Optimal Tree-Decompositions with Bags of Bounded Treewidth

TL;DR

This paper investigates the structure of various graph classes, proving bounds on tree-decompositions with bags of bounded treewidth, and shows that these bounds are tight for certain classes like 1-planar graphs.

Contribution

It establishes optimal bounds for tree-decompositions with bounded treewidth bags across multiple graph classes, including planar, genus, and minor-free graphs, and demonstrates limitations for 1-planar graphs.

Findings

Planar graphs have tree-decompositions with bags of treewidth at most 3.

Genus g graphs have tree-decompositions with bags of treewidth O(g).

1-planar graphs do not admit tree-decompositions with bags of bounded treewidth and width within an additive constant of optimal.

Abstract

We prove that several natural graph classes have tree-decompositions with minimum width such that each bag has bounded treewidth. For example, every planar graph has a tree-decomposition with minimum width such that each bag has treewidth at most 3. This treewidth bound is best possible. More generally, every graph of Euler genus $g$ has a tree-decomposition with minimum width such that each bag has treewidth in $O(g)$. This treewidth bound is best possible. Most generally, every $K_p$-minor-free graph has a tree-decomposition with minimum width such that each bag has treewidth at most some polynomial function $f(p)$. In such results, the assumption of an excluded minor is justified, since we show that analogous results do not hold for the class of 1-planar graphs, which is one of the simplest non-minor-closed monotone classes. In fact, we show that 1-planar graphs do not have tree-decompositions with width within an additive constant of optimal, and with bags of bounded treewidth. On the other hand, we show that 1-planar $n$-vertex graphs have tree-decompositions with width $O(\sqrt{n})$ (which is the asymptotically tight bound) and with bounded treewidth bags. Moreover, this result holds in the more general setting of bounded layered treewidth, where the union of a bounded number of bags has bounded treewidth.