An Optimal Algorithm for Product Structure in Planar Graphs

TL;DR

This paper presents a simple, linear-time algorithm for computing the product structure decomposition of planar graphs, improving efficiency over previous methods and enabling faster graph analysis.

Contribution

It introduces a linear-time algorithm for the Product Structure Theorem in planar graphs, enhancing computational efficiency over prior $O(n ext{log} n)$ algorithms.

Findings

Linear-time algorithm for product structure decomposition

Improved computational efficiency for planar graph analysis

Applicable to related graph decompositions

Abstract

The \emph{Product Structure Theorem} for planar graphs (Dujmovi\'c et al.\ \emph{JACM}, \textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple linear-time algorithm for finding this decomposition as well as several related decompositions. This improves on the previous $O(n\log n)$ time algorithm (Morin.\ \emph{Algorithmica}, \textbf{85}(5):1544--1558).