A graph G = (V,E) is recursive if every node of G has a finite number
of neighbors, and both V and E are recursive (i.e., decidable). We exa
mine the complexity of identifying the number of connected components
of an infinite recursive graph, and other related problems, both when
an upper bound to that value is given a priori or not. The problems th
at we deal with are unsolvable, but are recursive in some level of the
arithmetic hierarchy. Our measure of the complexity of these problems
is precise in two ways: the Turing degree of the oracle, and the numb
er of queries to that oracle. Although they are in several different l
evels of the arithmetic hierarchy, all problems addressed have the sam
e upper and lower bounds for the number of queries as the binary searc
h problem, both in the bounded and in the unbounded case.