Reconstructing the degree sequence of a sparse graph from a partial deck

TL;DR

This paper demonstrates that the degree sequence of a sparse graph can be reconstructed from a partial deck, even when a significant number of cards are missing, extending previous results to graphs with lower average degree.

Contribution

It proves that the degree sequence of a graph with average degree d can be reconstructed from a deck missing O(n/d^3) cards, generalizing prior work on complete decks.

Findings

Degree sequence can be reconstructed from partial decks in sparse graphs.

Reconstruction is possible even with a linear number of missing cards for certain graph classes.

Extends reconstruction results to graphs embeddable on fixed surfaces.

Abstract

The deck of a graph $G$ is the multiset of cards $\{G-v:v\in V(G)\}$. Myrvold (1992) showed that the degree sequence of a graph on $n\geq7$ vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree $d$ can reconstructed from any deck missing $O(n/d^3)$ cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.