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Given a set $S\subseteq\mathcal{X}$ of taxa, we use $\mathcal{C}\setminus S$ to denote the result of removing all elements of $S$ from each cluster in $\mathcal{C}$ and we use $\mathcal{C}|S$ to denote ${\mathcal{C}\setminus (\mathcal{X}\setminus S)}$ (the restriction of $\mathcal{C}$ to $S$). We say that a set $S \subseteq \mathcal{X}$ is an *ST-set* with respect to $\mathcal{C}$, if $S$ is not separated by $\mathcal{C}$ and any two clusters $C_1,C_2\in\mathcal{C}|S$ are compatible. Note that, unlike in [@cass], we allow the possibility that $S = \emptyset$ or $S = {{\mathcal X}}$. (We say that an ST-set $S$ is *trivial* if $S= \emptyset $ or $S= {{\mathcal X}}$). An ST-set $S$ is *maximal* if there is no ST-set $T$ with $S \subset T$. Informally, the maximal ST-sets are the result of repeatedly collapsing pairs of unseparated taxa for as long as possible; we can think of them as “islands of laminarity” within the cluster set. ST-sets first explicitly appeared in [@cass] but, as we shall see in due course, they implicitly arose earlier in the *recombination network* literature. An important feature of ST-sets is that there can in general be very many of them. For example, suppose $\mathcal{C}$ contains only $|\mathcal{X}|=n$ singleton clusters; then 