Colouring AT-Free Graphs

A vertex colouring assigns to each vertex of a graph a colour such that adjacent vertices have different colours. The algorithmic complexity of the Colouring problem, asking for the smallest number of colours needed to vertex-colour a given graph, is known for a large number of graph classes. Notably it is NP-complete in general, but polynomial time solvable for perfect graphs. A triple of vertices of a graph is called an asteroidal triple if between any two of the vertices there is a path avoiding all neighbours of the third one. Asteroidal triple-free graphs form a graph class with a lot of interesting structural and algorithmic properties. Broersma et al. (ICALP 1997) asked to find out the algorithmic complexity of Colouring on AT-free graphs. Even the algorithmic complexity of the k-Colouring problem, which asks whether a graph can be coloured with at most a fixed number k of colours, remained unknown for AT-free graphs. First progress was made recently by Stacho who presented an O(n) time algorithm for 3-colouring AT-free graphs (ISAAC 2010). In this paper we show that k-Colouring on AT-free graphs is in XP, i.e. polynomial time solvable for any fixed k. Even more, we present an algorithm using dynamic programming on an asteroidal decomposition which, for any fixed integers k and a, solves k-Colouring on any input graph G in time O(f(a, k) ·n), where a denotes the asteroidal number of G, and f(a, k) and g(a, k) are functions that do not depend on n. Hence for any fixed integer k, there is a polynomial time algorithm solving k-Colouring on graphs of bounded asteroidal number. The algorithm runs in time O(n) on AT-free graphs.