 Solution

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 $M\#N$.

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. ThusThis can be verified by choosing matching subdivisions of the boundary circles and counting the vertices and edges identified in the gluing.