Solving partition problems with colour-bipartitions

This paper was the culmination of my undergraduate research on monopolar graphs with Jing Huang. Having already shown how to find monopolar partitions of line graphs and claw-free graphs, we extended our technique to a class of graphs including those with no induced cycle of length 5 or greater. Our solution works for every graph class for which the monopolar partition problem is known to be polynomially-solvable, and also can be adapted to solve the related unipolar partition problem for general graphs.

This paper also proves an important relationship between the polar and monopolar partition problems: if a polar partition of a graph is already known, it can be decided in polynomial time whether the graph is monopolar.

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.

Ross Churchley and Jing Huang [1]

This research was completed in the summer of 2010, my last undergraduate research term. I am grateful to NSERC for funding my work with a Undergraduate Student Research Award, and to my supervisor and coauthor Jing Huang.

  1. Solving partition problems with colour-bipartitions. Graphs and Combinatorics 30(2). Ross Churchley and Jing Huang (2014). ↩︎