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Bipartizing with a Matching

  • Carlos V. G. C. Lima
  • Dieter Rautenbach
  • Uéverton S. Souza
  • Jayme L. Szwarcfiter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

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.

Keywords

Graph modification problems Edge bipartization Defective coloring Planar graphs 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Computer Science DepartmentFederal University of Minas GeraisBelo HorizonteBrazil
  2. 2.Institute of Optimization and Operations ResearchUlm UniversityUlmGermany
  3. 3.Institute of ComputingFluminense Federal UniversityNiteróiBrazil
  4. 4.PESC, COPPEFederal University of Rio de JaneiroRio de JaneiroBrazil

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