Subgraph-count characterization of monomial digraphs

Determine whether there exist finitely many digraphs D1,…,Dk such that any two monomial digraphs D(q;m,n) and D(q′;m′,n′) are isomorphic if and only if they contain the same number of copies of each Di as subdigraphs.

Background

The paper explains that counting cycles of lengths one through seven does not characterize monomial digraphs up to isomorphism. It asks whether a finite collection of subdigraph counts can provide a complete isomorphism criterion.

References

Problem 6. Are there digraphs D1, . . . , Dk such that any two monomial digraphs D = D(q; m, n) and D′ = D(q′; m′, n′) are isomorphic if and only if |D(Di)| = |D′(Di)| for each i = 1, . . . , k?

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Problem 6, Section 4.11