### Abstract

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f (d, n) (f_{a} (d, n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K (d, n). This paper proves that for any integers d ≥ 2 and n ≥ 1{Mathematical expression}where (φ{symbol} ȯ θ) (n) = ∑_{i | n} φ{symbol} (i) θ (n / i), i | n means i divides n, θ (i) = d^{i} + (- 1)^{i} d, φ{symbol} (1) = 1 and φ{symbol} (i) = i · ∏_{j = 1}^{r} (1 - 1 / p_{j}) for i ≥ 2, where p_{1}, ..., p_{r} are the distinct prime factors of i, not equal to 1.

Original language | English (US) |
---|---|

Pages (from-to) | 1589-1599 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 307 |

Issue number | 13 |

DOIs | |

State | Published - Jun 6 2007 |

### Fingerprint

### Keywords

- Acyclic subgraph
- Cycles
- Feedback number
- Feedback vertex set
- Kautz digraphs

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*307*(13), 1589-1599. https://doi.org/10.1016/j.disc.2006.09.010

**Feedback numbers of Kautz digraphs.** / Xu, Jun Ming; Wu, Ye Zhou; Huang, Jia; Yang, Chao.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 307, no. 13, pp. 1589-1599. https://doi.org/10.1016/j.disc.2006.09.010

}

TY - JOUR

T1 - Feedback numbers of Kautz digraphs

AU - Xu, Jun Ming

AU - Wu, Ye Zhou

AU - Huang, Jia

AU - Yang, Chao

PY - 2007/6/6

Y1 - 2007/6/6

N2 - A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f (d, n) (fa (d, n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K (d, n). This paper proves that for any integers d ≥ 2 and n ≥ 1{Mathematical expression}where (φ{symbol} ȯ θ) (n) = ∑i | n φ{symbol} (i) θ (n / i), i | n means i divides n, θ (i) = di + (- 1)i d, φ{symbol} (1) = 1 and φ{symbol} (i) = i · ∏j = 1r (1 - 1 / pj) for i ≥ 2, where p1, ..., pr are the distinct prime factors of i, not equal to 1.

AB - A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f (d, n) (fa (d, n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K (d, n). This paper proves that for any integers d ≥ 2 and n ≥ 1{Mathematical expression}where (φ{symbol} ȯ θ) (n) = ∑i | n φ{symbol} (i) θ (n / i), i | n means i divides n, θ (i) = di + (- 1)i d, φ{symbol} (1) = 1 and φ{symbol} (i) = i · ∏j = 1r (1 - 1 / pj) for i ≥ 2, where p1, ..., pr are the distinct prime factors of i, not equal to 1.

KW - Acyclic subgraph

KW - Cycles

KW - Feedback number

KW - Feedback vertex set

KW - Kautz digraphs

UR - http://www.scopus.com/inward/record.url?scp=34047182039&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34047182039&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2006.09.010

DO - 10.1016/j.disc.2006.09.010

M3 - Article

AN - SCOPUS:34047182039

VL - 307

SP - 1589

EP - 1599

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 13

ER -