Identifying Codes on Directed De Bruijn Graphs

For a directed graph 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 1and 2-identifiable and examines conditions under which it is not t-identifiable. This paper also proves that a t-identifying code for tidentifiable de Bruijn graphs must contain at least dn−1(d−1) vertices. Constructions are given to show that this lower bound is achievable for 1-identifying codes when n is odd, or n is even and d > 2, and for 2-identifying codes when n > 3. Further a construction is given proving that when n is even and d = 2 there is a 1-identifying code of ∗ [email protected] † [email protected] Approved for public release; distribution unlimited: 88ABW-2014-5829.