  For any graph $G$, we define the power $\pi(G)$ as the minimum of the largest
number of neighbors in a $\gamma$-set of $G$, of any vertex, taken over all
$\gamma$-sets of $G$. We show that $\gamma(G\square H)\geq
\frac{\pi(G)}{2\pi(G) -1}\gamma(G)\gamma(H)$. This implies that for any graphs
$G$ and $H$, $\gamma(G\square H)\geq
\frac{\gamma(G)}{2\gamma(G)-1}\gamma(G)\gamma(H)$, and if $G$ is claw-free or
$P_4$-free, $\gamma(G\square H)\geq \frac{2}{3}\gamma(G)\gamma(H)$, where
$\gamma(G)$ is the domination number of $G$.
