## Abstract

We continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998) 199-206). A paired-dominating set of a graph G with no isolated vertex is a domi-nating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G is the minimum cardinality of a paired-dominating set of G. For k ≥ 2, a k-packing in G is a set S of vertices of G that are pairwise at distance greater than k apart. The fc-packing number of G is the maximum cardinality of a k-packing in G. Haynes and Slater observed that the paired-domination number is bounded above by twice the domination number. We give a constructive characterization of the trees attaining this bound that uses labelings of the vertices. The key to our characterization is the observation that the trees with paired-domination number twice their domination number are precisely the trees with 2-packing number equal to their 3-packing number.

Original language | English |
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Pages (from-to) | 187-197 |

Number of pages | 11 |

Journal | Utilitas Mathematica |

Volume | 74 |

Publication status | Published - Nov 2007 |

Externally published | Yes |

## Keywords

- Domination
- Packing number
- Paired-domination

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics