A class of graphs approaching Vizing's conjecture

For any graph $G=(V,E)$, a subset $S\subseteq V$ \emph{dominates} $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written $\gamma(G)$. In 1963, V.G. Vizing conjectured that $\gamma(G \square H) \geq \gamma(G)\gamma(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we define classes of graphs $\mathcal{A}_n$, for $n\geq 0$, so that every graph belongs to some such class, and $\mathcal{A}_0$ corresponds to class $A$ of Bartsalkin and German. We prove that for any graph $G$ in class $\mathcal{A}_1$, $\gamma(G\square H)\geq \left(\gamma(G)-\sqrt{\gamma(G)}\right)\gamma(H)$.