On the pseudo-achromatic number problem

We study the parameterized complexity of the pseudo-achromatic number problem: Given an undirected graph and a parameter k, determine if the graph can be partitioned into k groups such that every two groups are connected by at least one edge. This problem has been extensively studied in graph theory and combinatorial optimization. We show that the problem has a kernel of at most (k−2)(k+1) vertices that is constructable in time , where n and m are the number of vertices and edges, respectively, in the graph, and k is the parameter. This directly implies that the problem is fixed-parameter tractable. We also study generalizations of the problem and show that they are parameterized intractable.