An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A 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 ΠB whose instance is a graph G
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and question is "Does
G
contain a realisation of
B
as an induced subgraph?". For several
B
's, the complexity of
ΠB
is known and here we give the complexity for several more. Our NP-completeness proofs for
ΠB
's rely on the NP-completeness proof of the following problem. Let
S
be a set of graphs and
d
be an integer. Let
ΓSd
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
S
and
x,y∈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
Γ∅3
is NP-complete. We give a simple criterion on a connected graph
H
to decide whether
Γ{H}+∞
is polynomial or NP-complete. The polynomial cases rely on the algorithm three-in-a-tree, due to Chudnovsky and Seymour.