Let us analyse the situation from the Berkovich viewpoint. As we have seen, the generic points of the special fiber are the reductions of canonical points of $\mathrm{X}$ : the vertices of the graph $R(\mathfrak{X})$ naturally live in $\mathrm{X}$. The same holds for the edges, but is a bit more subtle. As we have seen, blowing-up intersection points of components in the special fiber gives rise to new components, hence to new points of $\mathrm{X}$. Would we enlarge the ground field and blow-up indefinitely, the constellation of points in $\mathrm{X}$ that we draw converges to a graph which is isomorphic to $R(\mathfrak{X})$.

According to Berkovich [12], a far more precise result holds. Let us consider a neighborhood $\mathfrak{U}$ of a singular point of the special fiber, pretending it is isomorphic to the locus defined by the equation $xy-\pi$ in $\mathbf{A}^{2}$ ; so $\mathfrak{U}=\operatorname{Spec}(K^{\circ}[x,y]/(xy-\pi))$. Its generic fibre is the affinoid space $\mathrm{U}$ defined by the inequality $\left|{xy}\right|=\left|{\pi}\right|$ in the unit polydisk $\mathrm{B}^{2}=\mathscr{M}(K\langle x,y\rangle$. The affinoid algebra of $\mathrm{U}$ is the quotient

$$K\langle x,y\rangle/(xy-\pi)$$

whose elements $f$ are (non-uniquely) represented by a series

$$\sum_{m,n=0}^{\infty}a_{m,n}x^{m}y^{n},$$

with $a_{m,n}\rightarrow 0$ when $m+n\rightarrow\infty$. However, observing that $x$ is invertible in this algebra, with inverse $\pi^{-1}y$, so that $y=\pi x^{-1}$, we can replace each product $x^{m}y^{n}$ by $\pi^{n}x^{m-n}$, leading to an expression of the form

$$f=\sum_{n\in{\mathbf{Z}}}a_{n}x^{n},$$

where $\left|{a_{n}}\right|\rightarrow 0$ when $n\rightarrow+\infty$ and $\left|{a_{n}}\right|\pi^{-n}\rightarrow 0$ when $n\rightarrow-\infty$. Such an expression is now unique, and is called the Laurent expansion of $f$.

It leads to a natural family $(\gamma_{r})_{r\in[0,\log\left|{\pi}\right|^{-1}]}$ of multiplicative seminorms on the algebra $\mathscr{O}(\mathrm{U})$, parametrized by the unit interval in ${\mathbf{R}}$. Namely, for each real number $r\in[0,\log\left|{\pi}\right|^{-1}]$, we can set

$$\gamma_{r}(f)=\max_{n\in{\mathbf{Z}}}\left|{a_{n}}\right|e^{-rn},\qquad f=\sum_{n\in{\mathbf{Z}}}a_{n}x^{n}\in\mathscr{O}(\mathrm{U}).$$

Obviously, $\gamma_{r}$ is a norm on $\mathscr{O}(\mathrm{U})$ which extends the absolute value of $K$ ; its multiplicativity is proved analogously that of the Gauß norm. It is easy to check that the map $[0,\log\left|{\pi}\right|^{-1}]\rightarrow\mathrm{U}$ defined by $r\mapsto\gamma_{r}$ is continuous (this amounts to the fact that the maps $r\mapsto\gamma_{r}(f)$ are continuous), hence defines an parametrized path in the topological space $\mathrm{U}$.

Let $S(\mathfrak{U})$ be its image (with the induced distance) ; Berkovich calls it the skeleton of the formal scheme obtained by completing $\mathfrak{U}$ along its special fibre. A point $u$ in $\mathrm{U}$ has two coordinates $(x(u),y(u))$ in the completed residue field $\mathscr{H}(u)$ which are elements of absolute value $\leq 1$ satisfying $x(u)y(u)=\pi$. In particular,

$$r(u)=\log\left|{x(u)}\right|^{-1}\in[0,\log\left|{\pi}\right|^{-1}].$$

The map $\rho\colon u\mapsto\gamma_{r(u)}$ is a continuous from $\mathrm{U}$ to $S(\mathfrak{U})$.

Let us compute the image of $\gamma_{r}$ by this map. By definition of $\gamma_{r}$, one has

$$\left|{x(\gamma_{r})}\right|=\gamma_{r}(x)=e^{-r},$$

hence $r(\gamma_{r})=r$ and $\rho(\gamma_{r})=\gamma_{r}$. In other words, the map $\rho$ is a retraction of $\mathrm{U}$ onto the skeleton $S(\mathfrak{U})$.

The special fiber of $\mathfrak{U}$ is defined by the equation $xy=0$ in $\mathbf{A}^{2}_{\tilde{K}}$, hence has two components. One can check that the point $\gamma_{0}$ reduces to the generic point of the component with equation $y=0$, while $\gamma_{\log\left|{\pi}\right|^{-1}}$ reduces to the generic point of the component with equation $x=0$.

These constructions have to be done around each singular point of the special fiber of $\mathfrak{X}$, locally for the étale topology of $\mathfrak{X}$. Berkovich proves that they can be glued, so that the graph $R(\mathfrak{X})$ is again canonically interpreted as an actual metrized graph drawn on the analytic space $\mathrm{X}$ ; we write $\iota\colon R(\mathfrak{X})\hookrightarrow\mathrm{X}$ for the canonical embedding. The map $\iota$ admits a continuous retraction $\rho\colon\mathrm{X}\rightarrow R(\mathfrak{X})$.

Although we will not use this fact, we must mention that the retraction $\rho$ is a deformation retraction. (For any $t\in[0,1]$ and any $x\in\mathrm{X}$, $\rho_{t}(x)$ is the semi-norm $x^{1-t}\rho(x)^{t}$.)