### Abstract

In a set system F, a guard set of an F∈F is a subset B⊂F such that B intersects all those F'∈F which meet F but are not contained in F. Given a graph G, we consider set systems F whose intersection graph is G, and determine one such F in which the guard sets of all F∈F are as small as possible. We prove that the minimum-both in global and local sense-is attained by the dual of the clique hypergraph of G, a structure which also played an important role in the proof of the Perfect Graph Theorem. We also put some remarks concerning algorithmic complexity.

Original language | English |
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Pages (from-to) | 28-32 |

Number of pages | 5 |

Journal | European Journal of Combinatorics |

Volume | 32 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2011 |

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### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*European Journal of Combinatorics*,

*32*(1), 28-32. https://doi.org/10.1016/j.ejc.2010.08.001

**Optimal guard sets and the Helly property.** / Bacsó, Gábor; Tuza, Z.

Research output: Contribution to journal › Article

*European Journal of Combinatorics*, vol. 32, no. 1, pp. 28-32. https://doi.org/10.1016/j.ejc.2010.08.001

}

TY - JOUR

T1 - Optimal guard sets and the Helly property

AU - Bacsó, Gábor

AU - Tuza, Z.

PY - 2011/1

Y1 - 2011/1

N2 - In a set system F, a guard set of an F∈F is a subset B⊂F such that B intersects all those F'∈F which meet F but are not contained in F. Given a graph G, we consider set systems F whose intersection graph is G, and determine one such F in which the guard sets of all F∈F are as small as possible. We prove that the minimum-both in global and local sense-is attained by the dual of the clique hypergraph of G, a structure which also played an important role in the proof of the Perfect Graph Theorem. We also put some remarks concerning algorithmic complexity.

AB - In a set system F, a guard set of an F∈F is a subset B⊂F such that B intersects all those F'∈F which meet F but are not contained in F. Given a graph G, we consider set systems F whose intersection graph is G, and determine one such F in which the guard sets of all F∈F are as small as possible. We prove that the minimum-both in global and local sense-is attained by the dual of the clique hypergraph of G, a structure which also played an important role in the proof of the Perfect Graph Theorem. We also put some remarks concerning algorithmic complexity.

UR - http://www.scopus.com/inward/record.url?scp=77957788228&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957788228&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2010.08.001

DO - 10.1016/j.ejc.2010.08.001

M3 - Article

AN - SCOPUS:77957788228

VL - 32

SP - 28

EP - 32

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 1

ER -