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Bipartizing with a Matching

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Book cover Combinatorial Optimization and Applications (COCOA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11346))

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Abstract

We study the problem of determining whether a given graph \(G=(V,E)\) admits a matching M whose removal destroys all odd cycles of G (or equivalently whether \(G-M\) is bipartite). This problem is equivalent to determine whether G admits a (2, 1)-coloring, which is a 2-coloring of V(G) in which each color class induces a graph of maximum degree at most 1. We determine a dichotomy related to the NP-completeness of such a decision problem, where it is NP-complete even for 3-colorable planar graphs of maximum degree 4, while it is linear-time solvable for graphs of maximum degree at most 3. In addition, we present polynomial-time algorithms for many graph classes including those in which every odd-cycle subgraph is a triangle, graphs having bounded dominating sets, and \(P_5\)-free graphs. Additionally, we show that this problem is fixed-parameter tractable when parameterized by the clique-width, which implies that it is polynomial-time solvable for many interesting graph classes, such as distance-hereditary, outerplanar, and chordal graphs.

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, by the Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brasil (CNPq) - CNPq/DAAD2015SWE/290021/2015-4, and FAPERJ.

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Author information

Authors and Affiliations

  1. Computer Science Department, Federal University of Minas Gerais, Belo Horizonte, Brazil

    Carlos V. G. C. Lima

  2. Institute of Optimization and Operations Research, Ulm University, Ulm, Germany

    Dieter Rautenbach

  3. Institute of Computing, Fluminense Federal University, Niterói, Brazil

    Uéverton S. Souza

  4. PESC, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Jayme L. Szwarcfiter

Authors
  1. Carlos V. G. C. Lima
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  2. Dieter Rautenbach
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  3. Uéverton S. Souza
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  4. Jayme L. Szwarcfiter
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Corresponding author

Correspondence to Carlos V. G. C. Lima .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Kennesaw State University, Marietta, GA, USA

    Donghyun Kim

  2. Department of Mathematics and Computer Science, North Carolina Central University, Durham, NC, USA

    R. N. Uma

  3. Department of Computer Science, Georgia State University, Atlanta, GA, USA

    Alexander Zelikovsky

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Lima, C.V.G.C., Rautenbach, D., Souza, U.S., Szwarcfiter, J.L. (2018). Bipartizing with a Matching. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_14

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  • DOI: https://doi.org/10.1007/978-3-030-04651-4_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-04650-7

  • Online ISBN: 978-3-030-04651-4

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