### Abstract

An integer-valued function f on the set 2^{V} of all subsets of a finite set V is a connectivity function if it satisfies the following conditions: (1) f (X) + f (Y) ≥ f (X ∩ Y) + f (X ∪ Y) for all subsets X, Y of V, (2) f (X) = f (V {set minus} X) for all X ⊆ V, and (3) f (∅) = 0. Branch-width is defined for graphs, matroids, and more generally, connectivity functions. We show that for each constant k, there is a polynomial-time (in | V |) algorithm to decide whether the branch-width of a connectivity function f is at most k, if f is given by an oracle. This algorithm can be applied to branch-width, carving-width, and rank-width of graphs. In particular, we can recognize matroids M of branch-width at most k in polynomial (in | E (M) |) time if the matroid is given by an independence oracle.

Original language | English (US) |
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Pages (from-to) | 385-393 |

Number of pages | 9 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 97 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2007 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Keywords

- Branch-width
- Carving-width
- Connectivity function
- Rank-width
- Symmetric submodular function
- Tangle

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*97*(3), 385-393. https://doi.org/10.1016/j.jctb.2006.06.006