Short Paths in the Planar Graph Product Structure Theorem
Kevin Hendrey, David R. Wood
TL;DR
The paper resolves an open question about the Planar Graph Product Structure Theorem by proving that every $n$-vertex planar graph $G$ can be embedded in a product $H\boxtimes P\boxtimes K_c$ with $\mathrm{tw}(H)\le 3$, where the path $P$ can be chosen to be much shorter than $n$: specifically, $|V(P)|=O\big(\frac{1}{\epsilon} n^{(1+\epsilon)/2}\big)$ with $c=O(1/\epsilon)$ for any $\epsilon\in(0,1)$; and a stronger bound $|V(P)|=O\big(\frac{1}{\epsilon}\mathrm{tw}(G)\, n^{\epsilon}\big)$ is achieved. Taking $\epsilon=1/\log n$ yields $G\subseteq H\boxtimes P\boxtimes K_{O(\log n)}$ with treewidth within a factor $O(\log^2 n)$ of $\mathrm{tw}(G)$, which nearly matches known lower bounds. The approach builds a refined layered tree-decomposition via seeds and seed-based layering, connects tree-decompositions to graph products through $H$-partitions and layerings, and extends the framework to beyond-planar classes through shallow-minor arguments. The results unify and sharpen product-structure methods, facilitating applications in queue layouts, colorings, and beyond, while offering a rich set of open problems for tightening bounds and generalizing to other surfaces and minor-closed classes.
Abstract
The Planar Graph Product Structure Theorem of Dujmović et al. [J. ACM '20] says that every planar graph $G$ is contained in $H\boxtimes P\boxtimes K_3$ for some planar graph $H$ with treewidth at most 3 and some path $P$. This result has been the key to solving several old open problems. Several people have asked whether the Planar Graph Product Structure Theorem can be proved with good upper bounds on the length of $P$. No $o(n)$ upper bound was previously known for $n$-vertex planar graphs. We answer this question in the affirmative, by proving that for any $ε\in (0,1)$ every $n$-vertex planar graph is contained in $H\boxtimes P\boxtimes K_{O(1/ε)}$, for some planar graph $H$ with treewidth 3 and for some path $P$ of length $O(\frac{1}εn^{(1+ε)/2})$. This bound is almost tight since there is a lower bound of $Ω(n^{1/2})$ for certain $n$-vertex planar graphs. In fact, we prove a stronger result with $P$ of length $O(\frac{1}ε\,\textrm{tw}(G)\,n^ε)$, which is tight up to the $O(\frac{1}ε\,n^ε)$ factor for every $n$-vertex planar graph $G$. Finally, taking $ε=\frac{1}{\log n}$, we show that every $n$-vertex planar graph $G$ is contained in $H\boxtimes P\boxtimes K_{O(\log n)}$ for some planar graph $H$ with treewidth at most 3 and some path $P$ of length $O(\textrm{tw}(G)\,\log n)$. This result is particularly attractive since the treewidth of the product $H\boxtimes P\boxtimes K_{O(\log n)}$ is within a $O(\log^2n)$ factor of the treewidth of $G$.