Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

Abstract

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. This is optimal, as there is an infinite family of $O_2$-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 $O_k$-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 $O_k$-free graphs, and that deciding the $O_k$-freeness of sparse graphs is polynomial time solvable.