Acyclic graphs with at least $2\ell+1$ vertices are $\ell$-recognizable

TL;DR

This paper proves that acyclic graphs with at least 2ℓ+1 vertices are uniquely identifiable from their (n−ℓ)-decks, establishing conditions under which trees and forests are reconstructible from subgraph data.

Contribution

It demonstrates that the family of acyclic graphs with sufficiently many vertices is ℓ-recognizable, extending graph reconstruction results to broader classes of graphs.

Findings

Acyclic graphs with ≥ 2ℓ+1 vertices are ℓ-recognizable.

The family of trees is ℓ-recognizable when n ≥ 2ℓ+1 and ℓ ≠ 2.

Reconstruction fails for n=2ℓ cases.

Abstract

The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the family is also in the family. We prove that the family of $n$-vertex graphs with no cycles is $\ell$-recognizable when $n\ge2\ell+1$ (except for $(n,\ell)=(5,2)$). As a consequence, the family of $n$-vertex trees is $\ell$-recognizable when $n\ge2\ell+1$ and $\ell\ne2$. It is known that this fails when $n=2\ell$.