Realizability of arbitrary distributive lattices as minimum edge-cut lattices

Determine whether there exists a flow network whose family of minimum edge-cut sets forms an arbitrarily prescribed distributive lattice; equivalently, determine whether the closure under finite unions and intersections of the poset of finest minimum cuts containing individual vertices can be an arbitrary distributive lattice.

Background

The paper studies the partial order on network vertices induced by the inclusion relations among finest minimum cuts. It explains that a flow network can be reconstructed from certain Hasse diagrams and relates this reconstruction problem to earlier work on lattices formed by vertex or edge cut sets. Minimum cut sets are known to form a distributive sublattice of the lattice of cut sets, but the general realizability problem for distributive lattices is not settled.

The authors identify the unresolved problem as determining whether every abstract distributive lattice can arise as the lattice of minimum edge-cut sets of some flow network. In their formulation, this is equivalent to asking whether the closure of the family of finest minimum cuts under finite unions and intersections can realize an arbitrary distributive lattice. The reconstruction procedure proposed for Hasse diagrams is suggested as potentially relevant to this question.

References

The reconstruction of a flow network is possible for a few cases using the method in while it remains an open question whether there exists a flow network with its minimum edge-cut sets forming an arbitrarily given distributive lattice. We emphasize that this open question is equivalent to ask whether the closure of the poset $({S_x}_{x\in V(G)}, \subseteq)$ under finite unions and intersections is an arbitrary distributive lattice or not in our language, and the reconstruction proposed in this subsection might be a key to the open question.

— Canonical partial ordering from min-cuts and quantum entanglement in random tensor networks  (2506.23894 - Hu et al., 30 Jun 2025) in Section 2, subsection “Quotient graphs and Hasse diagrams”