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 -node -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 queries in expectation are both sufficient and necessary to adaptively reconstruct the graph. In contrast, we show that non-adaptive queries are required, even when . We also provide an query algorithm using only two rounds of adaptivity.
View on arXiv@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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