# Degree lists and connectedness are $3$-reconstructible for graphs with at least seven vertices

## TL;DR

This paper proves that for graphs with at least seven vertices, the degree list and connectedness can be uniquely reconstructed from specific subgraph collections, extending previous results and establishing sharp thresholds.

## Contribution

It establishes that degree lists and connectedness are 3-reconstructible for graphs with at least seven vertices, improving upon earlier 2-reconstructibility results.

## Key findings

- Degree list is 3-reconstructible for n ≥ 7
- Connectedness is 3-reconstructible for n ≥ 7
- Results extend and sharpen previous 2-reconstructibility findings

## Abstract

The $k$-deck of a graph is the multiset of its subgraphs induced by $k$ vertices. A graph or graph property is $l$-reconstructible if it is determined by the deck of subgraphs obtained by deleting $l$ vertices. We show that the degree list of an $n$-vertex graph is $3$-reconstructible when $n\ge7$, and the threshold on $n$ is sharp. Using this result, we show that when $n\ge7$ the $(n-3)$-deck also determines whether an $n$-vertex graph is connected; this is also sharp. These results extend the results of Chernyak and Manvel, respectively, that the degree list and connectedness are $2$-reconstructible when $n\ge6$, which are also sharp.