Reconstruction from smaller cards

Abstract The ℓ-deck of a graph G is the multiset of all induced subgraphs of G on ℓ vertices. We say that a graph is reconstructible from its ℓ-deck if no other graph has the same ℓ-deck. In 1957, Kelly showed that every tree with n ≥ 3 vertices can be reconstructed from its (n − 1) -deck, and Giles strengthened this in 1976, proving that trees on at least 6 vertices can be reconstructed from their (n − 2) -decks. Our main theorem states that trees are reconstructible from their (n − r) -decks for all r ≤ n /9 + o (n), making substantial progress towards a conjecture of Nýdl from 1990. In addition, we can recognise the connectedness of a graph from its ℓ-deck when ℓ ≥ 9 n /10, and reconstruct the degree sequence when 2n (2n) ℓ ≥ 2 n log (2 n). All of these results are significant improvements on previous bounds.