Zero Obstructed vertex induced subgraphs
Sorry this is too long for a comment :)

Maybe the complete graph is a trivial graph but it illustrates that e.g. graphs $G = (V,E)$ with large cliques of size $\Omega(\vert V \vert)$ also have exponentially many $0$ obstructed induced subgraphs. And I don't think that $N$ can be easily computed in such graphs. 

I also think that 3-regularity doesnt help: Let $A$, $B$ be a partition of $V$ each of size $\vert V \vert /2$. The induced subgraph consisting of nodes from either $A$ or $B$ is a cycle and the edges between $A$ and $B$ form a perfect matching. This graph is $3$-regular and every induced subgraph consisting of at least all nodes in $A$ or all nodes in $B$ is $0$ obstructed.