I'm not sure if this should be posted as an answer, but it became too long to post as a comment.

Let $X$ be a smooth projective geometrically connected curve of positive genus over a number field $K$. (The condition on the genus will be used below.) Let me explain some elements of the theory of models for $X$ over $O_K$.

Firstly, there exists a model of $X$ over $O_K$. In fact, there is a closed immersion $X\to\mathbf{P}^n_{K}$. The Zariski closure of $X$ in $\mathbf{P}^n_{O_K}$ via $X\to \mathbf{P}^n_K \subset \mathbf{P}^n_{O_K}$ gives a model for $X$ over $\mathcal{O}_K$. It is a projective model. It is irreducible and reduced as a scheme. Normalizing this scheme gives a normal (projective) model. Now, you can use "resolution of singularities", e.g., Lipman's theorem, to obtain a regular model for $X$ over $O_K$. Then, subsequently contracting all the $-1$-curves (also called exceptional curves) on this regular model you will obtain a regular projective model $\mathcal{X}_{min}$ for $X$ over $O_K$ which is "minimal". We call $\mathcal{X}_{min}$ the minimal regular model of $X$ over $O_K$. It is this minimality condition which is quite natural (for curves of positive genus).

Before continuing, let me give some references for the above paragraph. You can find them all in Liu's book.

For basic facts on the fibres of a model for $X$ over $O_K$ see Chapter 8.3.1.

Desingularization of a normal model is explained in Chapter 8.3.4. For example, a precise statement of Lipman's theorem can be found in Theorem 8.3.44.

Exceptional divisors on a model are defined in Definition 9.3.1.

There are two notions of minimality (Definition 9.3.12) which are shown to coincide in Corollary 9.3.24 (when the generic fibre has positive genus).

Finally, the existence of the minimal regular model is obtained in Therem 9.3.21.

To summarize, the minimal regular model exists and is unique. Thus, it is "natural" to look at the dual reduction graphs of this model.

The (geometric) fibres of $\mathcal{X}_{min}$ are, in general, very complicated. Of course, by Proposition 8.3.11, almost all of them are smooth. But, the geometric fibre over a "bad" place will be a singular curve with "complicated" singularities.

This brings us to semi-stable reduction. In fact, if your reduction isn't a smooth curve, you could hope for the next best thing: semi-stability.

A curve over an algebraically closed field is semi-stable if it is connected, reduced and has only ordinary double singularities. The model $\mathcal{X}_{min}$ doesn't have semi-stable geometric fibres in general, but a deep theorem of Grothendieck and Mumford states that there exists a finite field extension $L/K$ such that the minimal regular model of $X_L$ over $O_L$ is semi-stable, i.e., its geometric fibres are semi-stable. Considering the reduction graph of this model is also very natural; see Definition 10.3.17 in Liu's book.

(Caution: just because the singularities of the geometric fibres have "easy" singularities, doesn't mean the configuration of the irreducible fibres is easy. In fact, determining the configuration is a difficult problem in arithmetic geometry. For example, the semi-stable reduction of the modular curve $X_0(n)$ (for all $n$) has only been achieved recently by Jared Weinstein; see http://arxiv.org/abs/1010.4241 .)

There is also the notion of stability. This is stronger than semi-stability. The stable reduction of a curve is also "natural" to consider.

I'll finish with a quick note on elliptic curves.

Let $E/K$ be an elliptic curve. Then you can consider the minimal regular model and, for some suitable $L/K$, the semi-stable reduction of $E_L$ over $O_L$.

It is also natural to ask whether $E$ has a model over $O_K$ which extends the group structure and the smoothness property of $E/K$. Such a model doesn't exist in general if we demand properness. If we drop the properness condition, then such a model exists. (The finiteness conditions being as usual: the model is of finite type and separated over the base scheme Spec $O_K$.) You can then ask for a model which extends this group structure in the "best possible" way. This brings us to Neron models. Such a model for $E$ over $O_K$ always exists. In fact, you can show that the smooth locus $\mathcal{E}_{min}^{sm}$ of the minimal regular model $\mathcal E_{min}$ of $E$ over $O_K$ is the Neron model of $E$ over $O_K$; see Theorem 10.2 for a discussion of this beautiful theory.

`$\mathcal{X}/O_{K}$`

. Finiteness conditions (which one?) or properness (probably not for the model), regularness, etc... But, I will first take a look at [§10, Liu]. – jmc Oct 28 '12 at 21:27