A graph is $O_k$-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) $O_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices, which is sharp already for $k=2$. As a consequence, 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 these graphs, and in particular deciding the $O_k$-freeness of sparse graphs is polytime.

翻译：如果一个图表不包含一对一对一的顶点断裂和非对齐周期,则该图为无美元。我们证明“粗”(此处不包含大量完整的双边图作为子图)无美元图表在最多对数的对数中具有树宽度(甚至有反馈的顶点设定数字),而对于美元=2美元,该数字已经很尖锐。因此,大多数中央NP-完整的问题(如最大独立设置、最低紫外线覆盖、最低定界、最低颜色)可以在这些图形的多元时段中解决,特别是确定稀薄图的无值是多时段。