  To each quadratic number field $K$ and each quadratic polynomial $f$ with
$K$-coefficients, one can associate a finite directed graph $G(f,K)$ whose
vertices are the $K$-rational preperiodic points for $f$, and whose edges
reflect the action of $f$ on these points. This paper has two main goals. (1)
For an abstract directed graph $G$, classify the pairs $(K,f)$ such that the
isomorphism class of $G$ is realized by $G(f,K)$. We succeed completely for
many graphs $G$ by applying a variety of dynamical and Diophantine techniques.
(2) Give a complete description of the set of isomorphism classes of graphs
that can be realized by some $G(f,K)$. A conjecture of Morton and Silverman
implies that this set is finite. Based on our theoretical considerations and a
wealth of empirical evidence derived from an algorithm that is developed in
this paper, we speculate on a complete list of isomorphism classes of graphs
that arise from quadratic polynomials over quadratic fields.
