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Given is a surface represented through a triangle mesh (as commonly used in computer graphics, with vertices, edges and faces). The surface is known to be "watertight", i.e. no missing faces.

Is there a computationally efficient way to determine the genus (i.e. number of handles) of the surface represented by the mesh?

[I've tagged this as graph-theory because I believe this is a graph-theoretical problem, viewing the mesh as graph]

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$2-2g=$ number of vertices - number of edges + number of faces ($g$ is the genus and $2-2g$ is the Euler characteristic). In your case number of edges = $3/2$ times number of faces (triangles).

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    It's really a god-given (or Euler- and Poincaré-given?) miracle that the Euler characteristic is so easy to compute and contains all the topological information for a surface. – 2011-11-24