Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
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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.
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