  The cyclability of a graph is the maximum integer $k$ for which every $k$
vertices lie on a cycle. The algorithmic version of the problem, given a graph
$G$ and a non-negative integer $k,$ decide whether the cyclability of $G$ is at
least $k,$ is {\sf NP}-hard. We study the parametrized complexity of this
problem. We prove that this problem, parameterized by $k,$ is ${\sf
co\mbox{-}W[1]}$-hard and that its does not admit a polynomial kernel on planar
graphs, unless ${\sf NP}\subseteq{\sf co}\mbox{-}{\sf NP}/{\sf poly}$. On the
positive side, we give an {\sf FPT} algorithm for planar graphs that runs in
time $2^{2^{O(k^2\log k)}}\cdot n^2$. Our algorithm is based on a series of
graph-theoretical results on cyclic linkages in planar graphs.
