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.
@InProceedings{gawrychowski_et_al:LIPIcs.ESA.2024.59, author = {Gawrychowski, Pawe{\l} and Wasylkiewicz, Mateusz}, title = {{Finding Perfect Matchings in Bridgeless Cubic Multigraphs Without Dynamic (2-)connectivity}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {59:1--59:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.59}, URN = {urn:nbn:de:0030-drops-211301}, doi = {10.4230/LIPIcs.ESA.2024.59}, annote = {Keywords: perfect matching, cubic graphs, bridgeless graphs, link-cut tree} }
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