Optimal guard sets and the Helly property

Gábor Bacsó, Z. Tuza

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 languageEnglish
Pages (from-to)28-32
Number of pages5
JournalEuropean Journal of Combinatorics
Volume32
Issue number1
DOIs
Publication statusPublished - Jan 2011

Fingerprint

Set Systems
Algorithmic Complexity
Perfect Graphs
Intersection Graphs
Hypergraph
Clique
Intersect
Subset
Graph in graph theory
Theorem

ASJC Scopus subject areas

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

Cite this

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

In: European Journal of Combinatorics, Vol. 32, No. 1, 01.2011, p. 28-32.

Research output: Contribution to journalArticle

Bacsó, Gábor ; Tuza, Z. / Optimal guard sets and the Helly property. In: European Journal of Combinatorics. 2011 ; Vol. 32, No. 1. pp. 28-32.
@article{73f363704b184539a96047c1b737f439,
title = "Optimal guard sets and the Helly property",
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.",
author = "G{\'a}bor Bacs{\'o} and Z. Tuza",
year = "2011",
month = "1",
doi = "10.1016/j.ejc.2010.08.001",
language = "English",
volume = "32",
pages = "28--32",
journal = "European Journal of Combinatorics",
issn = "0195-6698",
publisher = "Academic Press Inc.",
number = "1",

}

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

VL - 32

SP - 28

EP - 32

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 1

ER -