### Abstract

Given an undirected graph, a bramble is a set of connected subgraphs (called bramble elements) such that every pair of subgraphs either contains a common node, or such that an edge (i,j) exists with node i belonging to one subgraph and node j belonging to the other. In this paper we examine the problem of finding the bramble number of a graph, along with a set of bramble elements that yields this number. The bramble number is the largest cardinality of a minimum hitting set over all bramble elements on this graph. A graph with bramble number k has a treewidth of k-1. We provide a branch-and-price-and-cut method that generates columns corresponding to bramble elements, and rows corresponding to hitting sets. We then examine the computational efficacy of our algorithm on a randomly generated data set.

Original language | English (US) |
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Pages (from-to) | 168-188 |

Number of pages | 21 |

Journal | Discrete Optimization |

Volume | 18 |

DOIs | |

State | Published - Nov 1 2015 |

Externally published | Yes |

### Keywords

- Bramble
- Branch-and-price
- Integer programming
- Treewidth

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Discrete Optimization*,

*18*, 168-188. https://doi.org/10.1016/j.disopt.2015.09.005