Sufficient conditions for a digraph to admit a $(1,\leq\ell)$-identifying code

TL;DR

This paper establishes sufficient conditions for certain digraphs to admit $(1, ext{leq}\ell)$-identifying codes, extending previous results and characterizing specific regular digraphs with such codes.

Contribution

It provides new sufficient conditions for digraphs with minimum in-degree 1 to admit identifying codes for specific $ ext{leq}\ell$ values, and characterizes all 2-in-regular digraphs with these codes.

Findings

Digraphs with minimum in-degree $ ext{leq}\ell$ admit codes under certain conditions.

Graphs with minimum degree $ ext{geq}\delta$ and girth $ ext{geq} ext{7}$ admit $(1, ext{leq}\delta)$-identifying codes.

All 1-in-regular digraphs with girth $ ext{geq} ext{5}$ have a $(1, ext{leq} ext{2})$-identifying code.

Abstract

A $(1,\le \ell)$-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 $\ell$ have different closed in-neighborhoods within $C$. In this paper, we give some sufficient conditions for a digraph of minimum in-degree $\delta^-\ge 1$ to admit a $(1,\le \ell)$-identifying code for $\ell=\delta^-, \delta^-+1$. As a corollary, we obtain the result by Laihonen that states that a graph of minimum degree $\delta\ge 2$ and girth at least 7 admits a $(1,\le \delta)$-identifying code. Moreover, we prove that every $1$-in-regular digraph has a $(1,\le 2)$-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 $(1,\le \ell)$-identifying code for $\ell=2,3$.