1 research outputs found
Identifying Codes on Directed De Bruijn Graphs
For a directed graph , a -identifying code is a subset with the property that for each vertex the set of vertices of
reachable from by a directed path of length at most is both
non-empty and unique. A graph is called {\it -identifiable} if there exists
a -identifying code. This paper shows that the de~Bruijn graph
is -identifiable if and only if . It
is also shown that a -identifying code for -identifiable de~Bruijn graphs
must contain at least vertices, and constructions are given to
show that this lower bound is achievable . Further a (possibly)
non-optimal construction is given when . Additionally, with respect to
we provide upper and lower bounds on the size of a
minimum \textit{-dominating set} (a subset with the property that every
vertex is at distance at most 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 )
is , and that if the minimum size of a {\it determining
set} (a subset with the property that the only automorphism that fixes
pointwise is the trivial automorphism) is .Comment: 27 pages, 4 figures; Revised definitions, notation, and clarity in
arguments. Added additional referenc