The Complexity of Colouring Circle Graphs

We study the complexity of the colouring problem for circle graphs. We will solve the two open questions of [Un88], where first results were presented. 1. Here we will present an algorithm which solves the 3-colouring problem of circle graphs in time O(n log(n)). In [Un88] we showed that the 4-colouring problem for circle graphs is NP-complete. 2. If the largest clique of a circle graph has size k then the 2.k1-colouring is NP-complete. Such circle graphs are 2k-colourable [Un88]. Further results and improvements of [Un88] complete the knowledge of the complexity of the colouring problem of circle graphs. Classification: algorithms and data structures, computationM complexity 1 I n t r o d u c t i o n A circle graph is an undirected graph which is an intersection graph of a set of chords in one circle. The vertex set of an intersection graph of a set of chords is that set of chords. Two vertices are joined by an edge iff the representing chords intersect each other. An example of such an intersection graph is given in Figure 1.