A Fast Algorithm for the Product Structure of Planar Graphs

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