Powers of planar graphs, product structure, and blocking partitions

TL;DR

This paper establishes a new product structure theorem for powers of planar and beyond planar graphs, introduces the concept of blocking partitions, and applies these to graph classes with bounded genus, solving several open problems.

Contribution

It proves a general product structure theorem for powers of planar and bounded genus graphs, introducing blocking partitions of independent interest.

Findings

Powers of planar graphs are contained in products of bounded treewidth graphs, paths, and cliques.

Every graph of Euler genus g has an ℓ-blocking partition with bounded parts.

4-regular graphs do not admit ℓ-blocking partitions with bounded size parts.

Abstract

We prove that the $k$-power of any planar graph $G$ is contained in $H\boxtimes P\boxtimes K_{f(\Delta(G),k)}$ for some graph $H$ with bounded treewidth, some path $P$, and some function $f$. This resolves an open problem of Ossona de Mendez. In fact, we prove a more general result in terms of shallow minors that implies similar results for many `beyond planar' graph classes, without dependence on $\Delta(G)$. For example, we prove that every $k$-planar graph is contained in $H\boxtimes P\boxtimes K_{f(k)}$ for some graph $H$ with bounded treewidth and some path $P$, and some function $f$. This resolves an open problem of Dujmovi\'c, Morin and Wood. We generalise all these results for graphs of bounded Euler genus, still with an absolute bound on the treewidth. At the heart of our proof is the following new concept of independent interest. An $\ell$-blocking partition of a graph $G$ is a partition of $V(G)$ into connected sets such that every path of length greater than $\ell$ in $G$ contains at least two vertices in one part. We prove that for some constant $\ell \ge 1$ every graph of Euler genus $g$ has an $\ell$-blocking partition with parts of size bounded by a function of $\Delta(G)$ and $g$. Motivated by this result, we study blocking partitions in their own right. We show that every graph $G$ has a $2$-blocking partition with parts of size bounded by a function of $\Delta(G)$ and $\textrm{tw}(G)$. On the other hand, we show that 4-regular graphs do not have $\ell$-blocking partitions with bounded size parts.