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.)

Keywords:

Bounds; Minimum degree two

Zbl 0997.05074
Full Text:

References:

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