Bipartizing with a Matching

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


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.


Graph modification problems Edge bipartization Defective coloring Planar graphs 


  1. 1.
    Andrews, J., Jacobson, M.: On a generalization of chromatic number. In: Proceedings of Sixteenth Southeastern International Conference on Combinatorics, Graph Theory and Computing (SEICCGTC 1985), vol. 47, pp. 18–33 (1985)Google Scholar
  2. 2.
    Angelini, P., et al.: Vertex-coloring with defects. J. Graph Algor. Appl. 21(3), 313–340 (2017). Scholar
  3. 3.
    Axenovich, M., Ueckerdt, T., Weiner, P.: Splitting planar graphs of girth 6 into two linear forests with short paths. J. Graph Theory 85(3), 601–618 (2017). Scholar
  4. 4.
    Bodlaender, H.L.: A partial \(k\)-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1–2), 1–45 (1998). Scholar
  5. 5.
    Bonamy, M., Dabrowski, K.K., Feghali, C., Johnson, M., Paulusma, D.: Independent feedback vertex set for \(P_5\)-free graphs. Algorithmica (2018).
  6. 6.
    Bondy, J.A., Locke, S.C.: Largest bipartite subgraphs in triangle-free graphs with maximum degree three. J. Graph Theory 10(4), 477–504 (1986)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borodin, O., Kostochka, A., Yancey, M.: On \(1\)-improper \(2\)-coloring of sparse graphs. Discrete Math. 313(22), 2638–2649 (2013). Scholar
  8. 8.
    Brandstädt, A., Dragan, F.F., Le, H., Mosca, R.: New graph classes of bounded clique-width. Theory Comput. Syst. 38(5), 623–645 (2005). Scholar
  9. 9.
    Brandstädt, A., Engelfriet, J., Le, H., Lozin, V.V.: Clique-width for \(4\)-vertex forbidden subgraphs. Theory Comput. Syst. 39(4), 561–590 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brandstädt, A., Klembt, T., Mahfud, S.: \(P_6\)- and triangle-free graphs revisited: structure and bounded clique-width. Discrete Math. Theor. Comput. Sci. 8, 173–188 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Burzyn, P., Bonomo, F., Durán, G.: NP-completeness results for edge modification problems. Discrete Appl. Math. 154(13), 1824–1844 (2006). Scholar
  12. 12.
    Camby, E., Schaudt, O.: A new characterization of \(P_k\)-free graphs. Algorithmica 75(1), 205–217 (2016). Scholar
  13. 13.
    Carneiro, A.D.A., Protti, F., Souza, U.S.: Deletion graph problems based on deadlock resolution. In: Cao, Y., Chen, J. (eds.) COCOON 2017. LNCS, vol. 10392, pp. 75–86. Springer, Cham (2017). Scholar
  14. 14.
    Choi, H.A., Nakajima, K., Rim, C.S.: Graph bipartization and via minimization. SIAM J. Discrete Math. 2(1), 38–47 (1989). Scholar
  15. 15.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cowen, L., Goddard, W., Jesurum, C.E.: Defective coloring revisited. J. Graph Theory 24(3), 205–219 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cowen, L.J., Cowen, R., Woodall, D.: Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency. J. Graph Theory 10(2), 187–195 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Diestel, R.: Graph Theory. Springer, Heidelberg (2017)CrossRefGoogle Scholar
  19. 19.
    Eaton, N., Hull, T.: Defective list colorings of planar graphs. Bull. Inst. Combin. Appl 25, 79–87 (1999)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Erdős, P.: On some extremal problems in graph theory. Israel J. Math. 3(2), 113–116 (1965)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Furmańczyk, H., Kubale, M., Radziszowski, S.: On bipartization of cubic graphs by removal of an independent set. Discrete Appl. Math. 209, 115–121 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci. 11(03), 423–443 (2000). Scholar
  23. 23.
    Harary, F., Jones, K.: Conditional colorability ii: bipartite variations. Congr. Numer. 50, 205–218 (1985)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64(1), 19–37 (2012). Scholar
  26. 26.
    Lima, C.V.G.C., Rautenbach, D., Souza, U.S., Szwarcfiter, J.L.: Decycling with a matching. Inf. Proc. Lett. 124, 26–29 (2017). Scholar
  27. 27.
    Liu, Y., Wang, J., You, J., Chen, J., Cao, Y.: Edge deletion problems: branching facilitated by modular decomposition. Theor. Comput. Sci. 573, 63–70 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lovász, L.: On decomposition of graphs. Studia Sci. Math. Hungar. 1, 237–238 (1966)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM 55(2), 1–29 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Protti, F., Souza, U.S.: Decycling a graph by the removal of a matching: characterizations for special classes. CoRR abs/1707.02473 (2017).
  31. 31.
    Robertson, N., Seymour, P.: Graph minors. ii. Algorithmic aspects of tree-width. J. Algorith. 7(3), 309–322 (1986). Scholar
  32. 32.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings 10th Symposium on Theory of Computing (STOC 1978), pp. 216–226. ACM Press, New York (1978).
  33. 33.
    Thorup, M.: All structured programs have small tree width and good register allocation. Inf. Comput. 142(2), 159–181 (1998). Scholar

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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