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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  1. Stephen Alstrup, Cyril Gavoille, Haim Kaplan, and Theis Rauhe. Nearest common ancestors: A survey and a new algorithm for a distributed environment. Theory Comput. Syst., 37(3):441-456, 2004. URL:
  2. Michael A. Bekos, Giordano Da Lozzo, Petr Hliněný, and Michael Kaufmann. Graph product structure for h-framed graphs. CoRR, abs/2204.11495, 2019. URL:
  3. Michael A. Bender and Martin Farach-Colton. The LCA problem revisited. In Gaston H. Gonnet, Daniel Panario, and Alfredo Viola, editors, LATIN 2000: Theoretical Informatics, 4th Latin American Symposium, Punta del Este, Uruguay, April 10-14, 2000, Proceedings, volume 1776 of Lecture Notes in Computer Science, pages 88-94. Springer, 2000. URL:
  4. Omer Berkman and Uzi Vishkin. Recursive star-tree parallel data structure. SIAM J. Comput., 22(2):221-242, 1993. URL:
  5. Prosenjit Bose, Vida Dujmović, Mehrnoosh Javarsineh, and Pat Morin. Asymptotically optimal vertex ranking of planar graphs. CoRR, abs/2007.06455, 2020. URL:
  6. Michal Debski, Stefan Felsner, Piotr Micek, and Felix Schröder. Improved bounds for centered colorings. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2212-2226. SIAM, 2020. URL:
  7. Reinhard Diestel. Graph Theory, Fifth Edition, volume 173 of Graduate texts in mathematics. Springer, 2017. URL:
  8. Vida Dujmović, Gwenaël Joret, Piotr Micek, Pat Morin, Torsten Ueckerdt, and David R. Wood. Planar graphs have bounded queue-number. J. ACM, 67(4):22:1-22:38, 2020. URL:
  9. Vida Dujmovic, Pat Morin, and David R. Wood. Graph product structure for non-minor-closed classes. CoRR, abs/1907.05168, 2019. URL:
  10. Vida Dujmović, Louis Esperet, Cyril Gavoille, Gwenaël Joret, Piotr Micek, and Pat Morin. Adjacency labelling for planar graphs (and beyond). J. ACM, 68(6):42:1-42:33, 2021. URL:
  11. Vida Dujmović, Louis Esperet, Gwenaël Joret, Bartosz Walczak, and David R. Wood. Planar graphs have bounded nonrepetitive chromatic number. CoRR, abs/1904.05269, 2019. URL:
  12. Louis Esperet, Gwenaël Joret, and Pat Morin. Sparse universal graphs for planarity. CoRR, abs/2010.05779, 2020. URL:
  13. Johannes Fischer and Volker Heun. Theoretical and practical improvements on the rmq-problem, with applications to LCA and LCE. In Moshe Lewenstein and Gabriel Valiente, editors, Combinatorial Pattern Matching, 17th Annual Symposium, CPM 2006, Barcelona, Spain, July 5-7, 2006, Proceedings, volume 4009 of Lecture Notes in Computer Science, pages 36-48. Springer, 2006. URL:
  14. Fănică Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B, 16:47-56, 1974. URL:
  15. Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. Comput., 13(2):338-355, 1984. URL:
  16. Pat Morin. A fast algorithm for the product structure of planar graphs. Algorithmica, 83(5):1544-1558, 2021. URL:
  17. Baruch Schieber and Uzi Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM J. Comput., 17(6):1253-1262, 1988. URL:
  18. Torsten Ueckerdt, David R. Wood, and Wendy Yi. An improved planar graph product structure theorem. CoRR, abs/2108.00198, 2021. URL:
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