Caterpillars with $n$ vertices are reconstructible from subgraphs with at most $n/2+1$ vertices
Alexandr V. Kostochka, Zishen Qu, Maddy Ritter, Douglas B. West
TL;DR
This work proves that for n ≥ 48, every n-vertex caterpillar is reconstructible from its m-deck when m > n/2, with the bound shown to be sharp via explicit examples. The authors build a two-pronged approach: a low-diameter analysis that recovers the spine and degree sequence from small-spine subgraphs, and a high-diameter strategy that anchors the reconstruction with high-degree vertices using maximal batons and tritons, supported by a robust counting framework. A sequence of case analyses—covering scenarios with four or more maximum-degree vertices, three or two maximum-degree vertices, and the unique-maximum-degree-vertex, as well as cases with at most three branch vertices—establishes reconstruction across the caterpillar family. The results advance the broader reconstruction program for trees, providing a sharp, structure-specific confirmation of Nydl’s conjecture for caterpillars and highlighting powerful deck-based counting techniques (batons, tritons, level-pairs) for spine-structure recovery.
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 Nýdl about trees.