Feedback numbers of de Bruijn digraphsI

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/ D 8>>< >>: 1 n X ijn diʹ n i for 2 n 4I dn n C O.ndn􀀀4/ for n 5; where i j n means i divides n, and ʹ.i/ is the Euler totient function.