The determination of defective elemets in a population by a series of group tests has received considerable attention. In this paper, the following natural generalization to graphs is studied. Given a graph G with vertex-set V and edge-set E, and an unknown edge . In order to find we choose a sequence of tests-sets A ⊆ V where after every test we are told whether has both end-vertices in A, one end-vertex, or lies outside. Find the minimum c(G) of tests required. c(G) is studied in detail for the complete graphs Kn and the complete bipartite graphs Km, n. Remarks are made on optimal graphs which achieve the information-theoretic lower bound and on a previously studied binary variant.