Large-Treewidth Graph Decompositions and Applications

TL;DR

This paper investigates how to decompose graphs into subgraphs with large treewidth, establishing tradeoffs between the number of subgraphs and their treewidth, and applying these results to improve algorithms and theoretical bounds.

Contribution

It introduces new theorems on graph decompositions with large treewidth and demonstrates their application in improving algorithmic and structural results.

Findings

Decomposition theorems relating number of subgraphs and treewidth bounds

Framework to bypass the Grid-Minor Theorem in applications

Improved parameters for Erdos-Posa-type results and FPT algorithms

Abstract

Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently large treewidth. We prove two theorems on the tradeoff between the number of the desired subgraphs $h$, and the desired lower bound $r$ on the treewidth of each subgraph. The theorems assert that, given a graph $G$ with treewidth $k$, a decomposition with parameters $h,r$ is feasible whenever $hr^2 \le k/\polylog(k)$, or $h^3r \le k/\polylog(k)$ holds. We then show a framework for using these theorems to bypass the well-known Grid-Minor Theorem of Robertson and Seymour in some applications. In particular, this leads to substantially improved parameters in some Erdos-Posa-type results, and faster algorithms for a class of fixed-parameter tractable problems.