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 \(G\).
Connected dominating set Dominating set Independent set
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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.
Arseneau L, Finbow A, Hartnell B, Maclean D, O’sullivan L (1997) On minimal connected dominating sets. JCMCC 24:185–191MathSciNetMATHGoogle Scholar