Exponential-in-n^2 lower bound for arbitrary-graph reconstruction under 1/2 edge-deletion noise

Prove that, in the edge deletion model with vertex-sampling probability p_v = 1 and edge-retention probability p_e = 1/2 (i.e., each edge is independently deleted with probability 1/2 and vertices are not subsampled), reconstructing an arbitrary undirected graph on n vertices from unlabeled traces requires exp(Ω(n^2)) traces in the worst case.

Background

The paper establishes that for arbitrary graphs, exp(Ω(n)) traces are necessary to distinguish certain non-isomorphic graphs when p_v = 1 and p_e = 1/2 in the edge deletion model, contrasting with much lower sample complexity for random graphs.

The authors hypothesize that this lower bound is not tight and that the true worst-case sample complexity should be exp(Ω(n^2)), which would match the trivial exp(O(n^2)) upper bound arising because one trace may equal the entire graph with that many samples.