Portraits of quadratic rational maps with a small critical cycle

Motivated by a uniform boundedness conjecture of Morton and Silverman, we study the graphs of pre-periodic points for maps in three families of dynamical systems, namely the collections of rational functions of degree two having a periodic critical point of period $n$, where $n\in\{2,3,4\}$. In particular, we provide a conjecturally complete list of possible graphs of rational pre-periodic points in the case $n=4$, analogous to well-known work of Poonen for $n=1$, and we strengthen earlier results of Canci and Vishkautsan for $n\in\{2,3\}$. In addition, we address the problem of determining the representability of a given graph in our list by infinitely many distinct linear conjugacy classes of maps.