  In any graph $G$, the domination number $\gamma(G)$ is at most the
independence number $\alpha(G)$. The Inverse Domination Conjecture says that,
in any isolate-free $G$, there exists pair of vertex-disjoint dominating sets
$D, D'$ with $|D|=\gamma(G)$ and $|D'| \leq \alpha(G)$. Here we prove that this
statement is true if the upper bound $\alpha(G)$ is replaced by
$\frac{3}{2}\alpha(G) - 1$ (and $G$ is not a clique). We also prove that the
conjecture holds whenever $\gamma(G)\leq 5$ or $|V(G)|\leq 16$.
