  We apply the techniques of computable model theory to the distance function
of a graph. This task leads us to adapt the definitions of several truth-table
reducibilities so that they apply to functions as well as to sets, and we prove
assorted theorems about the new reducibilities and about functions which have
nonincreasing computable approximations. Finally, we show that the spectrum of
the distance function can consist of an arbitrary single btt-degree which is
approximable from above, or of all such btt-degrees at once, or of the
bT-degrees of exactly those functions approximable from above in at most n
steps.
