Feedback numbers of de Bruijn digraphs

A subset of vertices of a graph G is called a feedback vertex set of G if its removal results in an acyclic subgraph. Let f (d, n) denote the minimum cardinality over all feedback vertex sets of the de Bruijn digraph B(d, n). This paper proves that for any integers d ≥ 2 and n ≥ 2 f (d, n) =  1 n ∑ i|n diφ (n i ) for 2 ≤ n ≤ 4; dn n + O(ndn−4) for n ≥ 5, where i | nmeans i divides n, and φ(i) is the Euler totient function. © 2009 Published by Elsevier Ltd