Let us say a graph is ${\mathcal{O}}_s$-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 ${\mathcal{O}}_2$-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that ${\mathcal{O}}_2$-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 ${\mathcal{O}}_s$-free graph G has at most $|G{|}^c$ induced paths, where $|G|$ is the number of vertices. This provides a poly-time algorithm to test if a graph is ${\mathcal{O}}_s$-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 ${\mathcal{O}}_s$-free graph G with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in $|G|$. 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.

假设一个图表是${\mathcal{氧}}_s$-自由，其中$s\ge 1$是一个整数，如果图中不存在成对顶点不相交且没有边连接的s 个循环。这种图的结构，即使当$s=2$，不太理解。例如，直到现在我们还不知道如何测试一个图是否是${\mathcal{氧}}_2$-多项式时间内自由；Ngoc Khang Le 提出了一个公开的猜想：${\mathcal{氧}}_2$-自由图仅具有多项式数量的诱导路径。

本文我们证明了Le的猜想；事实上，我们将向所有人证明$s\ge 1$， 那里存在$C>0$使得每个${\mathcal{氧}}_s$-自由图G最多有$|G{|}^C$诱导路径，其中$|G|$是顶点数。这提供了一个多时间算法来测试图是否是${\mathcal{氧}}_s$-free，对于所有固定的s。

证明分为三个部分。首先，Le 提供了一个简短而优美的证明，它可以将问题简化为证明没有长度为 4 的循环的图也能证明同样的事情。其次，Bonamy、Bonnet、Déprés、Esperet、Geniet、Hilaire、Thomassé 和 Wesolek 最近的结果表明，在每个${\mathcal{氧}}_s$- 自由图G，没有长度为 4 的环，有一组与每个环相交的顶点，其大小为对数$|G|$。第三，有一个论证使用了 Bonamy 等人的结果。推导出定理。最后是本文的主要内容。