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oai:drops-oai.dagstuhl.de:22694 2026-04-17T05:32:24Z ddc:004 open_access

Graph Reconstruction via MIS Queries Konrad, Christian O'Sullivan, Conor Traistaru, Victor Query Complexity Graph Reconstruction Maximal Independent Set Queries In the Graph Reconstruction (GR) problem, a player initially only knows the vertex set V of an input graph G = (V, E) and is required to learn its set of edges E. To this end, the player submits queries to an oracle and must deduce E from the oracle’s answers. Angluin and Chen [Journal of Computer and System Sciences, 2008] resolved the number of Independent Set (IS) queries necessary and sufficient for GR on m-edge graphs. In this setting, each query consists of a subset of vertices U ⊆ V, and the oracle responds with a boolean, indicating whether U is an independent set in G. They gave algorithms that use O(m ⋅ log n) IS queries, which is best possible. In this paper, we initiate the study of GR via Maximal Independent Set (MIS) queries, a more powerful variant of IS queries. Given a query U ⊆ V, the oracle responds with any, potentially adversarially chosen, maximal independent set I ⊆ U in the induced subgraph G[U]. We show that, for GR, MIS queries are strictly more powerful than IS queries when parametrized by the maximum degree Δ of the input graph. We give tight (up to poly-logarithmic factors) upper and lower bounds for this problem: 1) We observe that the simple strategy of taking uniform independent random samples of V and submitting those to the oracle yields a non-adaptive randomized algorithm that executes O(Δ² ⋅ log n) queries and succeeds with high probability. This should be contrasted with the fact that Ω(Δ ⋅ n ⋅ log(n/Δ)) IS queries are required for such graphs, which shows that MIS queries are strictly more powerful than IS queries. Interestingly, combining the strategy of taking uniform random samples of V with the probabilistic method, we show the existence of a deterministic non-adaptive algorithm that executes O(Δ³ ⋅ log(n/Δ)) queries. 2) Regarding lower bounds, we prove that the additional Δ factor when going from randomized non-adaptive algorithms to deterministic non-adaptive algorithms is necessary. We show that every non-adaptive deterministic algorithm requires Ω(Δ³ / log² Δ) queries. For arbitrary randomized adaptive algorithms, we show that Ω(Δ²) queries are necessary in graphs of maximum degree Δ, and that Ω(log n) queries are necessary, even when the input graph is an n-vertex cycle. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Christian Konrad and Conor O'Sullivan and Victor Traistaru 2025 Is Part Of LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2025.66 urn:nbn:de:0030-drops-226945 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.66 eng https://creativecommons.org/licenses/by/4.0/legalcode