  It has remained an open question for some time whether, given a set of not
necessarily binary (i.e. "nonbinary") trees T on a set of taxa X, it is
possible to determine in time f(r).poly(m) whether there exists a phylogenetic
network that displays all the trees in T, where r refers to the reticulation
number of the network and m=|X|+|T|. Here we show that this holds if one or
both of the following conditions holds: (1) |T| is bounded by a function of r;
(2) the maximum degree of the nodes in T is bounded by a function of r. These
sufficient conditions absorb and significantly extend known special cases,
namely when all the trees in T are binary, or T contains exactly two nonbinary
trees. We believe this result is an important step towards settling the issue
for an arbitrarily large and complex set of nonbinary trees. For completeness
we show that the problem is certainly solveable in polynomial time.
