  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.
