and $e_2$ are in different components of $T-e_3$. Such an edge exists because $e_1$ and $e_2$ are nonincident. Set ${\ensuremath{\mathcal{F}}}'$ is a complete set of pairwise parallel legal minimal cuts of $G$. Thus, there exists minimal cut $F' \in {\ensuremath{\mathcal{F}}}'$ where $e_3$ is the only edge of $T$ in $F'$. Since, minimal separators $F$ and $F'$ are in ${\ensuremath{\mathcal{F}}}$, they are parallel to each other and vertices $e_1$ and $e_2$ are in the same connected component of in $L({\ensuremath{\mathcal{P}}})-F'$. Thus, by Observation \[obs:path\], there exists a path between vertices $x$ and $x'$ in $G({\ensuremath{\mathcal{P}}})-F'$ and edges $e_1$ and $e_2$ are also in the same connected component of $G({\ensuremath{\mathcal{P}}})-F'$. But that is impossible, since, $F'$ is a legal minimal cut of $G$ and $e_1$ and $e_2$ are in different components of $T-e_3$.

Thus, every separator of ${\ensuremath{\mathcal{F}}}\setminus {\ensuremath{\mathcal{F}}}'$ is legal and the set ${\ensuremath{\mathcal{F}}}$ is a maximal set of pairwise minimal separators of $L({\ensuremath{\mathcal{P}}})$ where every separator in ${\ensuremath{\mathcal{F}}}$ is legal.

Lemmas \[lm:maximal\_seps\_equals\_maximal\_cuts\] and \[lm:maximal\_cuts\_equals\_maximal\_seps\] imply the following theorem.

\[thm:maximal\_seps\_equals\_maximal\_cuts\] There exists a maximal set ${\ensuremath{\mathcal{F}}}$ of pairwise parallel minimal separators of $L({\ensuremath{\mathcal{P}}})$ where every separator in ${\ensuremath{\mathcal{F}}}$ is legal if and only if there exists a complete set of pairwise 