Caterpillars with $n$ vertices are reconstructible from subgraphs with at most $n/2+1$ vertices

TL;DR

This paper proves that for large enough caterpillar trees with at least 48 vertices, the entire structure can be reconstructed from subgraphs with more than half of the vertices, confirming a special case of a longstanding conjecture.

Contribution

It establishes that $n$-vertex caterpillars are reconstructible from their $m$-deck when $m>n/2$, for all sufficiently large $n$, confirming a special case of Nydl's conjecture.

Findings

Reconstruction is possible for $n extgreater 48$ with $m>n/2$.

Sharpness of the result shown by counterexamples for $m=loor{n/2}$.

Confirms a special case of Nydl's 1990 conjecture for trees.

Abstract

The $\textit{$m$-deck}$ of an $n$-vertex graph is the multiset of unlabeled induced subgraphs with $m$ vertices. Caterpillars are trees in which all nonleaf vertices lie on a single path. We prove for $n\ge48$ that any $n$-vertex caterpillar is reconstructible (up to isomorphism) from its $m$-deck when $m>n/2$. The result is sharp, since for $n\ge6$ there are two $n$-vertex caterpillars having the same $\lfloor n/2 \rfloor$-deck. Our result proves the special case for caterpillars of a 1990 conjecture by N\'ydl about trees.