Independence and 2-domination in bipartite graphs

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) ≤ 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.