Identifying codes on directed de Bruijn graphs

For a directed graph G with the vertex set V(G), a t-identifying code is a subset S⊆V(G) with the property that for each vertex v∈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 t-identifiable if there exists a t-identifying code. This paper shows that the de Bruijn graph B→(d,n) is t-identifiable if and only if n≥2t−1. It is also shown that a t-identifying code for t-identifiable de Bruijn graphs must contain at least dn−1(d−1) vertices, and constructions are given to show that this lower bound is achievable when n≥2t. Further a (possibly) non-optimal construction is given when n=2t−1. Additionally, with respect to B→(d,n) we provide upper and lower bounds on the size of a minimum 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 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 dn−1(d−1), and that if d>n the minimum size of a determining set (a subset S with the property that the only automorphism that fixes S pointwise is the trivial automorphism) is d−1n.