We study the problem of combinatorial search for graphs under the additive model. The main result concerns the reconstruction of bounded degree graphs, i.e. graphs with the degree of all vertices bounded by a constant d. We show that such graphs can be reconstructed in O (dn) non-adaptive queries, that matches the information-theoretic lower bound. The proof is based on the technique of separating matrices. In particular, a new upper bound is obtained for d-separating matrices, that settles an open question stated by Lindström in [16]. Finally, we consider several particular classes of graphs. We show how an optimal non-adaptive solution of $O(n^2/\log n)$ queries for general graphs can be obtained.