Optimal tree-decompositions with bags of bounded pathwidth

2026-07-29

AI Summary

This work investigates how to construct optimal-width tree decompositions while bounding the pathwidth of subgraphs induced by each bag. Focusing on planar graphs and graph classes excluding a fixed two-vertex forest as a minor, the study combines tools from graph decomposition theory, pathwidth and treewidth analysis, minor exclusion, and structural graph theory to prove that every planar graph admits an optimal tree decomposition in which each bag induces a subgraph of pathwidth at most three—a bound shown to be tight. The paper further provides an exact characterization of bag structures satisfying a minimality condition. These results extend to graphs embeddable on any fixed surface and yield a new proof of the linear grid minor theorem for planar graphs.

Abstract

We show that every planar graph has a tree-decomposition with optimal width such that the subgraph induced by each bag has pathwidth at most 3. This bound is best possible, and for tree-decompositions that satisfy a certain minimality condition, we in fact give a precise description of the possible structures in each bag. Moreover, we show that the union of any $k$ bags has pathwidth $O(k)$. We also show that graphs excluding a fixed double-apex-forest minor have a tree-decomposition with optimal width such that the subgraph induced by each bag has bounded pathwidth. This includes graphs embeddable on any fixed surface. As a byproduct of our machinery, we give a new proof of the linear grid minor theorem for planar graphs.

Problem

• tree-decomposition
• pathwidth
• planar graphs
• bounded pathwidth
• graph minors

Innovation

• tree-decomposition
• pathwidth
• planar graphs
• graph minors
• grid minor theorem