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Optimal Tree-Decompositions with Bags of Bounded Treewidth

Kevin Hendrey, David R. Wood

TL;DR

The paper investigates when a graph class admits an optimal tree-decomposition whose bags induce subgraphs of bounded treewidth. It develops a framework of normal/basic/refined decompositions together with breakability and irreducibility to bound bag structure, achieving sharp results for planar graphs (bag TW ≤3) and graphs on fixed surfaces (bag TW ≤ max{4g+2,3}) as well as for K_p-minor-free graphs, while revealing inherent limits for 1-planar graphs. It also provides a width-O(√n) decomposition scheme based on layered treewidth that applies beyond minor-closed classes, including 1-planar graphs, and identifies a clear trade-off between optimal width and bag structure. These results enrich our understanding of the interplay between global width and local bag structure, with potential algorithmic benefits for problems solvable by dynamic programming on tree-decompositions. The work highlights both the power and the limits of bounded-bag approaches in broad graph families and lays out several compelling open questions.

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.

Optimal Tree-Decompositions with Bags of Bounded Treewidth

TL;DR

The paper investigates when a graph class admits an optimal tree-decomposition whose bags induce subgraphs of bounded treewidth. It develops a framework of normal/basic/refined decompositions together with breakability and irreducibility to bound bag structure, achieving sharp results for planar graphs (bag TW ≤3) and graphs on fixed surfaces (bag TW ≤ max{4g+2,3}) as well as for K_p-minor-free graphs, while revealing inherent limits for 1-planar graphs. It also provides a width-O(√n) decomposition scheme based on layered treewidth that applies beyond minor-closed classes, including 1-planar graphs, and identifies a clear trade-off between optimal width and bag structure. These results enrich our understanding of the interplay between global width and local bag structure, with potential algorithmic benefits for problems solvable by dynamic programming on tree-decompositions. The work highlights both the power and the limits of bounded-bag approaches in broad graph families and lays out several compelling open questions.

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 has a tree-decomposition with minimum width such that each bag has treewidth in . This treewidth bound is best possible. Most generally, every -minor-free graph has a tree-decomposition with minimum width such that each bag has treewidth at most some polynomial function . 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 -vertex graphs have tree-decompositions with width (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.

Paper Structure

This paper contains 10 sections, 36 theorems, 5 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Monotone Graph Classes
  3. Normal, Basic and Refined Tree-Decompositions
  4. Breakability and Reducibility
  5. Planar Graphs
  6. Graphs on Surfaces
  7. K_p-Minor-Free Graphs
  8. 1-Planar Graphs
  9. Width O(√n)
  10. Open Problems

Key Result

Theorem 1

Every planar graph has an optimal tree-decomposition in which every bag has treewidth at most 3.

Figures (2)

  • Figure 1: Non-separable planar graphs: (a) outerplanar, (b) wheel, (c) elongated triangular prism.
  • Figure 2: Subwalls $H_1,\dots,H_{k+1}$, and cycle collections $\mathcal{C}_1,\dots,\mathcal{C}_{k+1}$.

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 45 more