  Given a graph class $\mathcal{H}$, the task of the $\mathcal{H}$-Square Root
problem is to decide, whether an input graph $G$ has a square root $H$ from
$\mathcal{H}$. We are interested in the parameterized complexity of the problem
for classes $\mathcal{H}$ that are composed by the graphs at vertex deletion
distance at most $k$ from graphs of maximum degree at most one, that is, we are
looking for a square root $H$ such that there is a modulator $S$ of size $k$
such that $H-S$ is the disjoint union of isolated vertices and disjoint edges.
We show that different variants of the problems with constraints on the number
of isolated vertices and edges in $H-S$ are FPT when parameterized by $k$ by
demonstrating algorithms with running time $2^{2^{O(k)}}\cdot n^{O(1)}$. We
further show that the running time of our algorithms is asymptotically optimal
and it is unlikely that the double-exponential dependence on $k$ could be
avoided. In particular, we prove that the VC-$k$ Root problem, that asks
whether an input graph has a square root with vertex cover of size at most $k$,
cannot be solved in time $2^{2^{o(k)}}\cdot n^{O(1)}$ unless Exponential Time
Hypothesis fails. Moreover, we point out that VC-$k$ Root parameterized by $k$
does not admit a subexponential kernel unless $P=NP$.
