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Solving Partition Problems with Colour-Bipartitions

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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.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

    Ross Churchley

  2. Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, BC, V8W 3R4, Canada

    Jing Huang

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  1. Ross Churchley
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  2. Jing Huang
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Correspondence to Ross Churchley.

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Churchley, R., Huang, J. Solving Partition Problems with Colour-Bipartitions. Graphs and Combinatorics 30, 353–364 (2014). https://doi.org/10.1007/s00373-012-1270-z

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  • DOI: https://doi.org/10.1007/s00373-012-1270-z

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