Induced paths in graphs without anticomplete cycles

Let us say a graph is $O_s$-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 $O_2$-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that $O_2$-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 $O_s$-free graph G has at most $|G{|}^c$ induced paths, where $\left|G\right|$ is the number of vertices. This provides a poly-time algorithm to test if a graph is $O_s$-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, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek, that in every $O_s$-free graph G with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in $\left|G\right|$. 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.