Sufficient Conditions for a Digraph to Admit A (1, ≤ ℓ)-Identifying Code

A (1, ≤ ℓ)-identifying code in a digraph D is a subset C of vertices of D such that all distinct subsets of vertices of cardinality at most ℓ have distinct closed in-neighbourhoods within C. In this paper, we give some sufficient conditions for a digraph of minimum in-degree δ⁻ ≥ 1 to admit a (1, ≤ ℓ)-identifying code for ℓ ∈ {δ⁻, δ⁻ + 1}. As a corollary, we obtain the result by Laihonen that states that a graph of minimum degree δ ≥ 2 and girth at least 7 admits a (1, ≤ δ)-identifying code. Moreover, we prove that every 1-in-regular digraph has a (1, ≤ 2)-identifying code if and only if the girth of the digraph is at least 5. We also characterize all the 2-in-regular digraphs admitting a (1, ≤ ℓ)-identifying code for ℓ ∈ {2, 3}.