### Abstract

A seminal result of Reed et al. [Combinatorica, 16 (1996), pp. 535-554] says that a directed graph has either κ vertex-disjoint directed circuits or a set of at most f(κ) vertices meeting all directed circuits. This paper aims at generalizing their result to packing directed circuits through a prescribed set S of vertices. Such a circuit is called an S-circuit. Even et al. [Algorithmica, 20 (1998), pp. 151-174] showed a fractional version of packing S-circuits. In this paper, we show that the fractionality can be bounded by at most one-fifth: Given an integer κ and a vertex subset S, whose size may not depend on κ, we prove that either G has a 1/5-integral packing of k disjoint S-circuits, i.e., each vertex appears in at most five of these S-circuits, or G has a vertex set X of order at most f(k) (for some function f of κ) such that G - X has no such circuit. We also give a fixed-parameter tractable approximation algorithm for finding a 1/5-integral packing of S-circuits. This algorithm finds a 1/5-integral packing of size approximately κ in polynomial time if it has a 1/5-integral packing of size k for a given directed graph and an integer κ.

Original language | English |
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Pages (from-to) | 1121-1133 |

Number of pages | 13 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 26 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Oct 16 |

Externally published | Yes |

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### Keywords

- Disjoint circuits
- FPT approximability
- Feedback vertex sets

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*26*(3), 1121-1133. https://doi.org/10.1137/100786423