SIAM Journal on Discrete Mathematics
Graph classes: a survey
Linear Time Algorithms for Dominating Pairs in Asteroidal Triple-free Graphs
SIAM Journal on Computing
Domination and total domination on asteroidal triple-free graphs
Proceedings of the 5th Twente workshop on on Graphs and combinatorial optimization
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Independent Sets in Asteroidal Triple-Free Graphs
SIAM Journal on Discrete Mathematics
WG '97 Proceedings of the 23rd International Workshop on Graph-Theoretic Concepts in Computer Science
Feedback vertex set on AT-free graphs
Discrete Applied Mathematics
List coloring in the absence of a linear forest
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
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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(n4) 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 $\mathcal{O}(f(a,k) \cdot n^{g(a,k)})$, 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 $\mathcal{O}(n^{8k+2})$ on AT-free graphs.