of ${{\mathcal X}}$. We use $xy|z$ to denote the triplet with taxa $x,y$ on one side of the root and $z$ on the other side of the root. For triplets, the notion of “represent” can be formalized by the notion of “display” introduced above. However, for triplets “consistent with” is often used instead of “displayed by”. A triplet $xy|z$ is *consistent* with a phylogenetic network $N$ (and $N$ is *consistent* with $xy|z$) if $xy|z$ is displayed by $N$. See Figure \[fig:treestripletsclusters\] for an example. Given a phylogenetic tree $T$ on ${{\mathcal X}}$, we let  $Tr(T)$ denote the set of all rooted triplets on ${{\mathcal X}}$ that are consistent with $T$. For a set of phylogenetic trees $\mathcal{T}$, we let $Tr(\mathcal{T})$ denote the set of all rooted triplets that are consistent with some tree in $\mathcal{T}$, i.e. $Tr
(\mathcal{T})=\bigcup_{T\in\mathcal{T}}
Tr
(T)$. A set of triplets on ${{\mathcal X}}$ is *dense* if, for every size-3 subset $\{x,y,z\} \subseteq {{\mathcal X}}$, at least one of $xy|z$, $xz|y$, $yz|x$ is in the triplet set.

Let $R$ be a dense set of triplets on ${{\mathcal X}}$. Then, for every fixed $k \geq 0$, it is possible to determine in polynomial time whether a binary simple 