Optimal tree-decompositions with bags of bounded pathwidth
arXiv:2607.27601
The paper proves that every planar graph admits an optimal-width tree‑decomposition whose bags induce subgraphs of pathwidth at most three, and extends similar bounded‑pathwidth bag results to graphs excluding a fixed double‑apex‑forest minor.
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 bags has pathwidth . 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.