Treewidth 2 in the Planar Graph Product Structure Theorem

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 $H \boxtimes T \boxtimes K_c$.

Figures (2)

• Figure 1: Strong product of paths.
• Figure 2: (a) fan, (b) double-fan

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4: LNW
• proof
• Lemma 5: Thomassen95
• Lemma 6
• proof
• Corollary 1
• Theorem 7