  Let ${\cal F}$ be a family of graphs. In the ${\cal F}$-Completion problem,
we are given a graph $G$ and an integer $k$ as input, and asked whether at most
$k$ edges can be added to $G$ so that the resulting graph does not contain a
graph from ${\cal F}$ as an induced subgraph. It appeared recently that special
cases of ${\cal F}$-Completion, the problem of completing into a chordal graph
known as Minimum Fill-in, corresponding to the case of ${\cal
F}=\{C_4,C_5,C_6,\ldots\}$, and the problem of completing into a split graph,
i.e., the case of ${\cal F}=\{C_4, 2K_2, C_5\}$, are solvable in parameterized
subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$. The exploration of this
phenomenon is the main motivation for our research on ${\cal F}$-Completion.
  In this paper we prove that completions into several well studied classes of
graphs without long induced cycles also admit parameterized subexponential time
algorithms by showing that:
  - The problem Trivially Perfect Completion is solvable in parameterized
subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$, that is ${\cal
F}$-Completion for ${\cal F} =\{C_4, P_4\}$, a cycle and a path on four
vertices.
  - The problems known in the literature as Pseudosplit Completion, the case
where ${\cal F} = \{2K_2, C_4\}$, and Threshold Completion, where ${\cal F} =
\{2K_2, P_4, C_4\}$, are also solvable in time $2^{O(\sqrt{k}\log{k})}
n^{O(1)}$.
  We complement our algorithms for ${\cal F}$-Completion with the following
lower bounds:
  - For ${\cal F} = \{2K_2\}$, ${\cal F} = \{C_4\}$, ${\cal F} = \{P_4\}$, and
${\cal F} = \{2K_2, P_4\}$, ${\cal F}$-Completion cannot be solved in time
$2^{o(k)} n^{O(1)}$ unless the Exponential Time Hypothesis (ETH) fails.
  Our upper and lower bounds provide a complete picture of the subexponential
parameterized complexity of ${\cal F}$-Completion problems for ${\cal
F}\subseteq\{2K_2, C_4, P_4\}$.
