Powers of planar graphs, product structure, and blocking partitions

Abstract

We show that there exist a constant \(c\) and a function \(f\) such that the \(k\)-power of a planar graph with maximum degree \(\Delta\) is isomorphic to a subgraph of \(H \boxtimes P \boxtimes K_{f(\Delta, k)}\) for some graph \(H\) with treewidth at most \(c\) and some path \(P\). This is the first product structure theorem for \(k\)-powers of planar graphs, where the treewidth of \(H\) does not depend on \(k\). We actually prove a stronger result, which implies an analogous product structure theorem for other graph classes, including \(k\)-planar graphs (of arbitrary degree). Our proof uses a new concept of blocking partitions which is of independent interest. An \(\ell\)-blocking partition of a graph \(G\) is a partition of the vertex set of \(G\) into connected subsets such that every path in \(G\) of length greater than \(\ell\) contains two vertices in one set of the partition. The key lemma in our proof states that there exists a positive integer \(\ell\) such that every planar graph of maximum degree \(\Delta\) has an \(\ell\)-blocking partition with parts of size bounded in terms of \(\Delta\).