We investigate the number of maximal independent set queries required to reconstruct the edges of a hidden graph. We show that randomised adaptive algorithms need at least $\Omega(\Delta^2 \log(n / \Delta) / \log \Delta)$ queries to reconstruct $n$-vertex graphs of maximum degree $\Delta$ with success probability at least $1/2$, and we further improve this lower bound to $\Omega(\Delta^2 \log(n / \Delta))$ for randomised non-adaptive algorithms. We also prove that deterministic non-adaptive algorithms require at least $\Omega(\Delta^3 \log n / \log \Delta)$ queries. This improves bounds of Konrad, O'Sullivan, and Traistaru, and answers one of their questions. The proof of the lower bound for deterministic non-adaptive algorithms relies on a connection to cover-free families, for which we also improve known bounds.

翻译：我们研究了重构隐藏图边所需的最大独立集查询次数。结果表明，对于最大度为 $\Delta$ 的 $n$ 顶点图，随机自适应算法至少需要 $\Omega(\Delta^2 \log(n / \Delta) / \log \Delta)$ 次查询才能以至少 $1/2$ 的成功概率完成重构；对于随机非自适应算法，我们将此下界进一步改进为 $\Omega(\Delta^2 \log(n / \Delta))$。此外，我们证明了确定性非自适应算法至少需要 $\Omega(\Delta^3 \log n / \log \Delta)$ 次查询。该结果改进了 Konrad、O'Sullivan 和 Traistaru 的界，并回答了他们的一个开放性问题。确定性非自适应算法下界的证明依赖于覆盖自由族（cover-free families）的关联，我们也改进了该族的已知界。