Searching for an edge in a graph with restricted test sets

Consider the ( 2 , n ) group testing problem with test sets of cardinality at most 2. We determine the worst case number c 2 of tests for this restricted group testing problem. rmore, using a game theory approach we solve the generalization of this group testing problem to the following search problem, which was suggested by Aigner in [M. Aigner, Combinatorial Search, Wiley-Teubner, 1988]: Suppose a graph G ( V , E ) contains one defective edge e . We search for the endpoints of e by asking questions of the form “Is at least one of the vertices of X an endpoint of e ?”, where X is a subset of V with | X | ≤ 2 . What is the minimum number c 2 ( G ) of questions, which are needed in the worst case to identify e ? ive sharp upper and lower bounds for c 2 ( G ) . We also show that the determination of c 2 ( G ) is an NP-complete problem. Moreover, we establish some results on c 2 for random graphs.