Abstract :

An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is called essential, if each complementary region has a negative Euler characteristic. We show, by construction, given any metric graph, its metric can be re-scaled so that it can be essentially and isometrically embedded on a closed hyperbolic surface. The essential genus $g_e(G)$ of a metric graph $(G, d)$ is the lowest genus of a surface on which such an embedding of the graph is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ can be essentially and isometrically embedded (possibly after a re-scaling the metric $d$) on a surface of genus $g$.