Optimal Reconstruction of Graphs under the Additive Model

Abstract.

We study the problem of reconstructing unknown graphs under the additive combinatorial search 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) nonadaptive queries, which matches the information-theoretic lower bound. The proof is based on the technique of separating matrices. A central result here is a new upper bound for a general class of separating matrices. As a particular case, we obtain a tight upper bound for the class of d -separating matrices, which settles an open question stated by Lindström in [20]. Finally, we consider several particular classes of graphs. We show how an optimal nonadaptive solution of O(n ² / log n) queries for general graphs can be obtained. We also prove that trees with unbounded vertex degree can be reconstructed in a linear number of queries by a nonadaptive algorithm.