  Phylogenetic networks have gained prominence over the years due to their
ability to represent complex non-treelike evolutionary events such as
recombination or hybridization. Popular combinatorial objects used to construct
them are triplet systems and cluster systems, the motivation being that any
network $N$ induces a triplet system $\mathcal R(N)$ and a softwired cluster
system $\mathcal S(N)$. Since in real-world studies it cannot be guaranteed
that all triplets/softwired clusters induced by a network are available it is
of particular interest to understand whether subsets of $\mathcal R(N)$ or
$\mathcal S(N)$ allow one to uniquely reconstruct the underlying network $N$.
Here we show that even within the highly restricted yet biologically
interesting space of level-1 phylogenetic networks it is not always possible to
uniquely reconstruct a level-1 network $N$ even when all triplets in $\mathcal
R(N)$ or all clusters in $\mathcal S(N)$ are available. On the positive side,
we introduce a reasonably large subclass of level-1 networks the members of
which are uniquely determined by their induced triplet/softwired cluster
systems. Along the way, we also establish various enumerative results, both
positive and negative, including results which show that certain special
subclasses of level-1 networks $N$ can be uniquely reconstructed from proper
subsets of $\mathcal R(N)$ and $\mathcal S(N)$. We anticipate these results to
be of use in the design of, for example, algorithms for phylogenetic network
inference.
