The classification problem for graphs and lattices is wild

TL;DR

This paper demonstrates that classifying graphs and certain algebraic lattices up to isomorphism is as complex as classifying pairs of matrices up to simultaneous similarity, indicating a highly intricate problem.

Contribution

It establishes the wildness of the classification problem for graphs and specific algebraic lattices by linking it to the known complex problem of classifying matrix pairs.

Findings

Classification problem for graphs is wild.

Classification problem for algebraic lattices is wild.

Links to matrix pair classification complexity.

Abstract

We prove that the classification problem for graphs and several types of algebraic lattices (distributive, congruence and modular) up to isomorphism contains the classification problem for pairs of matrices up to simultaneous similarity.