A Fast Algorithm for the Product Structure of Planar Graphs

Dujmovi\'c et al (FOCS2019) recently proved that every planar graph $G$ is a subgraph of $H\boxtimes P$, where $\boxtimes$ denotes the strong graph product, $H$ is a graph of treewidth 8 and $P$ is a path. This result has found numerous applications to linear graph layouts, graph colouring, and graph labelling. The proof given by Dujmovi\'c et al is based on a similar decomposition of Pilipczuk and Siebertz (SODA2019) which is constructive and leads to an $O(n^2)$ time algorithm for finding $H$ and the mapping from $V(G)$ onto $V(H\boxtimes P)$. In this note, we show that this algorithm can be made to run in $O(n\log n)$ time.