Solving Partition Problems with Colour-Bipartitions
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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.
KeywordsMonopolar graphs Unipolar graphs Edge-coloured graphs Partition problems Polynomial algorithm
Mathematics Subject Classification (2000)05C70 05C85
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- 2.Brewster, R.: Vertex Colourings of Edge-Coloured Graphs. Ph.D. thesis, Simon Fraser University (1993)Google Scholar
- 4.Churchley, R.: Some Remarks on Monopolarity. Manuscript (2012)Google Scholar
- 5.Churchley, R.; Huang, J.: On the polarity and monopolarity of graphs. Submitted (2010)Google Scholar
- 8.Ekim, T.: Polarity of Claw-Free Graphs. Manuscript (2009)Google Scholar
- 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
- 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.Stacho, J.: Complexity of Generalized Colourings of Chordal Graphs. Ph.D. thesis, Simon Fraser University (2008)Google Scholar
- 22.West, D.: Introduction to Graph Theory. Prentice Hall, Englewood Cliffs, NJ (1996)Google Scholar