On the existence of a connected component of a graph
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.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1406.4786
 Bibcode:
 2014arXiv1406.4786G
 Keywords:

 Mathematics  Logic;
 03B30 (Primary);
 03D30;
 03F35;
 03D45 (Secondary)
 EPrint:
 25 pages, 3 figures. Versions 2 and 3 include additional results related to Weihrauch reducibility