On the existence of a connected component of a graph
  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. 
Kirill Gura, Jeffry L. Hirst, and Carl Mummert
math.LO