Identifying Codes on Directed De Bruijn Graphs

TL;DR

This paper characterizes $t$-identifying codes in directed de Bruijn graphs, establishing conditions for their existence, minimal sizes, and related concepts like dominating and resolving sets, with constructions and bounds provided.

Contribution

It provides a complete characterization of $t$-identifiable de Bruijn graphs and constructs minimal $t$-identifying codes, along with bounds on related graph parameters.

Findings

De~Bruijn graph $ extbf{$oldsymbol{ ext{ extbf{$f ilde{ extbf{B}}}(d,n)$}}}$ is $t$-identifiable iff $n extgreater= 2t-1.

Minimum size of a $t$-identifying code is $d^{n-1}(d-1)$ for $n extgreater= 2t$.

Bounds on the size of $t$-dominating, resolving, and determining sets are established.

Abstract

For a directed graph $G$, a $t$-identifying code is a subset $S\subseteq V(G)$ with the property that for each vertex $v\in V(G)$ the set of vertices of $S$ reachable from $v$ by a directed path of length at most $t$ is both non-empty and unique. A graph is called {\it $t$-identifiable} if there exists a $t$-identifying code. This paper shows that the de~Bruijn graph $\vec{\mathcal{B}}(d,n)$ is $t$-identifiable if and only if $n \geq 2t-1$. It is also shown that a $t$-identifying code for $t$-identifiable de~Bruijn graphs must contain at least $d^{n-1}(d-1)$ vertices, and constructions are given to show that this lower bound is achievable $n \geq 2t$. Further a (possibly) non-optimal construction is given when $n=2t-1$. Additionally, with respect to $\vec{\mathcal{B}}(d,n)$ we provide upper and lower bounds on the size of a minimum \textit{$t$-dominating set} (a subset with the property that every vertex is at distance at most $t$ from the subset), that the minimum size of a \textit{directed resolving set} (a subset with the property that every vertex of the graph can be distinguished by its directed distances to vertices of $S$) is $d^{n-1}(d-1)$, and that if $d>n$ the minimum size of a {\it determining set} (a subset $S$ with the property that the only automorphism that fixes $S$ pointwise is the trivial automorphism) is $\left\lceil \frac{d-1}{n}\right\rceil$.