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arXiv · 1905.05083

Identifying codes in line digraphs

C. Balbuena·C. Dalfó·B. Martínez-Barona
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Abstract

Given an integer $\ell\ge 1$, a $(1,\le \ell)$-identifying code in a digraph is a dominating subset $C$ of vertices such that all distinct subsets of vertices of cardinality at most $\ell$ have distinct closed in-neighbourhood within $C$. In this paper, we prove that every $k$-iterated line digraph of minimum in-degree at least 2 and $k\geq2$, or minimum in-degree at least 3 and $k\geq1$, admits a $(1,\le \ell)$-identifying code with $\ell\leq2$, and in any case it does not admit a $(1,\le \ell)$-identifying code for $\ell\geq3$. Moreover, we find that the identifying number of a line digraph is lower bounded by the size of the original digraph minus its order. Furthermore, this lower bound is attained for oriented graphs of minimum in-degree at least 2.

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C. Balbuena, C. Dalfó, B. Martínez-Barona. 2019-05-17. Identifying codes in line digraphs. https://arxiv.org/abs/1905.05083

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