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Graphs with large paired-domination number. (English) Zbl 1108.05069

Summary: In this paper, we continue the study of paired-domination in graphs introduced by T. W. Haynes and P. J. Slater [Networks 32, 199–206 (1998; Zbl 0997.05074)]. A paired-dominating set of a graph \(G\) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of \(G\), denoted by \(\gamma_{\text{pr}}(G)\), is the minimum cardinality of a paired-dominating set of \(G\). Let \(G\) be a connected graph of order \(n\) with minimum degree at least two. Haynes and Slater [loc. cit.] showed that if \(n \geq 6\), then \(\gamma_{\text{pr}}(G) \leq 2n/3\). In this paper, we show that there are exactly ten graphs that achieve equality in this bound. For \(n \geq 14\), we show that \(\gamma_{\text{pr}}(G) \leq 2(n-1)/3\) and we characterize the (infinite family of) graphs that achieve equality in this bound.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)

Citations:

Zbl 0997.05074
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References:

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