Abstract. We want to find the vertex sets of components of a graph G with a known vertex set V and unknown edge set E. We learn about G by sending an oracle a query set S ⊆ V , and the oracle tells us the vertices connected to S. The objective is to use the minimum number of queries to partition the vertex set into components. The problem is also known as interconnect diagnosis of wiring networks in VLSI. We present a deterministic algorithm using O(min{k, lg n}) queries and a randomized algorithm using expected O(min{k, lg k + lg lg n}) queries, where n is the number of vertices and k is the number of components. We also prove matching lower bounds.Key words. randomized algorithm, lower bound, fault diagnosis, graph, component, connection class.AMS subject classifications. 68Q25, 68R10, 05C85, 05C40, 94C121. Introduction. We study how to find the vertex sets of components of an unknown undirected graph G = (V, E) on a known vertex set V . Vertices u and v are connected if there is a path between them. The components of G are its maximal connected subgraphs. The connection relation is an equivalence relation on the vertex set V , and the vertex sets of the components are the equivalence classes of the connection relation, also called the connection classes. When we say "finding the components", we mean finding the connection classes. In our problem, we are given V but not E. We do not know the number of components or their sizes. The only operation we may use to obtain information about G is to query an oracle. For any query set S ⊆ V , the oracle will tell us Q(S), the set of vertices connected to vertices of S: