### Abstract

A cut in a graph G is the set of all edges between some set of vertices S and its complement S̄ = V(G) - S. A cut-cover of G is a collection of cuts whose union is E(G) and the total size of a cut-cover is the sum of the number of edges of the cuts in the cover. The cut-cover size of a graph G, denoted by cs(G), is the minimum total size of a cut-cover of G. We give general bounds on cs(G), find sharp bounds for classes of graphs such as 4-colorable graphs and random graphs. We also address algorithmic aspects and give sharp bounds for the sum of the cut-cover sizes of a graph and its complement. We close with a list of open problems.

Original language | English |
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Pages (from-to) | 129-148 |

Number of pages | 20 |

Journal | Discrete Mathematics |

Volume | 237 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Jun 28 2001 |

### Keywords

- Average distance
- Cut
- Geometric representation
- Minimum cover
- Nordhaus-Gaddum
- Random graphs

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Füredi, Z., & Kündgen, A. (2001). Covering a graph with cuts of minimum total size.

*Discrete Mathematics*,*237*(1-3), 129-148. https://doi.org/10.1016/S0012-365X(00)00367-8