Augmenting to 4-vertex connectivity is fixed-parameter tractable

We present fixed-parameter algorithms (FPT algorithms) for the $\lambda$-vertex connectivity augmentation ($\lambda$-VCA) problem for all values of $\lambda \le 4$; that is, we give an algorithm that given a graph $G$, a set $L$ of non-edges, and an integer $k$, determines in time $k^{\mathcal{O}(k)} \cdot n^{c}$ (for some constant $c$ independent of $k$) whether $G$ can be made $\lambda$-vertex connected by adding at most $k$ elements from $L$.