Let $X/K$ be a connected smooth projective curve of genus $g>1$. After passing to a finite extension $K^{\prime}$ of $K$ we may assume that $X$ has a canonical model ${\cal X}$ with stable reduction. The graph $\Gamma^{\prime}$ corresponding to the special fiber ${\cal X}_{0}$ is a retraction of $(X\otimes_{K}K^{\prime})^{an}$. The quotient graph $\Gamma=\Gamma^{\prime}/Gal(K^{\prime}/K)$ is a retraction of the analytic curve $X^{an}$ (see [Be1]). We define $B:=\Gamma$. Then $B^{sm}$ is a complement to a finite set. As in Section 3.2.2, a ${\bf Z}$-affine structure on a graph is the same as a length element (i.e. a metric). Therefore $\Gamma$ is a metrized graph. Notice also that the maximal number of edges of the graph corresponding to a genus $g$ curve is $3g-3$, which is the dimension of the moduli space of genus $g$ curves.

Notice that if in Section 4.2.1 subvariety $Z$ is a curve then its projection is a noncompact metrized graph with unbounded edges corresponding to punctures ${\overline{Z}}\setminus Z$.