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Journal of Combinatorial Optimization

, Volume 31, Issue 2, pp 713–724 | Cite as

2-Edge connected dominating sets and 2-Connected dominating sets of a graph

  • Hengzhe Li
  • Yuxing Yang
  • Baoyindureng Wu
Article

Abstract

A \(k\)-connected (resp. \(k\)-edge connected) dominating set \(D\) of a connected graph \(G\) is a subset of \(V(G)\) such that \(G[D]\) is \(k\)-connected (resp. \(k\)-edge connected) and each \(v\in V(G)\backslash D\) has at least one neighbor in \(D\). The \(k\) -connected domination number (resp. \(k\) -edge connected domination number) of a graph \(G\) is the minimum size of a \(k\)-connected (resp. \(k\)-edge connected) dominating set of \(G\), and denoted by \(\gamma _k(G)\) (resp. \(\gamma '_k(G)\)). In this paper, we investigate the relation of independence number and 2-connected (resp. 2-edge-connected) domination number, and prove that for a graph \(G\), if it is \(2\)-edge connected, then \(\gamma '_2(G)\le 4\alpha (G)-1\), and it is \(2\)-connected, then \(\gamma _2(G)\le 6\alpha (G)-3\), where \(\alpha (G)\) is the independent number of \(G\).

Keywords

Connected dominating set Dominating set Independent set 

Notes

Acknowledgments

This work is supported by NSFC (11161046), Xinjiang Young Talent Project (2013721012), and Research Found of Henan Normal University (qd13042). The authors are grateful to the referees for their careful reading and valuable comments.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China
  2. 2.College of Mathematics and System SciencesXinjiang UniversityUrumqiPeople’s Republic of China

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