Size reconstructibility of graphs

TL;DR

This paper investigates how many vertex-deleted subgraphs (cards) are needed to reconstruct the number of edges in a large graph, showing that fewer cards than previously known suffice for accurate reconstruction.

Contribution

It improves the bounds on the number of cards required to determine the number of edges in large graphs, reducing from two missing cards to roughly one-twentieth of the vertices.

Findings

Number of edges can be reconstructed from fewer cards than previously established.

For sufficiently large graphs, at most rac{1}{20}\sqrt{n} cards are needed.

The result extends the understanding of graph reconstructibility with fewer subgraphs.

Abstract

The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph $G$ can be computed from any deck missing 2 cards. We show that, for sufficiently large $n$, the number of edges can be computed from any deck missing at most $\frac1{20}\sqrt{n}$ cards.