Abstract
Here we show that, given a set of clusters C on a set of taxa X , where |X | = n, it is possible to determine in time f(k) · poly(n) whether there exists a level-≤ k network (i.e. a network where each biconnected component has reticulation number at most k) that represents all the clusters in C in the softwired sense, and if so to construct such a network. This extends a result from [20] which showed that the problem is polynomial-time solvable for fixed k. By defining "k-reticulation generators" analogous to "level-k generators", we then extend this fixed parameter tractability result to the problem where k refers not to the level but to the reticulation number of the whole network.