An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is called \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, given any metric graph, its metric can be rescaled 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 rescaling the metric $d$) on a surface of genus $g$.

Next, we study minimal embeddings, where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_e^{\max}(G)$ a graph $(G, d)$ is the largest genus of a surface on which the graph is minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$, where $g_e(G)$ and $g_e^{\max}(G)$ are realized.

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Last updated: 18 May 2024