$\Sigma(F_2) = abcfg | de$, $\Sigma(F_3) = ab | cdefg$, and $\Sigma(F_4) = abcde | fg$. Note that these splits are pairwise compatible.

Relationship to legal triangulation
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Let ${\ensuremath{\mathcal{P}}}$ be a profile of phylogenetic trees. Theorems \[thm:lt\] and \[thm:cuts\_equal\_compatibility\] together imply that if $G({\ensuremath{\mathcal{P}}})$ has a complete set ${\ensuremath{\mathcal{F}}}$ of pairwise parallel legal minimal cuts, there also exists a legal triangulation of $G({\ensuremath{\mathcal{P}}})$. As shown in [@Vakati11], a legal triangulation of $G({\ensuremath{\mathcal{P}}})$ can be derived from a compatible tree of ${\ensuremath{\mathcal{P}}}$. In this section, we show how to derive a legal triangulation of $G({\ensuremath{\mathcal{P}}})$ directly from ${\ensuremath{\mathcal{F}}}$ without building a compatible tree. This shows the relationship between complete sets of pairwise parallel legal minimal cuts and legal triangulations of display graphs. By Theorems \[thm:edge\_label\] and  \[thm:cuts\_seps\] and Lemma \[lm:parallel\], this also shows the relationship between restricted triangulations of edge label intersection graphs and legal triangulations of display graphs.

A complete set ${\ensuremath{\mathcal{F}}}$ of pairwise parallel legal minimal cuts of $G({\ensuremath{\mathcal{P}}})$ is *minimal* if no proper subset of ${\ensuremath{\mathcal{F}}}$ is also complete. Let ${\ensuremath{\mathcal{F}}}$ be a minimal complete set of pairwise parallel legal minimal cuts of $G({\ensuremath{\mathcal{P}}})$.

At a high level, we construct a legal triangulation of $G({\ensuremath{\mathcal{P}}})$ from ${\ensuremath{\mathcal{F}}}$ 