Optimally reconstructing caterpillars

TL;DR

This paper demonstrates that caterpillar graphs can be optimally reconstructed from approximately half of their induced subgraphs, improving previous results for trees and establishing tight bounds.

Contribution

It proves that caterpillar graphs can be reconstructed from a deck of size roughly half the number of vertices, which is asymptotically optimal.

Findings

Caterpillar graphs are reconstructible from their (1/2+o(1))n-deck.

This result improves upon the general tree reconstruction bound.

The bound for caterpillars is shown to be asymptotically tight.

Abstract

For a graph $G$, the $\ell$-deck of $G$ is the multiset of induced subgraphs on $G$ having $\ell$ vertices. Recently, Groenland et al. proved that any tree can be reconstructed from its $(8/9+o(1))n$-deck. For the particular case of caterpillar graphs, we show that the $(1/2+o(1))n$-deck suffices, which is asymptotically tight.