Shuffling the Deck: Invariant Theory and the Graph Reconstruction Conjecture

TL;DR

This paper surveys the graph reconstruction conjecture and introduces an algebraic approach using invariant theory to distinguish graphs via polynomial invariants.

Contribution

It presents an invariant-theoretic framework to translate the graph reconstruction problem into an algebraic context, offering new tools for analysis.

Findings

Proposes an algebraic method to study the conjecture

Shows polynomials can distinguish graphs and their decks

Bridges graph theory with invariant theory

Abstract

The graph reconstruction conjecture asserts that every simple graph on at least three vertices is uniquely determined by its deck of vertex-deleted subgraphs. In this expository article we survey the conjecture and present an invariant-theoretic approach to studying it. The aim is to be able to show that polynomials that distinguish between decks also distinguish between original graphs, thus translating a graph-theoretic problem into an algebraic one.