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Decomposition Theorems for Square-free 2-matchings in Bipartite Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2015)

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Abstract

A square-free 2-matching in an undirected graph is a simple 2-matching without cycles of length four. In bipartite graphs, the maximum square-free 2-matching problem is well-solved. Previous results include min-max theorems, polynomial combinatorial algorithms, polyhedral description with dual integrality, and discrete convex structure.

In this paper, we further investigate the structure of square-free 2-matchings in bipartite graphs to present new decomposition theorems, which serve as an analogue of the Dulmage-Mendelsohn decomposition for bipartite matchings and the Edmonds-Gallai decomposition for nonbipartite matchings. We exhibit two canonical minimizers of the set function in the min-max formula, and a characterization of the maximum square-free 2-matchings with the aid of these canonical minimizers.

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Acknowledgements

The author is thankful to Satoru Iwata for suggesting this research and Zoltán Király for informing him of the history of the two min-max theorems for the maximum square-free 2-matching problem in bipartite graphs. This work is partially supported by JSPS KAKENHI Grant Numbers 25280004 and 26280001.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Faculty of Science and Engineering, Hosei University, Tokyo, 184-8584, Japan

    Kenjiro Takazawa

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  1. Kenjiro Takazawa
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Correspondence to Kenjiro Takazawa .

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  1. TU München Institut für Informatik, Garching, Germany

    Ernst W. Mayr

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Takazawa, K. (2016). Decomposition Theorems for Square-free 2-matchings in Bipartite Graphs. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_27

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  • DOI: https://doi.org/10.1007/978-3-662-53174-7_27

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