Solving Vertex Integrity Faster than $2^n$

The vertex integrity of a graph $G$ is the minimum of $|S|+\max_{C\in\operatorname{cc}(G-S)}|V(C)|$ over all vertex sets $S\subseteq V(G)$, where the maximum is zero if $G-S$ is empty. We study its exact exponential complexity in terms of $n=|V(G)|$. First, we give a reduction from Vertex Cover on s