Embeddability of graphs and Weihrauch degrees

TL;DR

This paper analyzes the computational complexity of subgraph isomorphism problems for fixed countable graphs using Weihrauch and Wadge reducibility frameworks, building on prior reverse mathematics research.

Contribution

It introduces a detailed complexity analysis of subgraph isomorphism tasks within the Weihrauch degrees framework, addressing open questions from previous work.

Findings

Classifies the complexity of subgraph isomorphism problems

Establishes connections between graph embeddability and Weihrauch degrees

Provides new insights into the computational structure of graph problems

Abstract

We study the complexity of the following related computational tasks concerning a fixed countable graph G: 1. Does a countable graph H provided as input have a(n induced) subgraph isomorphic to G? 2. Given a countable graph H that has a(n induced) subgraph isomorphic to G, find such a subgraph. The framework for our investigations is given by effective Wadge reducibility and by Weihrauch reducibility. Our work follows on "Reverse mathematics and Weihrauch analysis motivated by finite complexity theory" (Computability, 2021) by BeMent, Hirst and Wallace, and we answer several of their open questions.