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 k-dominating set of G is called the k-domination number of G and is denoted by gamma(k)(G). In Blidia, Chellali and Favaron [Australas. J. Combin. 33 (2005), 317-327], they proved that a tree T satisfies alpha(T) <= gamma(2)(T) <= 3/2 alpha(T), where alpha(G) is the independence number of a graph G. They also claimed that they characterized the trees T with gamma(2)(T) = 3/2 alpha(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 gamma(2)(G) = 3/2 alpha(G) and point out that the characterization in the aforementioned paper of the trees with this property contains an error.