A classification of bicritical dynamic portraits

The dynamic of a rational map $f:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ is determined by the forward orbits of its critical points. Such a map is called {\em postcritically finite} if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle. These orbits can be encoded in a finite directed graph called a {\em dynamic portrait}. In this work, we classify which abstract bicritical dynamic portraits of degree $d\ge 2$ with at least 4 postcritical points are realizable exclusively by rational Thurston maps.