• DOI:10.7151/DMGT.2218
• Corpus ID: 119623662

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

@article{Balbuena2021SufficientCF,
  title={Sufficient Conditions for a Digraph to Admit A (1, ≤ ℓ)-Identifying Code},
  author={Camino Balbuena and Cristina Dalf'o and B. Mart'inez-Barona},
  journal={Discussiones Mathematicae Graph Theory},
  year={2021},
  volume={41},
  pages={853 - 872}
}

• C. Balbuena, C. Dalf'o, B. Mart'inez-Barona
• Published 1 February 2019
• Mathematics
• Discussiones Mathematicae Graph Theory

Abstract 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}.