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. Furthermore, 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? We derive 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.