Parameterized complexity of vertex colouring

Abstract

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 ÿxed-parameter tractable (linear time for each ÿxed k) for split + ke graphs and split − ke graphs, solvable in polynomial time for each ÿxed 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.

Minimum Fill-In and Treewidth of Split+ ke and Split+ kv Graphs

“…It is already known that for every hereditary family C for which maximum clique, maximum independent set and minimum vertex cover are polynomial time solvable, then the same problem can be solved in FPT time on the corresponding parametrized classes [5]. However in this case we are interested in listing all maximal independent sets and cliques in FPT time, not only finding the maximum.…”

“…Before to start, we would like to settle also a question left open by Cai in [5] about the existence of a uniformly linear algorithm to find a modulator for parametrized split graphs. The answer is affirmative, and follows from [14,13] where two linear time certification algorithms for split graphs are given.…”

“…Recently several authors studied the complexity of hard problems on such graph classes from a parametrized point of view [4,5,27,20,11]. In particular in [11] the whole idea is generalized and the parameter k is considered as a measurement of the "distance from triviality".…”

“…Such a set is called a modulator of the graph. Related to the problem of finding a modulator efficiently, in [5] Cai asks whether there is a recognition algorithm that is not only uniformly polynomial, but uniformly linear for parametrized split graphs. We can answer affirmatively to this question.…”

“…In [5], Cai shows that finding the chromatic number of split+ke and split−ke graphs is FPT, while it is W[1]-hard for split+kv graphs. He also shows that the same problem is solvable in linear time on bipartite+1v and bipartite+2e graphs, but NPcomplete for bipartite+2v and bipartite+3e graphs.…”