The sphere

A flat region of the plane can be continuously deformed into other genus-0 surfaces, such as a sphere. Specifically, the plane can be viewed as a sphere with one point removed — the "point at infinity".

Because homeomorphic surfaces preserve topological properties, any graph that can be embedded in the plane without edge intersections can also be embedded on a sphere in the same way.

The torus

By gluing opposite edges of a square, a torus is formed. A torus has genus 1, meaning it has one handle.

Some non-planar graphs can be embedded on a torus without edge intersections.

Example: The intersecting edges (2,5) and (1,6) of K_3,3 can be routed through opposite borders of the square, as they reconnect after the process of gluing and removing the crossings.

The Möbius strip

By twisting a square by 180° and gluing its vertical edges together, a Möbius strip is formed. This is a non-orientable surface with one crosscap.

Example: For K_3,3, flipping vertices 2 and 5 allows the edges (3,4) and (1,6) to be rerouted across the square borders, where they reconnect after the twist, therefore removing the crossings.