minimal separators of $G$ and let $H$ be a minimal triangulation of $G$.

1.  $G_{{\ensuremath{\mathcal{F}}}}$ is a minimal triangulation of $G$.

2.  Let $(T, B)$ be a clique tree of $G_{{\ensuremath{\mathcal{F}}}}$. There exists a minimal separator $F \in {\ensuremath{\mathcal{F}}}$ if and only if there exist two adjacent vertices $x$ and $y$ in $T$ where $B(x) \cap B(y) = F$.

3.  $\triangle_{H}$ is a maximal set of pairwise parallel minimal separators of $G$ and $G_{\triangle_H} = H$.

Let $T$ be a phylogenetic tree over label set ${\ensuremath{\mathcal{L}}}(T)$. Since there exists a bijective function from leaves of $T$ to ${\ensuremath{\mathcal{L}}}(T)$, we will represent leaves of $T$ by their labels. Let ${\ensuremath{\mathcal{P}}}= \{T_1, T_2, \cdots, T_k\}$ be a profile of $k$ phylogenetic trees. The *display graph* of profile ${\ensuremath{\mathcal{P}}}$, denoted by $G({\ensuremath{\mathcal{P}}})$, is a graph whose vertex set is $\bigcup_{i \in [k]} V(T_i)$ and edge set is $\bigcup_{j \in [k]} E(T_j)$. An example of a display graph is given in Fig. \[fig:example\]. A vertex $v$ of $G({\ensuremath{\mathcal{P}}})$ is a *leaf* vertex if $v \in {\ensuremath{\mathcal{L}}}({\ensuremath{\mathcal{P}}})$. Every other vertex of $G({\ensuremath{\mathcal{P}}})$ is an *internal* vertex. An edge of $G({\ensuremath{\mathcal{P}}})$ is an *internal* edge if both its endpoints are internal; otherwise, it is a *non-internal* 