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Optimal Graph Reconstruction by Counting Connected Components in Induced Subgraphs

Main:18 Pages
Bibliography:5 Pages
Abstract

The graph reconstruction problem has been extensively studied under various query models. In this paper, we propose a new query model regarding the number of connected components, which is one of the most basic and fundamental graph parameters. Formally, we consider the problem of reconstructing an nn-node mm-edge graph with oracle queries of the following form: provided with a subset of vertices, the oracle returns the number of connected components in the induced subgraph. We show Θ(mlognlogm)\Theta(\frac{m \log n}{\log m}) queries in expectation are both sufficient and necessary to adaptively reconstruct the graph. In contrast, we show that Ω(n2)\Omega(n^2) non-adaptive queries are required, even when m=O(n)m = O(n). We also provide an O(mlogn+nlog2n)O(m\log n + n\log^2 n) query algorithm using only two rounds of adaptivity.

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@article{black2025_2506.08405,
  title={ Optimal Graph Reconstruction by Counting Connected Components in Induced Subgraphs },
  author={ Hadley Black and Arya Mazumdar and Barna Saha and Yinzhan Xu },
  journal={arXiv preprint arXiv:2506.08405},
  year={ 2025 }
}
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