Treewidth 2 in the Planar Graph Product Structure Theorem

TL;DR

This paper proves that every planar graph can be represented within a product of two graphs with treewidth 2 and a complete graph of size 2, resolving a key question in graph structure theory.

Contribution

It establishes a tight product structure theorem for planar graphs involving graphs of treewidth 2, answering an open problem in the field.

Findings

Every planar graph is contained in H1×H2×K2 with tw(H1)=tw(H2)=2

The result is optimal; no similar bound holds with smaller parameters

Provides a counterexample showing the bound cannot be improved

Abstract

We prove that every planar graph is contained in $H_1\boxtimes H_2\boxtimes K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible: for any $c \in \mathbb{N}$, there is a planar graph $G$ such that for any tree $T$ and graph $H$ with $\text{tw}(H) \leqslant 2$, $G$ is not contained in $H \boxtimes T \boxtimes K_c$.