Encoding network classes by LCA relations

Characterize the network classes N for which two networks are isomorphic, with an isomorphism fixing the leaf set, if and only if they have identical LCA relations.

Background

The paper studies how LCA constraints determine canonical DAGs and networks, but it does not establish when the LCA relation uniquely determines a network up to leaf-fixing isomorphism within a prescribed class. This is posed as an encoding problem for network classes, including the possibility of restricting to special classes such as trees or other phylogenetic networks.

References

A further question arising in this context is that of ``encoding'' of networks, i.e., for which network classes $\mathcal{N}$ does it hold that $_N = _{N'}$ if and only if $N \sim N'$ for all $N, N' \in \mathcal{N}$, where $\sim$ denotes an isomorphism between $N$ and ${N'}$ that is the identity on the leaf-set?

— Inferring DAGs and Phylogenetic Networks from Least Common Ancestors  (2511.07965 - Lindeberg et al., 11 Nov 2025) in Section 8, Discussion and Outlook, paragraph “Limitation and Extension of Condition {X1} and {X2}”