Fig. 2.

An example for $\Theta (T,t_{\mu })\subset \Theta (T,{\widehat{t}}_T)$. Top Left: A gene tree $(T,\sigma )$ with extremal event labeling ${\widehat{t}}_T$, the corresponding orthology relation $\Theta (T,{\widehat{t}}_T)$ and map $\sigma (a_i)=A$, $\sigma (b_i)=B$ and $\sigma (c_i)=C$, $i=1,2$. Here we obtain $AB|C,AC|B\in \mathcal{S}(T,{\widehat{t}}_T,\sigma )$ as conflicting species triples, making $\mathcal{S}(T,{\widehat{t}}_T,\sigma )$ incompatible. Top Right and Bottom: The same tree $(T,\sigma )$ with another event labeling $t_{\mu }$ defined by the reconciliation map $\mu$ from $(T,\sigma )$ to the (tube-like) species tree S as shown at the bottom. The map $\mu$ is given implicitly by drawing T into S. The corresponding orthology relation $\Theta (T,t_{\mu })$ is shown below $(T,t_{\mu },\sigma )$. Clearly, since $\mu$ exists, $\mathcal{S}(T,t_{\mu },\sigma )=\{AB|C\}$ is compatible (cf. Prop. 1) (color figure online)