Close the necessary–sufficient gap for stable directed graphs

Characterize stable directed graphs graph-theoretically by determining whether the necessary cycle-cover property \(\mathcal N\) and the sufficient nested-cycle-cover property \(\mathcal S\) can be made equivalent for general directed graph topologies.

Background

For a strongly connected digraph, property N\mathcal N is necessary for the associated sparse matrix space to contain a Hurwitz matrix, whereas property S\mathcal S is sufficient. The paper emphasizes that these conditions coincide for symmetric patterns but may differ for directed patterns. A complete deterministic characterization of structural stability would therefore require closing this gap.

References

The gap between the necessary $\mathcal N$ and the sufficient $\mathcal S$ is the central obstacle to a complete graph-theoretic classification of stable digraphs, and closing it remains an {\em open problem}.

— Structural stability of systems and cycle covers in random graphs  (2610.01607 - Belabbas, 1 Oct 2026) in Section 1, subsection “Background: structural system theory”

A second direction concerns the gap between $\mathcal N$, $\mathcal S$, and stability itself: even deterministically, no graph-theoretic characterization of stable digraphs is known, and our results show that $\mathcal N$ and $\mathcal S$ separate on the directed step-graphon model (Example~\ref{ex:NS_separation}), mirroring the deterministic gap between them.

— Structural stability of systems and cycle covers in random graphs  (2610.01607 - Belabbas, 1 Oct 2026) in Section 10, “Conclusion and open problems”