### 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 |
---|---|

Pages (from-to) | 265-268 |

Number of pages | 4 |

Journal | Australasian Journal of Combinatorics |

Volume | 40 |

Publication status | Published - 2008 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*40*, 265-268.

**Independence and 2-domination in bipartite graphs.** / Fujisawa, Jun; Hansberg, Adriana; Kubo, Takahiro; Saito, Akira; Sugita, Masahide; Volkmann, Lutz.

Research output: Contribution to journal › Article

*Australasian Journal of Combinatorics*, vol. 40, pp. 265-268.

}

TY - JOUR

T1 - Independence and 2-domination in bipartite graphs

AU - Fujisawa, Jun

AU - Hansberg, Adriana

AU - Kubo, Takahiro

AU - Saito, Akira

AU - Sugita, Masahide

AU - Volkmann, Lutz

PY - 2008

Y1 - 2008

N2 - 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 γk(G). In Blidia, Chellali and Favaron [Australas. J. Combin. 33 (2005), 317-327], they proved that a tree T satisfies α(T) ≤ γ2(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 γ2(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 γ2(G) = 3/2α(G) and point out that the characterization in the aforementioned paper of the trees with this property contains an error.

AB - 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 γk(G). In Blidia, Chellali and Favaron [Australas. J. Combin. 33 (2005), 317-327], they proved that a tree T satisfies α(T) ≤ γ2(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 γ2(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 γ2(G) = 3/2α(G) and point out that the characterization in the aforementioned paper of the trees with this property contains an error.

UR - http://www.scopus.com/inward/record.url?scp=77953174773&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953174773&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77953174773

VL - 40

SP - 265

EP - 268

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

ER -