Characterization of regularly diagonal directed graphs

Characterize all regularly diagonal directed graphs, that is, all finite directed graphs whose eulerian magnitude homology is concentrated in bidegrees $(k,k)$.

Background

The paper proves a complete characterization of regularly diagonal undirected graphs under suitable diagonal nonvanishing hypotheses: such a graph must be complete. For directed graphs, the authors exhibit several additional families, including alternating orientations of bipartite graphs, directed trees, cones, joins, and transitive tournaments. These examples show that the directed setting is substantially richer and that the undirected characterization does not extend directly.

The authors explicitly state that they do not know a comparable characterization for directed graphs. In particular, although directed joins of complete graphs are regularly diagonal, not every regularly diagonal digraph arises in this way, since alternating orientations of bipartite graphs provide further examples. A complete structural classification therefore remains unresolved.