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Finding Perfect Matchings in Bridgeless Cubic Multigraphs Without Dynamic (2-)connectivity

Authors Paweł Gawrychowski , Mateusz Wasylkiewicz

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • perfect matching
  • cubic graphs
  • bridgeless graphs
  • link-cut tree

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Abstract

Petersen’s theorem, one of the earliest results in graph theory, states that every bridgeless cubic multigraph contains a perfect matching. While the original proof was neither constructive nor algorithmic, Biedl, Bose, Demaine, and Lubiw [J. Algorithms 38(1)] showed how to implement a later constructive proof by Frink in 𝒪(nlog⁴n) time using a fully dynamic 2-edge-connectivity structure. Then, Diks and Stańczyk [SOFSEM 2010] described a faster approach that only needs a fully dynamic connectivity structure and works in 𝒪(nlog²n) time. Both algorithms, while reasonable simple, utilize non-trivial (2-edge-)connectivity structures. We show that this is not necessary, and in fact a structure for maintaining a dynamic tree, e.g. link-cut trees, suffices to obtain a simple 𝒪(nlog n) time algorithm.

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Paweł Gawrychowski and Mateusz Wasylkiewicz. Finding Perfect Matchings in Bridgeless Cubic Multigraphs Without Dynamic (2-)connectivity. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.59

Author Details

Paweł Gawrychowski
  • University of Wrocław, Poland
Mateusz Wasylkiewicz
  • University of Wrocław, Poland

Funding

  • Wasylkiewicz, Mateusz: Partially supported by Polish National Science Center grant 2018/29/B/ST6/02633.

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