# Identifying codes in line digraphs

## TL;DR

This paper investigates the existence and bounds of identifying codes in line digraphs, establishing conditions under which such codes exist or do not, and relating their size to the original digraph's parameters.

## Contribution

It proves the existence of (1,≤2)-identifying codes in certain iterated line digraphs and establishes bounds relating the identifying number to the original digraph's size.

## Key findings

- (1,≤2)-identifying codes exist in specific iterated line digraphs.
- No (1,≤3)-identifying codes exist in these digraphs.
- Lower bounds on identifying number relate to the original digraph's size.

## 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.