  We consider a generalization of group testing where the potentially
contaminated sets are the members of a given hypergraph ${\cal F}=(V,E)$. This
generalization finds application in contexts where contaminations can be
conditioned by some kinds of social and geographical clusterings. We study
non-adaptive algorithms, two-stage algorithms, and three-stage algorithms.
Non-adaptive group testing algorithms are algorithms in which all tests are
decided beforehand and therefore can be performed in parallel, whereas
two-stage group testing algorithms and three-stage group testing algorithms are
algorithms that consist of two stages and three stages, respectively, with each
stage being a completely non-adaptive algorithm. In classical group testing,
the potentially infected sets are all subsets of up to a certain number of
elements of the given input set. For classical group testing, it is known that
there exists a correspondence between classical superimposed codes and
non-adaptive algorithms, and between two stage algorithms and selectors. Bounds
on the number of tests for those algorithms are derived from the bounds on the
dimensions of the corresponding combinatorial structures. Obviously, the upper
bounds for the classical case apply also to our group testing model. In the
present paper, we aim at improving on those upper bounds by leveraging on the
characteristics of the particular hypergraph at hand. In order to cope with our
version of the problem, we introduce new combinatorial structures that
generalize the notions of classical selectors and superimposed codes.
