$i \in \{1,2\}$ and $j \in \{1,2\}$ is non-empty. Let $a \in U_1 \cap V_1$, $b \in U_1 \cap V_2$, $c \in U_2 \cap V_1$ and $d \in U_2 \cap V_2$. Since $\{a, b\} \subseteq U_1$, there exists a path $\pi_1$ between leaf vertices $a$ and $b$ in $G({\ensuremath{\mathcal{P}}})-F_1$. But $a$ and $b$ are in different components of $G({\ensuremath{\mathcal{P}}})-F_2$. Thus, an edge $e_1$ in path $\pi_1$ in $G({\ensuremath{\mathcal{P}}})$ is in the cut $F_2$. Similarly, $\{c,d\} \subseteq U_2$ and there exists a path $\pi_2$ between labels $c$ and $d$ in $G({\ensuremath{\mathcal{P}}})-F_1$. Since $c$ and $d$ are in different components of $G({\ensuremath{\mathcal{P}}})-F_2$, cut $F_2$ contains an edge $e_2$ in path $\pi_2$. But paths $\pi_1$ and $\pi_2$ are in different components of $G({\ensuremath{\mathcal{P}}})-F_1$. So, edges $e_1$ and $e_2$ are in different components of $G({\ensuremath{\mathcal{P}}})-F_1$. Since $\{e_1, e_2\} \subseteq F_2$, the cuts $F_1$ and $F_2$ are not parallel, which is a contradiction.

\[lm:splits\_compatibility\] Let ${\ensuremath{\mathcal{F}}}$ be a complete set of pairwise parallel legal minimal cuts of the display graph of profile ${\ensuremath{\mathcal{P}}}$. The following statements hold.

(i) $\Sigma({\ensuremath{\mathcal{F}}})$ is compatible.

(ii) If $S$ is compatible tree for $\Sigma({\ensuremath{\mathcal{F}}})$, then $S$ is a compatible tree for ${\ensuremath{\mathcal{P}}}$.

(iii) There exists a compatible tree $S$ of 