  For two integers $r, \ell \geq 0$, a graph $G = (V, E)$ is an
$(r,\ell)$-graph if $V$ can be partitioned into $r$ independent sets and $\ell$
cliques. In the parameterized $(r,\ell)$-Vertex Deletion problem, given a graph
$G$ and an integer $k$, one has to decide whether at most $k$ vertices can be
removed from $G$ to obtain an $(r,\ell)$-graph. This problem is NP-hard if
$r+\ell \geq 1$ and encompasses several relevant problems such as Vertex Cover
and Odd Cycle Transversal. The parameterized complexity of $(r,\ell)$-Vertex
Deletion was known for all values of $(r,\ell)$ except for $(2,1)$, $(1,2)$,
and $(2,2)$. We prove that each of these three cases is FPT and, furthermore,
solvable in single-exponential time, which is asymptotically optimal in terms
of $k$. We consider as well the version of $(r,\ell)$-Vertex Deletion where the
set of vertices to be removed has to induce an independent set, and provide
also a parameterized complexity dichotomy for this problem.
