2-domination in Bipartite Graphs with Odd Independence Number

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 cardinality of a smallest k-dominating set of G is called the k-domination number of G and is denoted by γk(G). The independence number of a graph G is denoted by α(G). In [Australas. J. Combin. 40 (2008), 265–268], Fujisawa, Hansberg, Kubo, Saito, Sugita and Volkmann proved that a connected bipartite graph G satisfies γ2(G) ≤ ⌊ 3α(G) 2 ⌋ . They also characterized the bipartite graphs G with γ2(G) = 3α(G) 2 and therefore α(G) even. In this note, we give a characterization of the bipartite graphs G with α(G) odd satisfying γ2(G) = 3α(G)−1 2 .