论文标题
平面图产品结构的快速算法
A Fast Algorithm for the Product Structure of Planar Graphs
论文作者
论文摘要
Dujmović等人（focs2019）最近证明，每个平面图$ g $都是$ h \ boxtimes p $的子图，其中$ \ boxtimes $表示强的图形产品，$ h $是Treewidth 8和$ p $的图。该结果发现了许多用于线性图布局，图形着色和图形标记的应用程序。 Dujmović等人给出的证明是基于Pilipczuk和Siebertz（Soda2019）的类似分解，它们具有建设性，并导致了$ O（N^2）$ TIME ALGORITHM查找$ H $的时间算法，并从$ v（g）$ V（H \ boxtimes p）$中的$ v（g）$。在本说明中，我们表明该算法可以以$ O（n \ log n）$时间运行。
Dujmović 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 Dujmović 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.
