Trees with at least $6\ell +11$ vertices are $\ell$-reconstructible

Abstract

The $(n-\ell )$-deck of an $n$-vertex graph is the multiset of (unlabeled) subgraphs obtained from it by deleting $\ell$ vertices. An $n$-vertex graph is $\ell$-reconstructible if it is determined by its $(n-\ell )$-deck, meaning that no other graph has the same deck. We prove that every tree with at least $6\ell +11$ vertices is $\ell$-reconstructible.