  An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible}
edges and \emph{real} edges. A \emph{realisation} of an s-graph $B$ is any
graph obtained by subdividing subdivisible edges of $B$ into paths of arbitrary
length (at least one). Given an s-graph $B$, we study the decision problem
$\Pi_B$ whose instance is a graph $G$ and question is "Does $G$ contain a
realisation of $B$ as an induced subgraph?". For several $B$'s, the complexity
of $\Pi_B$ is known and here we give the complexity for several more. Our
NP-completeness proofs for $\Pi_B$'s rely on the NP-completeness proof of the
following problem. Let $\cal S$ be a set of graphs and $d$ be an integer. Let
$\Gamma_{\cal S}^d$ be the problem whose instance is $(G, x, y)$ where $G$ is a
graph whose maximum degree is at most d, with no induced subgraph in $\cal S$
and $x, y \in V(G)$ are two non-adjacent vertices of degree 2. The question is
"Does $G$ contain an induced cycle passing through $x, y$?". Among several
results, we prove that $\Gamma^3_{\emptyset}$ is NP-complete. We give a simple
criterion on a connected graph $H$ to decide whether $\Gamma^{+\infty}_{\{H\}}$
is polynomial or NP-complete. The polynomial cases rely on the algorithm
three-in-a-tree, due to Chudnovsky and Seymour.
