On the Existence of $t$-Identifying Codes in Undirected De Bruijn Graphs

TL;DR

This paper establishes conditions under which undirected de Bruijn graphs possess $t$-identifying codes, expanding understanding of their structural properties and identifying cases where such codes exist.

Contribution

It proves the existence of $t$-identifying codes in undirected de Bruijn graphs for various parameters, and identifies open cases and graph eccentricity.

Findings

$ ext{B}(d,n)$ is $t$-identifiable for $d ext{≥} 3$, $n ext{≥} 2t$, $t ext{≥} 1$

Identifiability holds for specific cases with $d ext{≥} 3$, $n ext{≥} 3$, $t=2$, and $d=2$, $n ext{≥} 3$, $t=1$

Eccentricity of undirected non-binary de Bruijn graph is $n$

Abstract

This paper proves the existence of $t$-identifying codes on the class of undirected de Bruijn graphs with string length $n$ and alphabet size $d$, referred to as $\mathcal{B}(d,n)$. It is shown that $\mathcal{B}(d,n)$ is $t$-identifiable whenever $d \geq 3$ and $n \geq 2t$, and $t \geq 1$. We also show that $\mathcal{B}(d,n)$ is $t$-identifiable if either $d \geq 3$, $n \geq 3$, and $t=2$, or if $d = 2$, $n \geq 3$, and $t=1$. The remaining cases remain open. Additionally, we show that the eccentricity of the undirected non-binary de Bruijn graph is $n$.