Reconstruction from smaller cards

TL;DR

This paper advances the understanding of graph reconstruction by showing trees can be reconstructed from smaller decks than previously known, and provides bounds for recognizing connectedness and degree sequences from partial information.

Contribution

The authors prove that trees are reconstructible from their $(n-r)$-decks for all $r o n/9$, and establish new bounds for recognizing connectedness and degree sequences from partial decks.

Findings

Trees are reconstructible from their $(n-r)$-decks for all $r o n/9$.

Connectedness can be recognized from the $rac{9n}{10}$-deck.

Degree sequences can be reconstructed from the $ ext{sqrt}(2n ext{log}(2n))$-deck.

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 $n\ge3$ 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.