  We give the first $2$-approximation algorithm for the cluster vertex deletion
problem. This is tight, since approximating the problem within any constant
factor smaller than $2$ is UGC-hard. Our algorithm combines the previous
approaches, based on the local ratio technique and the management of true
twins, with a novel construction of a 'good' cost function on the vertices at
distance at most $2$ from any vertex of the input graph.
  As an additional contribution, we also study cluster vertex deletion from the
polyhedral perspective, where we prove almost matching upper and lower bounds
on how well linear programming relaxations can approximate the problem.
