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.
Solving partition problems with colour-bipartitions. Ross Churchley and Jing Huang. Graphs and Combinatorics 30 (2), 353–364. March 2014.