exists a cut $F \in {\ensuremath{\mathcal{F}}}$ where $e$ is the only edge of $T$ in $F$.

For the display graph $G({\ensuremath{\mathcal{P}}})$ of Fig. \[fig:example\], let ${\ensuremath{\mathcal{F}}}= \{F_1,F_2,F_3,F_4\}$, where $F_1 = \{\{1,2\}, \{5,6\}\}$, $F_2 = \{\{2,3\}, \{6,7 \}\{5,6\}\}$, $F_3=\{\{4,5\}, \{1,2\},\{1,c\}\}$ and $F_4=\{\{6,7\},\{2,f\}\}$. Then, ${\ensuremath{\mathcal{F}}}$ is a complete set of pairwise parallel legal minimal cuts.

\[lm:maximal\_seps\_equals\_maximal\_cuts\] If there exists a maximal set ${\ensuremath{\mathcal{F}}}$ of pairwise parallel minimal separators of $L({\ensuremath{\mathcal{P}}})$ where every separator $F \in {\ensuremath{\mathcal{F}}}$ is legal, then there exists a complete set of pairwise parallel legal minimal cuts for $G({\ensuremath{\mathcal{P}}})$.

We will show that for every internal edge $e=\{u,v\}$ of an input tree $T$ there exists a minimal separator in ${\ensuremath{\mathcal{F}}}$ which contains only vertex $e$ from $L(T)$. Then from Theorem \[thm:cuts\_seps\] and Lemma \[lm:parallel\] it follows that ${\ensuremath{\mathcal{F}}}$ is a complete set of pairwise parallel legal minimal cuts for display graph $G({\ensuremath{\mathcal{P}}})$.

As shown in proof of Theorem \[thm:edge\_label\], $L({\ensuremath{\mathcal{P}}})_{{\ensuremath{\mathcal{F}}}}$ is a restricted minimal triangulation of $L({\ensuremath{\mathcal{P}}})$. Let $(S, B)$ be a clique tree of $L({\ensuremath{\mathcal{P}}})_{{\ensuremath{\mathcal{F}}}}$. By definition, the vertices in each of the sets $\hat{K}(u)$ and $\hat{K}(v)$ form a clique in $L({\ensuremath{\mathcal{P}}})$. Consider any vertex $p$ of $S$ where $\hat{K}(u) \subseteq B(p)$ and any vertex $q$ of $S$ 