Optimal guard sets and the Helly property

Authors:
Gábor Bacsó;Zsolt Tuza
Affiliations:
Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u.13-17, Hungary;Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u.13-17, Hungary and Department of Computer Science and Systems Technology, University of Pannonia, H-8200 ...
Venue:
European Journal of Combinatorics
Year:
2011

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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.