$T_N$ of $N$ such that $(u,v)$ represents $C$ w.r.t $T_N$. Note that, in general, an edge of $N$ might represent multiple clusters, and a cluster might be represented by multiple edges of $N$.

Given a set of clusters ${{\mathcal C}}$ on ${{\mathcal X}}$, throughout the article we assume that, for any taxon $x \in {{\mathcal X}}$, ${{\mathcal C}}$ contains at least one cluster $C$ containing $x$. For a set $\mathcal{C}$ of clusters on $\mathcal{X}$ we define $r(\mathcal{C})$ as $\min \{ r(N) | N \text{ represents } \mathcal{C} \}$, we sometimes refer to this as the *reticulation number* of $\mathcal{C}$. The related concept of [*level*]{} requires some more background. A directed acyclic graph is *connected* (also called “weakly connected”) if there is an undirected path (ignoring edge orientations) between each pair of nodes. A node (edge) of a directed graph is called a *cut-node* (*cut-edge*) if its removal disconnects the graph. A directed graph is *biconnected* if it contains no cut-nodes. A biconnected subgraph $B$ of a directed graph $G$ is said to be a *biconnected component* if there is no biconnected subgraph $B' \neq 
B$ of $G$ that contains $B$. A phylogenetic network is said to be a *${\mbox{level-}\leq k}$ 