  Given a dense triplet set $\mathcal{T}$, there arise two interesting
questions: Does there exists any phylogenetic network consistent with
$\mathcal{T}$? And if so, can we find an effective algorithm to construct one?
For cases of networks of levels $k=0$ or 1 or 2, these questions were answered
with effective polynomial algorithms. For higher levels $k$, partial answers
were recently obtained with an $O(|\mathcal{T}|^{k+1})$ time algorithm for
simple networks. In this paper we give a complete answer to the general case.
The main idea is to use a special property of SN-sets in a level-k network. As
a consequence, we can also find the level-k network with the minimum number of
reticulations in polynomial time.
