exists a restricted triangulation $H$ of $L({\ensuremath{\mathcal{P}}})$. We can assume that $H$ is minimal triangulation, since if it is not, a restricted minimal triangulation of $L({\ensuremath{\mathcal{P}}})$ can be obtained by repeatedly deleting fill-in edges from $H$ until it is a minimal triangulation. Let ${\ensuremath{\mathcal{F}}}= \triangle_{H}$. From Theorem \[thm:parallel\_minimal\_ct\], ${\ensuremath{\mathcal{F}}}$ is a maximal set of pairwise parallel minimal separators of $L({\ensuremath{\mathcal{P}}})$ and $L({\ensuremath{\mathcal{P}}})_{{\ensuremath{\mathcal{F}}}} = H$. Assume that ${\ensuremath{\mathcal{F}}}$ contains a separator $F$ that is not legal. Let $\{e, e'\} \subseteq F$ where $\{e, e'\} \subseteq E(T)$ for some input tree $T$ and $e \cap e' = \emptyset$. Vertices of $F$ form a clique in $H$. Thus, $H$ contains the edge $\{e, e'\}$. Since $\{e, e'\}$ is not a valid edge, $H$ is not a restricted triangulation, which is a contradiction. Hence, every separator in ${\ensuremath{\mathcal{F}}}$ is legal.

Let ${\ensuremath{\mathcal{F}}}$ be a maximal set of pairwise parallel minimal separators of $L({\ensuremath{\mathcal{P}}})$ where every separator in ${\ensuremath{\mathcal{F}}}$ is legal. From Theorem \[thm:parallel\_minimal\_ct\], $L({\ensuremath{\mathcal{P}}})_{{\ensuremath{\mathcal{F}}}}$ is a minimal triangulation of $L({\ensuremath{\mathcal{F}}})$. If $\{e, e'\} \in E(L({\ensuremath{\mathcal{P}}})_{{\ensuremath{\mathcal{F}}}})$ is a fill-in edge, then $e \cap e' = \emptyset$ and there exists a minimal separator $F \in {\ensuremath{\mathcal{F}}}$ where $\{e, e'\} \subseteq F$. Since separator $F$ is legal, 