Bounds on Feedback Numbers of de Bruijn Graphs

Abstract

The feedback number of a graph $G$ is the minimum number of vertices whose removal from $G$ results in an acyclic subgraph. We use $f(d,n)$ to denote the feedback number of the de Bruijn graph $UB(d,n)$. R. Královic and P. Ruzicka [Minimum feedback vertex sets in shuffle-based interconnection networks. Information Processing Letters, 86 (4) (2003), 191-196] proved that $f(2,n) = \lceil \frac{2^{n}-2}{3} \rceil$. This paper gives the upper bound on $f(d,n)$ for $d \ge 3$, that is, $f(d,n) \leq d^n \left(1 - \left( \frac{d}{1+d}\right)^{d-1} \right) + \binom{n+d-2}{d-2}$.