Maximum cardinality subgraphs meeting a Jaccard overlap threshold
The problem is NP-hard if $\alpha$ is part of the input. We can reduce the Steiner tree problem to this. Given a graph $G$ and terminal vertices $T$, consider $G_0$ to be a complete graph on $T$, and $G_1=G$.

Let $|T|=k$ and the size of the minimum Steiner tree of $T$ in $G$ be $h$.

If $\alpha=k/h$, we claim that $V_0'=T$. Assume not, then $|V_0'|&lt;k$. 
$|V_0'|+|V_1'|\geq k+h$, so $|V_1'|&gt; h$. But then $|V_0'|/|V_1'|&lt;\alpha$. A contradiction. Also, $|V_1'|=h$, as anything larger violates $|V_0'|/|V_1'|\geq \alpha$. $V_1'$ also gives us the vertices of a minimum Steiner tree on $T$. We can obtain $h$ by guess every possible $h$, there are only $n$ possibilities. Hence we reduce Steiner tree problem to your problem.