Optimal guard sets and the Helly property

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.