Definition:Genus of Surface

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Let $S$ be a surface.

Let $G = \left({V, E}\right)$ be a graph which is embedded in $S$.

Let $G$ be such that each of its faces is a simple closed curve.

Let $\chi \left({G}\right) = v - e + f = 2 - 2 p$ be the Euler characteristic of $G$ where:

$v = \left|{V}\right|$ is the number of vertices
$e = \left|{E}\right|$ is the number of edges
$f$ is the number of faces.

Then $p$ is known as the genus of $S$.