As a main tool in the proof, we need strictly semistable models of $X$ and their canonical skeletons. This construction is due to Berkovich in [Ber99]. We recall here only the case of a smooth projective curve $X$ over $K$ for which we can also refer to [Thu05].

A ${K^{\circ}}$-model $\mathscr{X}$ of $X$ as in 2.1 is called strictly semistable if there is an open covering of $\mathscr{X}$ by open subsets $\mathcal{U}$ such that there are étale morphisms $\mathcal{U}\to\Spec({K^{\circ}}[x,y]/(xy-\rho_{\mathcal{U}}))$ for some $\rho_{\mathcal{U}}\in K^{\circ\circ}$. Applying the construction in [Thu05, §2.2] to the associated formal scheme $\hat{\mathscr{X}}$, we get a canonical skeleton $S(\mathscr{X})\subseteq X^{\mathrm{an}}$ with a proper strong deformation retraction $\tau\colon{X^{{\mathrm{an}}}}\to S(\mathscr{X})$. The skeleton $S(\mathscr{X})$ carries a canonical structure of a metrized graph. We note that the generic fiber of the formal scheme $\hat{\mathcal{U}}$ intersects $S(\mathscr{X})$ in an edge of length $v(\rho_{\mathcal{U}})$. By using the reduction map, the vertices of $S(\mathscr{X})$ correspond to the irreducible components of the special fiber $\mathscr{X}_{s}$ and the open edges of $S(\mathscr{X})$ correspond to the singular points of $\mathscr{X}_{s}$.