Upper bounds for inverse domination in graphs

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\u27 with |D|=\gamma(G) and |D\u27| \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