Definition:Genus of Surface
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Definition
Let $S$ be a surface.
Let $G = \struct {V, E}$ be a graph which is embedded in $S$.
Let $G$ be such that each of its faces is a simple closed curve.
Let $\map \chi G = v - e + f = 2 - 2 p$ be the Euler characteristic of $G$ where:
- $v = \size V$ is the number of vertices
- $e = \size E$ is the number of edges
- $f$ is the number of faces.
Then $p$ is known as the genus of $S$.
Also see
- Results about genera of surfaces can be found here.
Linguistic Note
The plural of genus is genera.
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)