Reconstruction from smaller cards

The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of
$G$ on $\ell$ vertices. We say that a graph is reconstructible from its
$\ell$-deck if no other graph has the same $\ell$-deck. In 1957, Kelly showed
that every tree with $n\ge3$ 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
$r\le n/{9}+o(n)$, making substantial progress towards a conjecture of N\'ydl
from 1990. In addition, we can recognise the connectedness of a graph from its
$\ell$-deck when $\ell\ge 9n/10$, and reconstruct the degree sequence when
$\ell\ge\sqrt{2n\log(2n)}$. All of these results are significant improvements
on previous bounds.