Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:{\mathbb{P}}_K^1\to {\mathbb{P}}_K^1$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism ${\mathbb{P}}_{R_S}^1\to {\mathbb{P}}_{R_S}^1$. A finite Galois invariant subset $X\subset {\mathbb{P}}_K^1(\bar{K})$ has good reduction outside $S$ if its closure in ${\mathbb{P}}_{R_S}^1$ is 'etale over $R_S$. We study triples $(f,Y,X)$ with $X=Y\cup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many ${\text{PGL}}_2(R_S)$-equivalence classes of triples with $\text{deg}(f)=d$ and ${\sum }_{P\in Y}{e}_f(P)\ge 2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Y\to X$ is specified, and we give an exhaustive analysis for degree $2$ maps on ${\mathbb{P}}^1$ when $Y=X$.