A Note on the Computational Complexity of Unsmoothened Vertex Attack Tolerance.

We have previously introduced vertex attack tolerance (VAT) and unsmoothened VAT (UVAT), denoted respectively as $tau(G) = min_{S subset V} frac{|S|}{|V-S-C_{max}(V-S)|+1}$ and $hat{tau}(G) = min_{S subset V} frac{|S|}{|V-S-C_{max}(V-S)|}$, where $C_{max}(V-S)$ is the largest connected component in $V-S$, as appropriate mathematical measures of resilience in the face of targeted node attacks for arbitrary degree networks. Here we prove the hardness of approximating $hat{tau}$ under various plausible computational complexity hypotheses.