I'm trying to understand a paragraph from Keith Conrad's expository paper on the dihedral group. The paragraph is:

> 
> Pick two adjacent vertices of a regular $n$-gon, and call them $A$ and $B$ as in the figure below. An element $g$ of $D\_n$ is a rigid motion taking the $n$-gon back to itself, and it must carry vertices to vertices (**how are vertices unlike other points in terms of distance relationships with all points in the polygon?**) and $g$ must preserve adjacency of vertices, so $g(A)$ and $g(B)$ are adjacent vertices of the polygon.
> 
> 
> 

The point I've bolded is what is causing me confusion. I can't figure out how vertices differ from other points. Is the idea that a symmetry has to send vertices to vertices, and adjacent vertices have to be mapped to adjacent vertices? I'm struggling to answer this in a way that ties back to the definition of a rigid motion.

Can someone help me to understand this?