Graphs and Combinatorics

, Volume 30, Issue 2, pp 353–364

Solving Partition Problems with Colour-Bipartitions

Original Paper

Abstract

Polar, monopolar, and unipolar graphs are defined in terms of the existence of certain vertex partitions. Although it is polynomial to determine whether a graph is unipolar and to find whenever possible a unipolar partition, the problems of recognizing polar and monopolar graphs are both NP-complete in general. These problems have recently been studied for chordal, claw-free, and permutation graphs. Polynomial time algorithms have been found for solving the problems for these classes of graphs, with one exception: polarity recognition remains NP-complete in claw-free graphs. In this paper, we connect these problems to edge-coloured homomorphism problems. We show that finding unipolar partitions in general and finding monopolar partitions for certain classes of graphs can be efficiently reduced to a polynomial-time solvable 2-edge-coloured homomorphism problem, which we call the colour-bipartition problem. This approach unifies the currently known results on monopolarity and extends them to new classes of graphs.

Keywords

Monopolar graphs Unipolar graphs Edge-coloured graphs Partition problems Polynomial algorithm 

Mathematics Subject Classification (2000)

05C70 05C85 

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References

  1. 1.
    Bawar Z., Brewster R., Marcotte D.: Homomorphism duality in edge-coloured graphs. Ann. Sci. Math. Québec 29(1), 21–34 (2005)MATHMathSciNetGoogle Scholar
  2. 2.
    Brewster, R.: Vertex Colourings of Edge-Coloured Graphs. Ph.D. thesis, Simon Fraser University (1993)Google Scholar
  3. 3.
    Chernyak Z.A., Chernyak A.A.: About recognizing (α, β) classes of polar graphs. Discret. Math. 62, 133–138 (1986)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Churchley, R.: Some Remarks on Monopolarity. Manuscript (2012)Google Scholar
  5. 5.
    Churchley, R.; Huang, J.: On the polarity and monopolarity of graphs. Submitted (2010)Google Scholar
  6. 6.
    Churchley R., Huang J.: Line-polar graphs: characterization and recognition. SIAM J. Discret. Math. 25, 1269–1284 (2011)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Churchley R., Huang J.: List-monopolar partitions of claw-free graphs. Discret. Math. 312(17), 2545–2549 (2012)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Ekim, T.: Polarity of Claw-Free Graphs. Manuscript (2009)Google Scholar
  9. 9.
    Ekim, T., Heggernes, P., Meister, D.: Polar permutation graphs. In: Combinatorial Algorithms: 20th International Workshop, IWOCA 20009, Lecture Notes in Computer Science. vol. 5874, 218–229 (2009)Google Scholar
  10. 10.
    Ekim T., Hell P., Stacho J., de Werra D.: Polarity of chordal graphs. Discret. Appl. Math. 156(13), 1652–1660 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ekim T., Huang J.: Recognizing line-polar bipartite graphs in time O(n). Discret. Appl. Math. 158(15), 1593–1598 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Ekim T., Mahadev N.V.R., de Werra D.: Polar cographs. Discret. Appl. Math. 156(10), 1652–1660 (2008)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Even S., Itai A., Shamir A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5(4), 691–703 (1976)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Farrugia A.: Vertex-partitioning into fixed additive induced-hereditary properties is NP-hard. Electr. J. Comb. 11(1), 46 (2004)MathSciNetGoogle Scholar
  15. 15.
    Gallai T.: Transitiv orientierbare graphen. Acta. Math. Acad. Sci. Hung. 18, 25–66 (1967)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Gavril F.: An efficiently solvable graph partition problem to which many problems are reducible. Inf. Process. Lett. 45, 285–290 (1993)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Huang J., Xu B.: A forbidden subgraph characterization of line-polar bipartite graphs. Discret. Appl. Math. 158(6), 666–680 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Le, V.B., Nevries, R.: Recognizing polar planar graphs using new results for monopolarity. In: Algorithms and Computation: 22nd International Symposium. ISAAC 2011, Lecture Notes in Computer Science. vol. 7074, 120–129 (2011)Google Scholar
  19. 19.
    Stacho, J.: Complexity of Generalized Colourings of Chordal Graphs. Ph.D. thesis, Simon Fraser University (2008)Google Scholar
  20. 20.
    Tyshkevich R.I., Chernyak A.A.: Algorithms for the canonical decomposition of a graph and recognizing polarity. Izvestia Akad. Nauk BSSR, ser. Fiz. Mat. Nauk 6, 16–23 (1985)MathSciNetGoogle Scholar
  21. 21.
    Tyshkevich R.I., Chernyak A.A.: Decomposition of graphs. Kibernetika (Kiev) 2, 67–74 (1985)MathSciNetGoogle Scholar
  22. 22.
    West, D.: Introduction to Graph Theory. Prentice Hall, Englewood Cliffs, NJ (1996)Google Scholar

Copyright information

© Springer Japan 2012

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada

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