### Abstract

For a positive integer k, a set of vertices S in a graph G is said to be a k-dominating set if each vertex x in V(G)-S has at least k neighbors in S. The order of a smallest fc-dominating set of G is called the k-domination number of G and is denoted by γ_{k}(G). In Blidia, Chellali and Favaron [Australas. J. Combin. 33 (2005), 317-327], they proved that a tree T satisfies α(T) ≤ γ_{2}(T) ≤ 3/2α(T), where α(G) is the independence number of a graph G. They also claimed that they characterized the trees T with γ_{2}(T). In this note, we will show that the second inequality is even valid for bipartite graphs. Further, we give a characterization of the bipartite graphs G satisfying γ_{2}(G) = 3/2α(G) and point out that the characterization in the aforementioned paper of the trees with this property contains an error.

Original language | English |
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Pages (from-to) | 265-268 |

Number of pages | 4 |

Journal | Australasian Journal of Combinatorics |

Volume | 40 |

Publication status | Published - 2008 Dec 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*40*, 265-268.