The determination of defective elements in a population by a series of group tests goes back to questions arising in connection with medical examinations during the second world war. Some natural generalizations to graphs have been studied by Aigner, Andreae, Chang, Hwang, Lin and others. In this paper we investigate the following binary variant of search problem: Given a graph $G$ with vertex-set $V$ and edge-set $E$, and an unknown edge $e^*\in E$. In order to find $e^*$ we choose a sequence of test-sets $A\subseteq V$ where after every test we are told whether $e^*$ has at least one end-vertex in $A$ or none. We Eire asked to find the minimum number $\bar c(G)$ of tests required. Chang, Hwang and Lin have studied $\bar c$ for $G=K_{m,n}$ and $G=K_n$ in [5,6]. We extend some of their results to some other classes of graphs, which include planar graphs.