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    Identifying Codes on Directed De Bruijn Graphs

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    For a directed graph GG, a tt-identifying code is a subset SβŠ†V(G)S\subseteq V(G) with the property that for each vertex v∈V(G)v\in V(G) the set of vertices of SS reachable from vv by a directed path of length at most tt is both non-empty and unique. A graph is called {\it tt-identifiable} if there exists a tt-identifying code. This paper shows that the de~Bruijn graph Bβƒ—(d,n)\vec{\mathcal{B}}(d,n) is tt-identifiable if and only if nβ‰₯2tβˆ’1n \geq 2t-1. It is also shown that a tt-identifying code for tt-identifiable de~Bruijn graphs must contain at least dnβˆ’1(dβˆ’1)d^{n-1}(d-1) vertices, and constructions are given to show that this lower bound is achievable nβ‰₯2tn \geq 2t. Further a (possibly) non-optimal construction is given when n=2tβˆ’1n=2t-1. Additionally, with respect to Bβƒ—(d,n)\vec{\mathcal{B}}(d,n) we provide upper and lower bounds on the size of a minimum \textit{tt-dominating set} (a subset with the property that every vertex is at distance at most tt 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 SS) is dnβˆ’1(dβˆ’1)d^{n-1}(d-1), and that if d>nd>n the minimum size of a {\it determining set} (a subset SS with the property that the only automorphism that fixes SS pointwise is the trivial automorphism) is ⌈dβˆ’1nβŒ‰\left\lceil \frac{d-1}{n}\right\rceil.Comment: 27 pages, 4 figures; Revised definitions, notation, and clarity in arguments. Added additional referenc
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