Fig. 3

Consider the “true” history (left) that is also shown in Fig. 1. The center-left gene tree \((T;t,\sigma )\) is biologically feasible and obtained as the observable part of the true history. There is no reconciliation map for \((T;t,\sigma )\) to any species tree according to Def. 2 because \(\mathcal {S}(T;t,\sigma )\) is inconsistent (cf. Thm. 5.4). The graph in the lower-center depicts the orthology-relation that comprises all pairs (x, y) of vertices for which \(t({\text {lca}}(x,y)) =\mathfrak {s}\). The center-right gene tree \((T';t,\sigma )\) is non-binary and can directly be computed from the orthology-relation. Although \(\mathcal {S}(T';t,\sigma )\) is inconsistent, there is a valid reconciliation map \(\mu\) to a species tree for \((T';t,\sigma )\) according to Def. 2 (right). Note, both trees \((T;t,\sigma )\) and \((T';t,\sigma )\) satisfy axioms (O1)–(O3) and even (O3.A). However, the reconciliation map \(\mu\) does not satisfy the extra Condition (M2.iv), since \(\mu (z)\) and \(\mu (a')=A\) are comparable, although z and \(a'\) are children of a common speciation vertex. Therefore, Axioms (O1)–(O3) and (O3.A) do not imply (M2.iv). Moreover, Thm. 5.7 implies that there is no restricted reconciliation map for \((T;t,\sigma )\) as well as \((T';t,\sigma )\) and any species tree, since \(\mathcal {S}(T;t,\sigma )\) and \(\mathcal {S}(T';t,\sigma )\) are inconsistent. See text for further details