Dujmovic 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 Dujmovic 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.

中文翻译：

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平面图乘积结构的一种快速算法

Dujmovic 等人 (FOCS2019) 最近证明了每个平面图 $G$ 都是 $H\boxtimes P$ 的子图，其中 $\boxtimes$ 表示强图积，$H$ 是树宽为 8 和 $P$ 的图是一条路径。这一结果已在线性图形布局、图形着色和图形标记方面得到了大量应用。Dujmovic 等人给出的证明基于 Pilipczuk 和 Siebertz (SODA2019) 的类似分解，该分解具有建设性，并导致 $O(n^2)$ 时间算法用于查找 $H$ 和 $V(G )$ 到 $V(H\boxtimes P)$。在本笔记中，我们展示了该算法可以在 $O(n\log n)$ 时间内运行。