  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
polynomial time result from "On the elusiveness of clusters" by Kelk,
Scornavacca and Van Iersel(2011). By generalizing the concept of "level-k
generator" to general networks, 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.
