On the existence of a connected component of a graph

TL;DR

This paper explores the logical and computability aspects of countable graph theory, establishing equivalences with foundational systems and analyzing the complexity of finding connected components.

Contribution

It characterizes the logical strength of the principle that every countable graph has a connected component and relates it to reverse mathematics and Weihrauch degrees.

Findings

The existence of a connected component is equivalent to ACA_0 over RCA_0.

Decomposition into connected components is Weihrauch equivalent to finding a single component.

For finitely many components, the problem's strength varies with the formulation, relating to Σ^0_2 induction.

Abstract

We study the reverse mathematics and computability of countable graph theory, obtaining the following results. The principle that every countable graph has a connected component is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$. The problem of decomposing a countable graph into connected components is strongly Weihrauch equivalent to the problem of finding a single component, and each is equivalent to its infinite parallelization. For graphs with finitely many connected components, the existence of a connected component is either provable in $\mathsf{RCA}_0$ or is equivalent to induction for $\Sigma^0_2$ formulas, depending on the formulation of the bound on the number of components.