as follows. Consider any cut $F \in {\ensuremath{\mathcal{F}}}$. We build a pair $D_F=(X,Y)$ where $X$ and $Y$ are subsets of $E(F)$ and are vertex separators of $G({\ensuremath{\mathcal{P}}})$. Let $A$ and $B$ be the connected components of $G({\ensuremath{\mathcal{P}}})-F$. Also, let $A'$, $B'$ be the subgraphs induced in $G({\ensuremath{\mathcal{P}}})$ by the vertex sets $V(A) \cup \{X \cap Y\}$ and $V(B) \cup \{X \cap Y\}$ respectively. To legally triangulate $G({\ensuremath{\mathcal{P}}})$ we first triangulate the subgraph of $G({\ensuremath{\mathcal{P}}})$ induced by the vertex set $X \cup Y$ and then triangulate the subgraphs $A'$ and $B'$. To triangulate either of those subgraphs, we again use vertex separators built from endpoints of a different cut. We make sure that, for every set $D_I$ for some $I \in {\ensuremath{\mathcal{F}}}$ built after $D_F$, both the sets of $D_I$ are subsets of either $V(A')$ or $V(B')$ but not both.

We now give the details of our construction. We consider the elements of ${\ensuremath{\mathcal{F}}}$ in some arbitrary, but fixed order, and use a set $W$ to record all such cuts $F \in {\ensuremath{\mathcal{F}}}$ for which $D_F$ has already been constructed. Initially $W$ is empty. For each successive cut $F$ in ${\ensuremath{\mathcal{F}}}$, we do the following. Let $F' \subseteq F$ be the 