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 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.
| 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 |
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ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
Cite this
Independence and 2-domination in bipartite graphs. / Fujisawa, Jun; Hansberg, Adriana; Kubo, Takahiro; Saito, Akira; Sugita, Masahide; Volkmann, Lutz.
In: Australasian Journal of Combinatorics, Vol. 40, 2008, p. 265-268.Research output: Contribution to journal › Article
}
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 -