 Vertex link in a polygonal schema

Replace each corner of a polygonal schema by a small closed interval cutting across that corner. The endpoint identifications induced by paired polygon edges glue these intervals into a graph at each vertex equivalence class. This is its vertex link. A circle link gives a disc neighbourhood; a closed interval link gives a half-disc neighbourhood in a surface with boundary.

For a complete pairing of the sides of a single polygon, every link endpoint is paired exactly once. In each vertex equivalence class the link graph is connected, since the corner endpoint identifications generate precisely that class, and all its vertices have degree two. It is therefore a circle. For partial pairings, a connected link can instead be a closed interval when it has two unpaired endpoints. Counting free polygon edges alone does not count boundary components of a surface; their endpoints must first be identified.