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An Optimal Algorithm for Product Structure in Planar Graphs

Authors Prosenjit Bose , Pat Morin , Saeed Odak

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Author Details

Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Pat Morin
  • School of Computer Science, Carleton University, Ottawa, Canada
Saeed Odak
  • Department of Computer Science and Electrical Engineering, University of Ottawa, Canada


This research was initiated at the BIRS 21w5235 Workshop on Graph Product Structure Theory, held November 21-26, 2021 at the Banff International Research Station. The authors are grateful to the workshop organizers and participants for providing a stimulating research environment. We are especially grateful to Vida Dujmović for sharing Theorem 1.b with us.

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Prosenjit Bose, Pat Morin, and Saeed Odak. An Optimal Algorithm for Product Structure in Planar Graphs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 19:1-19:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


The Product Structure Theorem for planar graphs (Dujmović et al. JACM, 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(nlog n) time algorithm (Morin. Algorithmica, 85(5):1544-1558).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graph algorithms
  • Planar graphs
  • product structure


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