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
$\left[b\right]{\mathbb{P}}_{{R}_{S}}^{1}\to {\mathbb{P}}_{{R}_{S}}^{1}$. A finite Galois
invariant subset
$X\subset {\mathbb{P}}_{K}^{1}\left(\stackrel{\u0304}{K}\right)$ has
good reduction outside
$S$
if its closure in
${\mathbb{P}}_{{R}_{S}}^{1}$
is étale over
${R}_{S}$.
We study triples
$\left(f,Y,X\right)$
with
$X=Y\cup f\left(Y\right)$. We prove
that for a fixed
$K$,
$S$, and
$d$, there are only finitely
many
${PGL}_{2}\left({R}_{S}\right)$equivalence
classes of triples with
$deg\left(f\right)=d$
and
${\sum}_{P\in Y}{e}_{f}\left(P\right)\ge 2d+1$ 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
$Y=X$.
