## Februaru 1, 2019:

# Graph Reconstruction and Verification

## Pedro Matias

How efficiently can we find an unknown graph using distance or shortest path queries between its vertices? We assume that the unknown graph G is connected, unweighted, and has bounded degree. In the reconstruction problem, the goal is to find the graph G. In the verification problem, we are given a hypothetical graph Ĝ and want to check whether G is equal to Ĝ.
We provide a randomized algorithm for reconstruction using Õ(n^{3/2}) distance queries, based on Voronoi cell decomposition. Next, we analyze natural greedy algorithms for reconstruction using a shortest path oracle and also for verification using either oracle, and show that their query complexity is n^{1+o(1)}. We further improve the query complexity when the graph is chordal or outerplanar. Finally, we show some lower bounds, and consider an approximate version of the reconstruction problem.

Paper by Sampath Kannan, Claire Mathieu and Hang Zhou [TALG 18]
https://dl.acm.org/citation.cfm?id=3199606