network* if each biconnected component has reticulation number less than or equal to $k$.[^3] A ${\mbox{level-}\leq k}$ network is called a *simple level-$\leq k$ network* if the removal of a cut-node or a cut-edge creates two or more connected components of which at most one is non-trivial (i.e. contains at least one edge). A (simple) level-$\leq k$ network $N$ is called a (simple) *level-$k$ network* if the maximum reticulation number among the biconnected components of $N$ is precisely $k$. For example, the network in Figure \[fig:bcc\] is a level-2 network while the one in Figure \[fig:gallednetwork\] is a simple level-4 network. Note that a tree is a level-0 network. For a set $\mathcal{C}$ of clusters on $\mathcal{X}$ we define $\ell(\mathcal{C})$, the *level* of $\mathcal{C}$, as the smallest $k \geq 0$ such that there exists a level-$k$ network that represents $\mathcal{C}$. It is immediate that for every cluster set $\mathcal{C}$ $r(\mathcal{C}) \geq \ell(\mathcal{C})$, because a level-$k$ network always contains at least one biconnected component containing $k$ reticulations.

We say that two clusters $C_1,C_2\subset\mathcal{X}$ are *compatible* if either $C_1\cap C_2=\emptyset$ or $C_1\subseteq C_2$ or $C_2\subseteq C_1$. Consider a set of clusters $\mathcal{C}$. The *incompatibility graph* $IG(\mathcal{C})$ of $\mathcal{C}$ is the undirected graph 