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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识别线图中的代码

给定整数 $\ell\ge 1$，有向图中的 $(1,\le \ell)$ 标识代码是顶点的支配子集 $C$，使得所有不同的基数顶点子集至多为 $\ ell$ 在 $C$ 内有明显的封闭邻里。在本文中，我们证明了最小入度至少为 2 和 $k\geq2$，或最小入度至少为 3 和 $k\geq1$ 的每一个 $k$-迭代的线有向图，承认一个 $(1 ,\le \ell)$ 与 $\ell\leq2$ 的识别代码，并且在任何情况下它都不允许 $\ell\geq3$ 的 $(1,\le \ell)$ 识别代码。此外，我们发现一个有向图的标识号的下界是原始有向图的大小减去它的阶数。此外，这个下界是针对最小入度至少为 2 的有向图获得的。