  Given a $k$-vertex-connected graph $G$ and a set $S$ of extra edges (links),
the goal of the $k$-vertex-connectivity augmentation problem is to find a set
$S' \subseteq S$ of minimum size such that adding $S'$ to $G$ makes it
$(k+1)$-vertex-connected. Unlike the edge-connectivity augmentation problem,
research for the vertex-connectivity version has been sparse.
  In this work we present the first polynomial time approximation algorithm
that improves the known ratio of 2 for $2$-vertex-connectivity augmentation,
for the case in which $G$ is a cycle. This is the first step for attacking the
more general problem of augmenting a $2$-connected graph.
  Our algorithm is based on local search and attains an approximation ratio of
$1.8704$. To derive it, we prove novel results on the structure of minimal
solutions.
