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 $f:mathbb{P}^1_Ktomathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $mathbb{P}^1_{R_S}tomathbb{P}^1_{R_S}$. A finite Galois invariant subset $Xsubsetmathbb{P}^1_K(bar{K})$ has good reduction outside $S$ if its closure in $mathbb{P}^1_{R_S}$ is etale over $R_S$. We study triples $(f,Y,X)$ with $X=Ycup f(Y)$. 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 $sum_{Pin Y}e_f(P)ge2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Yto X$ is specified, and we give an exhaustive analysis for degree $2$ maps on $mathbb{P}^1$ when $Y=X$.