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. In this paper, we initiate the study of GR via Maximal Independent Set (MIS) queries, a more powerful variant of Independent Set (IS) queries. Given a query $U \subseteq V$, the oracle responds with any, potentially adversarially chosen, maximal independent set $I \subseteq U$ 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 $\Delta$ 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 $O(\Delta^2 \cdot \log n)$ queries and succeeds with high probability. Furthermore, 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 $O(\Delta^3 \cdot \log(\frac{n}{\Delta}))$ queries. 2. Regarding lower bounds, we prove that the additional $\Delta$ factor when going from randomized non-adaptive algorithms to deterministic non-adaptive algorithms is necessary. We show that every non-adaptive deterministic algorithm requires $\Omega(\Delta^3 / \log^2 \Delta)$ queries. For arbitrary randomized adaptive algorithms, we show that $\Omega(\Delta^2)$ queries are necessary in graphs of maximum degree $\Delta$, and that $\Omega(\log n)$ queries are necessary, even when the input graph is an $n$-vertex cycle.

翻译：在图重构（GR）问题中，玩家初始仅知道输入图 $G=(V, E)$ 的顶点集 $V$，需要学习其边集 $E$。为此，玩家向一个预言机提交查询，并必须从预言机的回答中推断出 $E$。本文首次研究了通过最大独立集（MIS）查询进行图重构的问题，这是独立集（IS）查询的一个更强大的变体。给定一个查询 $U \subseteq V$，预言机返回诱导子图 $G[U]$ 中的任意一个（可能是对抗性选择的）最大独立集 $I \subseteq U$。我们证明，在图重构问题中，当以输入图的最大度 $\Delta$ 为参数时，MIS 查询严格强于 IS 查询。我们给出了该问题紧确（忽略多对数因子）的上界和下界：1. 我们观察到，对 $V$ 进行均匀独立随机采样并将样本提交给预言机的简单策略，产生了一个非自适应随机化算法，该算法执行 $O(\Delta^2 \cdot \log n)$ 次查询并以高概率成功。此外，结合对 $V$ 进行均匀随机采样的策略与概率方法，我们证明存在一个确定性的非自适应算法，执行 $O(\Delta^3 \cdot \log(\frac{n}{\Delta}))$ 次查询。2. 关于下界，我们证明了从随机化非自适应算法到确定性非自适应算法所需的额外 $\Delta$ 因子是必要的。我们证明每个非自适应确定性算法都需要 $\Omega(\Delta^3 / \log^2 \Delta)$ 次查询。对于任意的随机化自适应算法，我们证明在最大度为 $\Delta$ 的图中需要 $\Omega(\Delta^2)$ 次查询，并且即使输入图是一个 $n$ 顶点环，也需要 $\Omega(\log n)$ 次查询。