A note on Vizing's conjecture

Let $\gamma (G)$ denote the domination number of graph $G$. Let $G$ and $H$ be graphs and $G\mathrm{\square }H$ their Cartesian product. For $h\in V(H)$ define $G_h=\{(g,h)\mathrm{\mid }g\in V(G)\}$ and call this set a $G$-layer of $G\mathrm{\square }H$. We prove the following special case of Vizing’s conjecture. Let $D$ be a dominating set of $G\mathrm{\square }H$. If there exist minimum dominating sets $D_1$ and $D_2$ of $G$ such that for every $h\in V(H)$, the projection of $D\cap G_h$ to $G$ is contained in $D_1$ or $D_2$, then $\mathrm{\mid }D\mathrm{\mid }\ge \gamma (G)\gamma (H)$.