Characterizing extremal digraphs for identifying codes and extremal cases of Bondys theorem on induced subsets

An identifying code of a (di)graph $G$ is a dominating subset $C$ of the vertices of $G$ such that all distinct vertices of $G$ have distinct (in)neighbourhoods within $C$. In this paper, we classify all finite digraphs which only admit their whole vertex set in any identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well known theorem of A. Bondy on set systems we classify the extremal cases for this theorem.

💡 Research Summary

The paper investigates the extremal situation in which a directed graph (digraph) admits no identifying code other than the whole vertex set. An identifying code C ⊆ V of a digraph G = (V, A) must be a dominating set and must give each vertex a unique closed in‑neighbourhood intersected with C. The authors call a digraph “extremal” when every proper subset of V fails to satisfy one of these two conditions.

Finite digraphs.
The authors prove that the only finite digraphs with this property belong to two very specific families.

1. Complete oriented graphs (tournaments). For every unordered pair {u, v} exactly one of the arcs (u → v) or (v → u) is present. In such a tournament, removing any vertex makes at least two remaining vertices share the same closed in‑neighbourhood, so the remaining set cannot be an identifying code.
2. Bidirectional complete graphs. For every unordered pair {u, v} both arcs (u → v) and (v → u) are present. Again, deletion of any vertex collapses the distinctness of closed in‑neighbourhoods.

The proof proceeds by first showing that in either family every vertex’s closed in‑neighbourhood equals the whole vertex set, which forces the whole set to be the only possible code. Then a converse argument demonstrates that any digraph not belonging to these families admits a proper identifying code, typically by exploiting a vertex with a private in‑neighbour or by constructing a small dominating set that already separates all vertices.

Infinite oriented graphs.
For infinite graphs the situation is more delicate because domination can be achieved with infinite but sparse subsets. The authors introduce the notion of an infinite complete oriented tree: a rooted, acyclic digraph in which each vertex has infinitely many ancestors, and for every level L_k (vertices at distance k from the root) each vertex in L_{k‑1} has an outgoing arc to every vertex in L_k. In such a structure, any proper subset of vertices fails to dominate all vertices or fails to keep the closed in‑neighbourhoods distinct, so again the whole vertex set is the unique identifying code. The paper proves that these trees, together with the finite families above, exhaust all infinite oriented graphs with the extremal property.

Connection to Bondy’s theorem.
Bondy’s classic result on set systems states that for any family of n distinct subsets of a ground set, one can delete a single element so that the resulting family remains pairwise distinct. The extremal case—where deletion of any element makes two subsets identical—corresponds precisely to the extremal digraphs described earlier. By interpreting each vertex v as the set N⁻