$(V,E)$ that has node set $V=\mathcal{C}$ and edge set $E=\{\{C_1,C_2\}\enspace |\enspace C_1$ $C_2$ $\mathcal{C}$}. We say that a set of taxa ${{\mathcal X}}' \subseteq\mathcal{X}$ is *separated* (by $\mathcal{C}$) if there exists a cluster $C\in\mathcal{C}$ that is incompatible with ${{\mathcal X}}'$, and *unseparated* otherwise.

We say that a set of clusters $\mathcal{C}$ on $\mathcal{X}$ is [*separating*]{} if it separates all sets of taxa ${{\mathcal X}}'$ such that ${{\mathcal X}}' \subset {{\mathcal X}}$ and $|{{\mathcal X}}'| \geq 2$. We say that a set of clusters $\mathcal{C}$ on $\mathcal{X}$ is [*tangled*]{} [@HusonRuppScornavacca10] if:

1.  $IG(\mathcal{C})$ is connected and has more than one node;

2.  every pair of taxa $x$ and $y$ in $\mathcal{X}$ is separated by $\mathcal{C}$.

Remember that here we assume that any taxon of ${{\mathcal X}}$ is contained in at least one cluster $C\in {{\mathcal C}}$. Then, it can easily be verified that, given a tangled set of clusters $\mathcal{C}$ on ${{\mathcal X}}$, $\mathcal{C}$ is separating. The incompatibility graph and the concept of tangled clusters are important because they highlight an important difference between (the computation of) $r(\mathcal{C})$ and $\ell(\mathcal{C})$. In [@cass] the authors show that, if $\ell(\mathcal{C})=k$, then a level-$k$ network that represents $\mathcal{C}$ can be constructed by combining in 