Here is a stacky variant of Will Savin's answer, with a more precise result. The reference is my paper:

*Problèmes de Skolem sur les champs algébriques*, Compo. Math. 125 (2001), 1—30

(1) First, note the following: Let $F$ be a local field, $Y$ an $F$-scheme of finite type, $C\to Y$ a stable curve of genus $g$ over $Y$, and $\Gamma$ a stable graph of genus $g$. Let $U$ (resp. $U'\subset U$) be the set of points $y\in Y(F)$ such that $C_y$ has (potential) reduction type $\Gamma$ (resp. has stable reduction on $O_F$, with reduction type $\Gamma$). Then both $U$ and $U'$ are open in $Y(F)$, for the valuation topology. (Exercise).

(2) Now let $I'\subset I$ correspond to singular curves (i.e. we remove the one-point graph from $I$).

For each $i\in I'$, fix a prime $p_i$ (they have to be pairwise distinct!). I will make the assumption that

(*) *there exists a smooth curve over $\mathbb{Q}_{p_i}$ with potential reduction graph $\Gamma_i$*.

(I am pretty sure that this is always true; otherwise, choose $p_i$ accordingly).

Put $R:=\mathbb{Z}\left[(1/p_i)_{i\in I'}\right]$.

**Theorem.** *There is a number field $K$, and a stable curve $\mathscr{X}$ over $O_K$ such that:*

• $\mathscr{X}$ is smooth over $R\otimes_{\mathbb{Z}}O_K$;

• for each $i\in I'$ and each prime $\mathfrak{p}$ of $O_K$ above $p_i$, $\mathscr{X}$ has reduction graph $\Gamma_i$ at $\mathfrak{p}$.

Assume further that for each $i\in I'$ there is a stable curve over $\mathbb{F}_{p_i}$ with graph $\Gamma_i$. (This holds if $p_i$ is large enough). *Then we can take $K$ totally split at each $p_i$.*

**Proof**: Consider the moduli stack $\mathscr{M}_g$ (resp. $\overline{\mathscr{M}}_g$) of smooth (resp. stable) curves of genus $g$.

Let us prove the first claim. For each $i$, let $\Omega_i\subset{\mathscr{M}}_g(\mathbb{Q}_{p_i})$ be the subcategory of those smooth curves with (potential) reduction graph $\Gamma_i$. By (1), this is $p_i$-adically open in ${\mathscr{M}}_g(\mathbb{Q}_{p_i})$, in the sense of Definition 2.2 in the paper. Moreover, it is not empty, due to assumption (*).

It is straightforward to check that $\mathscr{M}_{g,R}\to\mathrm{Spec}(R)$ with the local data $(\Omega_i)_{i\in I'}$ constitutes a Skolem datum in the sense of Definition 0.6. (We use the fact that $\mathscr{M}_g$ is smooth with geometrically connected fibers over $\mathrm{Spec}(\mathbb{Z})$). Now apply Theorem 0.7: this almost gives the result (with $K$ totally split at each $p_i$) except that the curve may not have stable reduction outside $\mathrm{Spec}(R)$. To fix this, just enlarge $K$, possibly losing splitness.

For the second claim, we do the same, replacing $\Omega_i$ by $\Omega'_i$ consisting of curves with stable reduction of type $\Gamma_i$ over $\mathbb{Z}_{p_i}$: the extra assumption on $p_i$ guarantees that $\Omega'_i\neq\emptyset$. QED