For the connected sum of surfaces, remove an embedded open disc from each surface and identify the resulting boundary circles by a homeomorphism. The resulting closed connected surface is denoted .
Removing a disc decreases the Euler characteristic by one. Gluing the remaining surfaces along their common boundary subtracts the characteristic of a circle, which is zero. Thus
(1)
This can be verified by choosing matching subdivisions of the boundary circles and counting the vertices and edges identified in the gluing.

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