  Node-connectivity augmentation is a fundamental network design problem. We
are given a $k$-node connected graph $G$ together with an additional set of
links, and the goal is to add a cheap subset of links to $G$ to make it
$(k+1)$-node connected.
  In this work, we characterize completely the computational complexity status
of the problem, by showing hardness for all values of $k$ which were not
addressed previously in the literature.
  We then focus on $k$-node connectivity augmentation for $k=n-4$, which
corresponds to the highest value of $k$ for which the problem is NP-hard. We
improve over the previously best known approximation bounds for this problem,
by developing a $\frac{3}{2}$-approximation algorithm for the weighted setting,
and a $\frac{4}{3}$-approximation algorithm for the unweighted setting.
