Independence and 2-domination in bipartite graphs
Jun Fujisawa, Adriana Hansberg, Takahiro Kubo, Akira Saito, Masahide Sugita, Lutz Volkmann · 2008
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 k-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) ≤ 3α(T), where α(G) is the independence number 2 of a graph G. They also claimed that they characterized the trees T with γ2(T) = 3α(T). In this note, we will show that the second inequality 2 is even valid for bipartite graphs. Further, we give a characterization of the bipartite graphs G satisfying γ2(G) = 3α(G) and point out that 2 the characterization in the aforementioned paper of the trees with this property contains an error.