Three-colourability and forbidden subgraphs. II: polynomial algorithms

In this paper we study the chromatic number for graphs with forbidden induced subgraphs. We focus our interest on graph classes (de2ned in terms of forbidden induced subgraphs) for which the question of 3-colourability can be decided in polynomial time and, if so, a proper 3-colouring can be determined also in polynomial time. Note that the 3-colourability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. triangle-free and K1;5-free (Discrete Math. 162 (1–3) (1996) 313). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-colouring of a given graph in general. We present three di7erent approaches to reach our goal. The 2rst approach is purely a structural analysis of the graph class in consideration; the second one is a structural analysis of only the non-perfect K4-free members of the considered graph class; 2nally the last approach is based on propositional logic and bounded dominating subgraphs. c © 2002 Elsevier Science B.V. All rights reserved. MSC: 05C15; 05C75; 05C85