  A directed graph $D$ is semicomplete if for every pair $x,y$ of vertices of
$D,$ there is at least one arc between $x$ and $y.$ \viol{Thus, a tournament is
a semicomplete digraph.} In the Directed Component Order Connectivity (DCOC)
problem, given a digraph $D=(V,A)$ and a pair of natural numbers $k$ and
$\ell$, we are to decide whether there is a subset $X$ of $V$ of size $k$ such
that the largest strong connectivity component in $D-X$ has at most $\ell$
vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem
for $\ell=1.$ We study parametered complexity of DCOC for general and
semicomplete digraphs with the following parameters: $k, \ell,\ell+k$ and
$n-\ell$. In particular, we prove that DCOC with parameter $k$ on semicomplete
digraphs can be solved in time $O^*(2^{16k})$ but not in time $O^*(2^{o(k)})$
unless the Exponential Time Hypothesis (ETH) fails. \gutin{The upper bound
$O^*(2^{16k})$ implies the upper bound $O^*(2^{16(n-\ell)})$ for the parameter
$n-\ell.$ We complement the latter by showing that there is no algorithm of
time complexity $O^*(2^{o({n-\ell})})$ unless ETH fails.} Finally, we improve
\viol{(in dependency on $\ell$)} the upper bound of G{\"{o}}ke, Marx and Mnich
(2019) for the time complexity of DCOC with parameter $\ell+k$ on general
digraphs from $O^*(2^{O(k\ell\log (k\ell))})$ to $O^*(2^{O(k\log (k\ell))}).$
Note that Drange, Dregi and van 't Hof (2016) proved that even for the
undirected version of DCOC on split graphs there is no algorithm of running
time $O^*(2^{o(k\log \ell)})$ unless ETH fails and it is a long-standing
problem to decide whether Directed Feedback Vertex Set admits an algorithm of
time complexity $O^*(2^{o(k\log k)}).$
