Sharp Graham-Sequence Reconstruction for Spiders

For a graph G, let a_j(G) be the number of vertices in its jth iterated line graph. For an n-vertex spider with r >= 3 legs, we prove reconstruction within the class of spiders from the prefix ending at index floor((n-r-1)/2)+2. This bound is sharp whenever n-r-1 >= 2, even with n and r given. A marked-core inversion recovers the number of surviving legs at each step; symmetry eliminates the unknown assignment of those legs. Trees with exactly three leaves are recognizable from a_0 and a_2 and are therefore reconstructed among all trees through index floor(n/2). Separately, two exact implementations verify Graham-sequence separation for all 1,346,024 unlabelled trees of order at most twenty. Neither result resolves Graham's conjecture for arbitrary trees. The full text of the closely related 2024 chapter by King, Lane-Harvard and Milligan was unavailable; overlap with its detailed results has not been excluded, and no absolute priority is claimed. This is an unrefereed preprint. AI assistance was used in developing the arguments, writing the manuscript and implementing the checks; the author is responsible for the claims.