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.
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
Arumugam, S., Velammal, S.: Connected edge domination in graph. Tamkang J. Math. 24(1), 55–64 (2009)
Bondy, J.A., Murty, U.S.R.: Graph theory. In: GTM 244. Springer, Cham (2008)
Bhaskarudu, P.: Matching domination of Kronecker product of two graphs. Int. J. Adv. Res. Ideas Innov. Technol. 2(5), 937–944 (2017)
Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)
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)
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)
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)
Forcade, R.: Smallest maximal matching in the graph of the d-dimensional cube. J. Combin. Theory Ser. B 14, 153–156 (1973)
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)
Gao, H., Gao, W.: Maximum matching and kernelization of edge dominating set. Inform. Process. Lett. 136, 21–24 (2018)
Hedetniemi, S., Mitchell, S.: Edge domination in tree. Graph Theory. Comput. 19, 489–509 (1977)
Kulli, V., Sigarkanti, S.: The connected edge domination number of a graph. Department of Mathematics (1988)
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)
Sampathkumar, E., Walikar, H.: The connected domination number of a graph. J. Math. Phy. Sci. 13, 607–613 (1979)
Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)
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
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
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.
About this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-023-02626-y