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.

The second author was partially supported by an NSF postdoctoral research fellowship.