Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

TL;DR

This paper proves that sparse graphs without large induced cycle packings have logarithmically bounded treewidth, enabling efficient algorithms for many NP-complete problems in these graph classes.

Contribution

It establishes a tight bound on the treewidth of sparse, -free graphs, leading to quasi-polynomial and polynomial algorithms for key problems.

Findings

Treewidth of -free sparse graphs is logarithmic in the number of vertices.

Maximum Independent Set and 3-Coloring are solvable in quasi-polynomial time in these graphs.

Most NP-complete problems become polynomial-time solvable in sparse -free graphs.

Abstract

A graph is $\mathcal{O}_k$-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) $\mathcal{O}_k$-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 $\mathcal{O}_2$-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 $\mathcal{O}_k$-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 $\mathcal{O}_k$-free graphs, and that deciding the $\mathcal{O}_k$-freeness of sparse graphs is polynomial time solvable.