A graph is $\mathcal{O}_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $\mathcal{O}_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of $\mathcal{O}_2$-free graphs without $K_{2,3}$ as a subgraph and whose treewidth is (at least) logarithmic. Using our result, we show that Maximum Independent Set and 3-Coloring in $\mathcal{O}_k$-free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse $\mathcal{O}_k$-free graphs, and that deciding the $\mathcal{O}_k$-freeness of sparse graphs is polynomial time solvable.

翻译：如果一个图不包含$k$个两两顶点不相交且互不相邻的圈，则称该图为$\mathcal{O}_k$-free图。我们证明“稀疏”（此处指不包含大完全二部图作为子图）的$\mathcal{O}_k$-free图的树宽（甚至反馈顶点集数）最多为顶点数的对数。这一结果是紧的，因为存在一个无穷族$\mathcal{O}_2$-free图，不包含$K_{2,3}$作为子图，其树宽（至少）为对数。利用这一结果，我们证明在$\mathcal{O}_k$-free图上，最大独立集问题和3-着色问题可以在拟多项式时间内求解。其他推论包括：大多数核心NP完全问题（如最大独立集、最小顶点覆盖、最小支配集、最小着色）可以在稀疏$\mathcal{O}_k$-free图上于多项式时间内求解，且稀疏图的$\mathcal{O}_k$-free性质判定问题可在多项式时间内解决。