On the existence of a connected component of a graph

@article{Gura2015OnTE,
  title={On the existence of a connected component of a graph},
  author={Kirill Gura and Jeffry L. Hirst and Carl Mummert},
  journal={Comput.},
  year={2015},
  volume={4},
  pages={103-117}
}
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… 

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