  An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is
\emph{essential}, if each complementary region has a negative Euler
characteristic. We show, by construction, that given any metric graph, its
metric can be rescaled so that it admits an essential and isometric embedding
on a closed hyperbolic surface. The essential genus $g_e(G)$ of $(G, d)$ is the
lowest genus of a surface on which such an embedding is possible. In the next
result, we establish a formula to compute $g_e(G)$. Furthermore, we show that
for every integer $g\geq g_e(G)$, $(G, d)$ admits such an embedding (possibly
after a rescaling of $d$) on a surface of genus $g$.
  Next, we study minimal embeddings where each complementary region has Euler
characteristic $-1$. The maximum essential genus $g_e^{\max}(G)$ of $(G, d)$ 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.
