  We provide a complete classification of possible graphs of rational
preperiodic points of quadratic rational functions defined over the rationals
with a rational periodic critical point of period 3, under two assumptions:
that these functions have no periodic points of period at least 5 and the
conjectured enumeration of points on a certain genus 6 affine plane curve. We
show that there are exactly six such possible graphs, and that rational
functions satisfying the conditions above have at most eleven rational
preperiodic points.
