$C$ is the set of leaf descendants of $v$. Alternatively, we say that $N$ represents $C$ “in the softwired sense” if $N$ displays some tree $T$ on ${{\mathcal X}}$ such that $T$ represents $C$. In this article we only consider the softwired notion of cluster representation and henceforth assume this implicitly. A network represents a set of clusters $\mathcal{C}$ if it represents every cluster in $\mathcal{C}$ (and possibly more). The set of softwired clusters of a network can be obtained as follows. For a network $N$, we say that a *switching* of $N$ is obtained by, for each reticulation node, deleting all but one of its incoming edges. Given a network $N$ and a switching $T_N$ of $N$, we say that an edge $(u,v)$ of $N$ represents a cluster $C$ w.r.t. $T_N$ if $(u,v)$ is an edge of $T_N$ and $C$ is the set of leaf descendants of $v$ in $T_N$. The set of softwired clusters of $N$ is the set of clusters represented by all edges of $N$ w.r.t. $T_N$, where $T_N$ ranges over all possible switchings [@HusonRuppScornavacca10]. It is also natural to define that an edge $(u,v)$ of $N$ represents a cluster $C$ if there exists some switching 