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. Kr a lovic 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, $f(d,n)\leq d^n\left(1-\left(\frac{d}{1+d}\right)^{d-1}\right)+\binom{n+d-2}{d-2}$.