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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics