  Let us say a graph is $s\mathcal{O}$-free, where $s\ge 1$ is an integer, if
there do not exist $s$ cycles of the graph that are pairwise vertex-disjoint
and have no edges joining them. The structure of such graphs, even when $s=2$,
is not well understood. For instance, until now we did not know how to test
whether a graph is $2\mathcal{O}$-free in polynomial time; and there was an
open conjecture, due to Ngoc Khang Le, that $2\mathcal{O}$-free graphs have
only a polynomial number of induced paths.
  In this paper we prove Le's conjecture; indeed, we will show that for all
$s\ge 1$, there exists $c>0$ such that every $s\mathcal{O}$-free graph $G$ has
at most $|G|^c$ induced paths. This provides a poly-time algorithm to test if a
graph is $s\mathcal{O}$-free, for all fixed $s$.
  The proof has three parts. First, there is a short and beautiful proof, due
to Le, that reduces the question to proving the same thing for graphs with no
cycles of length four. Second, there is a recent result of Bonamy, Bonnet,
D\'epr\'es, Esperet, Geniet, Hilaire, Thomass\'e and Wesolek, that in every
$s\mathcal{O}$-free graph $G$ with no cycle of length four, there is a set of
vertices that intersects every cycle, with size logarithmic in $|G|$. And
third, there is an argument that uses the result of Bonamy et al. to deduce the
theorem. The last is the main content of this paper.
