Matroid structure for restricted network classes

Determine whether there exist network classes N for which the family of inclusion-minimal closure-generating subsets of a realizable relation R forms a matroid.

Background

For a realizable relation R, the authors define the family of subsets contained in inclusion-minimal subsets R' whose closure equals the closure of R. Although the general setting does not exhibit a matroid structure, they ask whether restricting the underlying realizing networks to particular classes could restore one. This question concerns both the existence of suitable classes and the induced closure-generating family.

References

Moreover, one may restrict attention to relations $R$ that can be realized by particular classes of networks, for example, phylogenetic trees. This leads to the question of whether there exist network classes $\mathcal{N}$ for which the tuple

(R, \mathcal{F}_R), \quad \text{where } \mathcal{F}_R = { R'' \subseteq R' \mid R' \subseteq R \text{ is inclusion-minimal with } (R') = (R) },

forms a matroid, with $R$ a relation realized by some network $N \in \mathcal{N}$.

— Inferring DAGs and Phylogenetic Networks from Least Common Ancestors  (2511.07965 - Lindeberg et al., 11 Nov 2025) in Section 8, Discussion and Outlook, paragraph “Matroids”