  For a class $\mathcal{G}$ of graphs, the problem SUBGRAPH COMPLEMENT TO
$\mathcal{G}$ asks whether one can find a subset $S$ of vertices of the input
graph $G$ such that complementing the subgraph induced by $S$ in $G$ results in
a graph in $\mathcal{G}$. We investigate the complexity of the problem when
$\mathcal{G}$ is $H$-free for $H$ being a complete graph, a star, a path, or a
cycle. We obtain the following results:
  - When $H$ is a $K_t$ (a complete graph on $t$ vertices) for any fixed $t\geq
1$, the problem is solvable in polynomial-time. This applies even when
$\mathcal{G}$ is a subclass of $K_t$-free graphs recognizable in
polynomial-time, for example, the class of $(t-2)$-degenerate graphs.
  - When $H$ is a $K_{1,t}$ (a star graph on $t+1$ vertices), we obtain that
the problem is NP-complete for every $t\geq 5$. This, along with known results,
leaves only two unresolved cases - $K_{1,3}$ and $K_{1,4}$.
  - When $H$ is a $P_t$ (a path on $t$ vertices), we obtain that the problem is
NP-complete for every $t\geq 7$, leaving behind only two unresolved cases -
$P_5$ and $P_6$.
  - When $H$ is a $C_t$ (a cycle on $t$ vertices), we obtain that the problem
is NP-complete for every $t\geq 8$, leaving behind four unresolved cases -
$C_4, C_5, C_6,$ and $C_7$.
  Further, we prove that these hard problems do not admit subexponential-time
algorithms (algorithms running in time $2^{o(|V(G)|)}$), assuming the
Exponential Time Hypothesis. A simple complementation argument implies that
results for $\mathcal{G}$ are applicable for $\overline{\mathcal{G}}$, thereby
obtaining similar results for $H$ being the complement of a complete graph, a
star, a path, or a cycle. Our results generalize two main results and resolve
one open question by Fomin et al. (Algorithmica, 2020).
