2}\)-isolation in graphs. Discrete Appl. Math. 304, 365–374 (2021)

    Article  MathSciNet  Google Scholar 

  12. Zhang, G., Wu, B.: Isolation of cycles and trees in graphs. J. **njiang Univ. (Nat. Sci. Ed. Chin. Eng.) 39(1), 1–6 (2022)

    Google Scholar 

Download references

Acknowledgements

This research was supported Science and Technology Commission of Shanghai Municipality (STCSM) Grant 18dz2271000.

Author information

Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200241, China

    **gru Yan

Authors
  1. **gru Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to **gru Yan.

Additional information

Communicated by Xueliang Li.

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

Yan, J. Isolation of the Diamond Graph. Bull. Malays. Math. Sci. Soc. 45, 1169–1181 (2022). https://doi.org/10.1007/s40840-022-01248-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-022-01248-6

Keywords

Mathematics Subject Classification

Search

Navigation