Let $G$ be a graph and let $\mathrm{cl}(G)$ be the number of distinct induced cycle lengths in $G$. We show that for $c,t\in \mathbb N$, every graph $G$ that does not contain an induced subgraph isomorphic to $K_{t+1}$ or $K_{t,t}$ and satisfies $\mathrm{cl}(G) \le c$ has bounded treewidth. As a consequence, we obtain a polynomial-time algorithm for deciding whether a graph $G$ contains induced cycles of at least three distinct lengths.

翻译：令$G$为一个图，且令$\mathrm{cl}(G)$表示$G$中不同诱导环长度的数量。我们证明，对于$c,t\in \mathbb N$，每一个不包含同构于$K_{t+1}$或$K_{t,t}$的诱导子图，且满足$\mathrm{cl}(G) \le c$的图$G$，其树宽是有界的。由此，我们获得了一个多项式时间算法，用于判定一个图$G$是否包含至少三个不同长度的诱导环。