Parameterized Complexity of Vertex Colouring

For a family F of graphs and a nonnegative integer k, F + ke and F − ke, respectively, denote the families of graphs that can be obtained from F graphs by adding and deleting at most k edges, and F + kv denotes the family of graphs that can be made into F graphs by deleting at most k vertices. This paper is mainly concerned with the parameterized complexity of the vertex colouring problem on F + ke, F − ke and F + kv for various families F of graphs. In particular, it is shown that the vertex colouring problem is 5xed-parameter tractable (linear time for each 5xed k) for split + ke graphs and split − ke graphs, solvable in polynomial time for each 5xed k but W [1]-hard for split + kv graphs. Furthermore, the problem is solvable in linear time for bipartite + 1v graphs and bipartite + 2e graphs but, surprisingly, NP-complete for bipartite + 2v graphs and bipartite + 3e graphs. ? 2002 Published by Elsevier Science B.V. MSC: 05C15; 05C85; 68Q17; 68Q25; 68R10