Graph Reconstruction with a Connected Components Oracle
Juha Harviainen, Pekka Parviainen · arXiv (Cornell University) · 2025
In the Graph Reconstruction (GR) problem, the goal is to recover a hidden graph by utilizing some oracle that provides limited access to the structure of the graph. The interest is in characterizing how strong different oracles are when the complexity of an algorithm is measured in the number of performed queries. We study a novel oracle that returns the set of connected components (CC) on the subgraph induced by the queried subset of vertices. Our main contributions are as follows: 1) For a hidden graph with n vertices, m edges, maximum degree Δ, and treewidth k, GR can be solved in 𝒪(min{m/log m, Δ², k²} ⋅ log n) CC queries by an adaptive randomized algorithm. 2) For a hidden graph with n vertices and degeneracy d, GR can be solved in 𝒪(d² log² n) CC queries by an adaptive randomized algorithm. 3) There are hidden graphs with n vertices, m edges, maximum degree Δ, treewidth k, and degeneracy d such that Ω(m), Ω(Δ²), Ω(k²), and Ω(d²) CC queries are required for solving GR.