Almost every $n$-vertex graph is determined by its $3 \log_2{n}$-vertex subgraphs

TL;DR

This paper proves that for almost all graphs with n vertices, the collection of all induced subgraphs on about 3 log2(n) vertices uniquely determines the graph up to isomorphism, simplifying graph isomorphism testing.

Contribution

It establishes that a small, logarithmic-sized subset of subgraphs suffices to identify almost every n-vertex graph uniquely, advancing understanding of graph reconstruction.

Findings

Almost every n-vertex graph is determined by its 3 log2(n)-vertex subgraphs.

The multiset of these subgraphs suffices for graph isomorphism testing.

This result holds with high probability for random graphs.

Abstract

The paper shows that almost every $n$-vertex graph is such that the multiset of its induced subgraphs on $3 \log_2{n}$ vertices is sufficient to determine it up to isomorphism. Therefore, for checking the isomorphism of a pair of $n$-vertex graphs, almost surely the multiset of their $3 \log_2{n}$-vertex subgraphs is sufficient .