Taiwanese Journal of Mathematics

Bounds on Feedback Numbers of de Bruijn Graphs

Xirong Xu, Jun-Ming Xu, and Yongchang Cao

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

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Taiwanese J. Math., Volume 15, Number 3 (2011), 1101-1113.

First available in Project Euclid: 18 July 2017

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Primary: 05C85: Graph algorithms [See also 68R10, 68W05] 05C85: Graph algorithms [See also 68R10, 68W05] 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35]

graph theory feedback vertex set feedback number de Bruijn graphs cycles scyclic subgraph networks


Xu, Xirong; Xu, Jun-Ming; Cao, Yongchang. Bounds on Feedback Numbers of de Bruijn Graphs. Taiwanese J. Math. 15 (2011), no. 3, 1101--1113. doi:10.11650/twjm/1500406287. https://projecteuclid.org/euclid.twjm/1500406287

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