Summary: The domination number of a graph \(G\) is the smallest order, \(\gamma (G)\), of a dominating set for \(G\). A conjecture of \textit{V. G. Vizing} [Vychisl. Sist. 9, 30-43 (1963; Zbl 0194.25203)] states that for every pair of graphs \(G\) and \(H\), \(\gamma (G \square H) \geq \gamma (G) \gamma (H)\), where \(G \square H\) denotes the Cartesian product of \(G\) and \(H\). We show that if the vertex set of \(G\) can be partitioned in a certain way then the above inequality holds for every graph \(H\). The class of graphs \(G\) which have this type of partitioning includes those whose 2-packing number is no smaller than \(\gamma (G) - 1\) as well as the collection of graphs considered by \textit{A. M. Bartsalkin} and \textit{L. F. German} in [Izv. Akad. Nauk Mold. SSR, Ser. Fiz.-Tekh. Math. Nauk 1979, No. 1, 5-8 (1979; Zbl 0457.05053)]. A crucial part of the proof depends on the well-known fact that the domination number of any connected graph of order at least two is no more than half its order.