Skip to main content
SpringerLink
Log in

2-(Edge-)Connected Edge Domination Number and Matching Number

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

This article has been updated

Abstract

A k-connected (resp. k-edge-connected) edge dominating set D of a connected graph G is a subset of E(G) such that G[D] is k-connected (resp. k-edge-connected) and each \(e\in E(G)\setminus D\) has at least one neighbor in D. The k-connected edge domination number (resp. k-edge-connected edge domination number) of a graph G is the minimum size of a k-connected (resp. k-edge-connected) edge dominating set of G, and denoted by \(\gamma _k(G)\) (resp. \(\gamma '_k(G)\)). In this paper, we investigate the relationship between matching number and 2-connected (resp. 2-edge-connected) edge domination number, and prove that for a graph G, if it is 2-edge-connected, then \(\gamma '_2(G)\le 5\alpha '(G)-2\), and if it is 2-connected, then \(\gamma _2(G)\le 4\alpha '(G)-1\), where \(\alpha '(G)\) is the matching number of G.

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

Access this article

Log in via an institution

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

Data Availibility Statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Change history

  • 24 March 2023

    The original version is updated due to spell error in the article title.

References

  1. Arumugam, S., Velammal, S.: Connected edge domination in graph. Tamkang J. Math. 24(1), 55–64 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Bondy, J.A., Murty, U.S.R.: Graph theory. In: GTM 244. Springer, Cham (2008)

  3. Bhaskarudu, P.: Matching domination of Kronecker product of two graphs. Int. J. Adv. Res. Ideas Innov. Technol. 2(5), 937–944 (2017)

  4. Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blum, J., Ding, M., Thaeler, A., Cheng, X.: Connected dominating set in sensor networks and MANETs. In: Handbook of Combinatorial Optimization, pp. 329–369. Springer, New York (2005)

  6. Du, H., Ding, L., Wu, W., Kim, D., Pardalos, P., Willson, J.: Connected dominating set in wireless networks. In: Handbook of Combinatorial Optimization, pp. 783–833. Springer, New York (2013)

  7. Dai, F., Wu, J.: On constructing \(k\)-connected \(k\)-dominating set in wireless ad hoc and sensor networks. J. Parallel Distrib. Comput. 66(7), 947–958 (2006)

    Article  MATH  Google Scholar 

  8. Forcade, R.: Smallest maximal matching in the graph of the d-dimensional cube. J. Combin. Theory Ser. B 14, 153–156 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fink, J., Jacobson, M.: \(n\)-Domination in graphs. In: Graph Theory with Applications to Algorithms and Computer Science, pp. 283–300. Wiley, New Jersery (1985)

  10. Gao, H., Gao, W.: Maximum matching and kernelization of edge dominating set. Inform. Process. Lett. 136, 21–24 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hedetniemi, S., Mitchell, S.: Edge domination in tree. Graph Theory. Comput. 19, 489–509 (1977)

    MathSciNet  MATH  Google Scholar 

  12. Kulli, V., Sigarkanti, S.: The connected edge domination number of a graph. Department of Mathematics (1988)

  13. Li, H., Yang, Y., Wu, B.: \(2\)-edge-connected dominating sets and \(2\)-connected dominating sets of a graph. J. Comb. Optim. 31, 713–724 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sampathkumar, E., Walikar, H.: The connected domination number of a graph. J. Math. Phy. Sci. 13, 607–613 (1979)

    MathSciNet  MATH  Google Scholar 

  15. Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the referees and the editor for their valuable comments and suggestions, which greatly improved the presentation of this paper.

Funding

Supported by NSFC No.11071130, “Fundamental Research Funds for the Central Universities", and “Foundation of Department of Science and Technology of Henan (HNGD2022060)".

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China

    Hengzhe Li & Ankang Wei

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Shenggui Zhang

Authors
  1. Hengzhe Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ankang Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shenggui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Author Contributions: All authors contributed to the study conception and design. All authors commented on previous versions of the manuscript and all authors read and approved the final manuscript. HL: Methodology, Funding acquisition, Writing-Original Draft. AW: Methodology, Validation, Writing. SZ: Validation, Supervision-Review and Editing.

Corresponding author

Correspondence to Hengzhe Li.

Ethics declarations

Conflict of Interest

The authors have no relevant financia lo rnon-financial interests to disclose.

Additional information

Publisher's Note

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

Supported by NSFC No.11071130, “Fundamental Research Funds for the Central Universities", and “Foundation of Department of Science and Technology of Henan (HNGD2022060)".

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Wei, A. & Zhang, S. 2-(Edge-)Connected Edge Domination Number and Matching Number. Graphs and Combinatorics 39, 31 (2023). https://doi.org/10.1007/s00373-023-02626-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-023-02626-y

Keywords

Mathematics Subject Classification

Search

Navigation