\begin{definition}[Definition:Genus of Surface]
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 $S$.
\end{definition}