# Can an undirected graph cycle have only two vertices?

I've always seen cycles in graphs described as containing three or more vertices. I had a question posed to me today that I now pose to you: Is it valid to consider a pair of vertices connected by a single edge in an undirected graph to be a cycle?

More concretely, given the following undirected graph:

a---b
\ /
c
|
d


Obviously, (c, b, a, c) is a cycle. Is (c, d, c)? Why or why not?

• The edges in a cycle must be distinct, whatever its length. $cabac$ isn't a cycle either. – MJD Sep 2 '14 at 1:21
• Oh, of course. Thank you! – asciiphil Sep 2 '14 at 11:11

Page 8: "If $P=x_0 ... x_{k-1}$ is a path and $k \geq 3$, then the graph $C:=P+x_{k-1}x_0$ is called a cycle."
Thus $cdc$ is not a cycle because $cd$ is not a path with more than two vertices.
But what about $cdcdc$? It would be a path if $cdcd$ would. However, according Diestel definition of path (page 6) all vertices of a path are different.