## Abstract

Let G be a graph and let v be a vertex of G. The open neighbourhood N(V) of v is the set of all vertices adjacent with v in G, while the closed neighbourhood of v is N(v) ∪ {v}. A packing of a graph G is a set of vertices whose closed neighbourhoods are pairwise disjoint. Equivalently, a packing of a graph G is a set of vertices whose elements are pairwise at distance at least 3 apart in G. The lower packing number of G, denoted ρ_{L}(G), is the minimum cardinality of a maximal packing of G while the (upper) packing number of G, denoted ρ(G), is the maximum cardinality among all packings of G. An open packing of G is a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number of G, denoted ρ^{o}_{L}(G), is the minimum cardinality of a maximal open packing of G while the (upper) open packing number of G, denoted ρ^{o}(G), is the maximum cardinality among all open packings of G. We present upper bounds on the packing number and the lower packing number of a tree. Bounds relating the packing numbers and open packing numbers of a tree are established.

Original language | English |
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Pages (from-to) | 145-155 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 186 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 15 May 1998 |

Externally published | Yes |

## Keywords

- Bounds
- Open packing
- Packing
- Trees

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics