Graphs and Combinatorics

, Volume 30, Issue 2, pp 353–364 | Cite as

Solving Partition Problems with Colour-Bipartitions

  • Ross Churchley
  • Jing Huang
Original Paper


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.


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

Mathematics Subject Classification (2000)

05C70 05C85 


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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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