A graph is $O_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in $O_k$-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish 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. This is proven sharp as there is an infinite family of $O_2$-free graphs without $K_{3,3}$-subgraph and whose treewidth is (at least) logarithmic. 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 $O_k$-free graphs, and that deciding the $O_k$-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不相邻的圈，则称该图为$O_k$-free图。我们证明，在$O_k$-free图中，最大独立集问题和3-染色问题可以在拟多项式时间内求解。作为主要技术成果，我们证明：稀疏（即不包含大完全二部子图）的$O_k$-free图具有至多与顶点数成对数关系的树宽（甚至是反馈顶点集数）。这一结果被证明是紧的，因为存在一个无限族$O_2$-free图，不含$K_{3,3}$子图且其树宽（至少）为对数级别。其他推论包括：在稀疏的$O_k$-free图中，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、最小染色）可在多项式时间内求解，且判定稀疏图的$O_k$-free性质是多项式时间可解的。