Reconstruction from smaller cards

Abstract

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 $n\ge 3$ 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 $r\le n/9+o(n)$, making substantial progress towards a conjecture of Nýdl from 1990. In addition, we can recognise the connectedness of a graph from its $\ell$-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.

29 pages, improvements to exposition and proof clarity