Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

A graph is Ok-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) Ok-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 O2-free graphs without K2,3 as a subgraph and whose treewidth is (at least) logarithmic.Using our result, we show that Maximum Independent Set and 3-Coloring in Ok-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 Ok-free graphs, and that deciding the Ok-freeness of sparse graphs is polynomial time solvable.