Independence and 2-domination in bipartite graphs

Jun Fujisawa, Adriana Hansberg, Takahiro Kubo, Akira Saito, Masahide Sugita, Lutz Volkmann

Research output: Contribution to journalArticle

18 Citations (Scopus)

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 languageEnglish
Pages (from-to)265-268
Number of pages4
JournalAustralasian Journal of Combinatorics
Volume40
Publication statusPublished - 2008
Externally publishedYes

Fingerprint

Domination
Bipartite Graph
Dominating Set
Independence number
Domination number
Graph in graph theory
Valid
Integer
Vertex of a graph
Independence

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

Cite this

Fujisawa, J., Hansberg, A., Kubo, T., Saito, A., Sugita, M., & Volkmann, L. (2008). Independence and 2-domination in bipartite graphs. 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.

In: Australasian Journal of Combinatorics, Vol. 40, 2008, p. 265-268.

Research output: Contribution to journalArticle

Fujisawa, J, Hansberg, A, Kubo, T, Saito, A, Sugita, M & Volkmann, L 2008, 'Independence and 2-domination in bipartite graphs', Australasian Journal of Combinatorics, vol. 40, pp. 265-268.
Fujisawa J, Hansberg A, Kubo T, Saito A, Sugita M, Volkmann L. Independence and 2-domination in bipartite graphs. Australasian Journal of Combinatorics. 2008;40:265-268.
Fujisawa, Jun ; Hansberg, Adriana ; Kubo, Takahiro ; Saito, Akira ; Sugita, Masahide ; Volkmann, Lutz. / Independence and 2-domination in bipartite graphs. In: Australasian Journal of Combinatorics. 2008 ; Vol. 40. pp. 265-268.
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